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1-1z" fill="currentColor"></path></svg></a></li><li class="breadcrumbs__item"><span class="breadcrumbs__link">Compute Engine</span></li><li class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link">Arithmetic</span></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Arithmetic</h1></header><h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="constants">Constants<a href="#constants" class="hash-link" aria-label="Direct link to Constants" title="Direct link to Constants" translate="no"></a></h2>
<div class="symbols-table first-column-header" style="--first-col-width:16ch"><table><thead><tr><th style="text-align:left">Symbol</th><th style="text-align:left">Value</th><th style="text-align:left"></th></tr></thead><tbody><tr><td style="text-align:left"><code>ExponentialE</code></td><td style="text-align:left"><code class="language-math math-inline">\approx 2.7182818284\ldots</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q82435" target="_blank" rel="noopener noreferrer" class="">Euler's number</a></td></tr><tr><td style="text-align:left"><code>MachineEpsilon</code></td><td style="text-align:left"><code class="language-math math-inline"> 2^{−52}</code></td><td style="text-align:left">The difference between 1 and the next larger floating point number. <br>See <a href="https://en.wikipedia.org/wiki/Machine_epsilon" target="_blank" rel="noopener noreferrer" class="">Machine Epsilon on Wikipedia</a></td></tr><tr><td style="text-align:left"><code>CatalanConstant</code></td><td style="text-align:left"><code class="language-math math-inline">\approx 0.9159655941\ldots </code></td><td style="text-align:left"><code class="language-math math-inline">\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2}</code> <br> See <a href="https://en.wikipedia.org/wiki/Catalan%27s_constant" target="_blank" rel="noopener noreferrer" class="">Catalan's Constant on Wikipedia</a></td></tr><tr><td style="text-align:left"><code>GoldenRatio</code></td><td style="text-align:left"><code class="language-math math-inline">\approx 1.6180339887\ldots</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{1+\sqrt{5}}{2}</code> <br>See <a href="https://en.wikipedia.org/wiki/Golden_ratio" target="_blank" rel="noopener noreferrer" class="">Golden Ratio on Wikipedia</a></td></tr><tr><td style="text-align:left"><code>EulerGamma</code></td><td style="text-align:left"><code class="language-math math-inline">\approx 0.5772156649\ldots </code></td><td style="text-align:left">See <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant" target="_blank" rel="noopener noreferrer" class="">Euler-Mascheroni Constant on Wikipedia</a></td></tr></tbody></table></div>
<div class="readMore_KWn8"><a href="/compute-engine/reference/trigonometry/"><p>See also <strong>Trigonometry</strong> for <code>Pi</code> and related constants<svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div>
<div class="readMore_KWn8"><a href="/compute-engine/reference/complex/"><p>See also <strong>Complex</strong> for <code>ImaginaryUnit</code><svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="functions">Functions<a href="#functions" class="hash-link" aria-label="Direct link to Functions" title="Direct link to Functions" translate="no"></a></h2>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="arithmetic-functions">Arithmetic Functions<a href="#arithmetic-functions" class="hash-link" aria-label="Direct link to Arithmetic Functions" title="Direct link to Arithmetic Functions" translate="no"></a></h3>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Notation</th><th style="text-align:left"></th></tr></thead><tbody><tr><td style="text-align:left"><code>Add</code></td><td style="text-align:left"><code class="language-math math-inline"> a + b</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q32043" target="_blank" rel="noopener noreferrer" class="">Addition</a></td></tr><tr><td style="text-align:left"><code>Subtract</code></td><td style="text-align:left"><code class="language-math math-inline"> a - b</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q32043" target="_blank" rel="noopener noreferrer" class="">Subtraction</a></td></tr><tr><td style="text-align:left"><code>Negate</code></td><td style="text-align:left"><code class="language-math math-inline">-a</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q715358" target="_blank" rel="noopener noreferrer" class="">Additive inverse</a></td></tr><tr><td style="text-align:left"><code>Multiply</code></td><td style="text-align:left"><code class="language-math math-inline">a\times b</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q40276" target="_blank" rel="noopener noreferrer" class="">Multiplication</a></td></tr><tr><td style="text-align:left"><code>Divide</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{a}{b}</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q1226939" target="_blank" rel="noopener noreferrer" class="">Divide</a></td></tr><tr><td style="text-align:left"><code>Power</code></td><td style="text-align:left"><code class="language-math math-inline">a^b</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q33456" target="_blank" rel="noopener noreferrer" class="">Exponentiation</a></td></tr><tr><td style="text-align:left"><code>Root</code></td><td style="text-align:left"><code class="language-math math-inline">\sqrt[n]{x}=x^{\frac1n}</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q601053" target="_blank" rel="noopener noreferrer" class="">n<sup>th</sup> root</a></td></tr><tr><td style="text-align:left"><code>Sqrt</code></td><td style="text-align:left"><code class="language-math math-inline">\sqrt{x}=x^{\frac12}</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q134237" target="_blank" rel="noopener noreferrer" class="">Square root</a></td></tr><tr><td style="text-align:left"><code>Square</code></td><td style="text-align:left"><code class="language-math math-inline">x^2</code></td><td style="text-align:left"></td></tr></tbody></table></div>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="non-finite-results">Non-Finite Results<a href="#non-finite-results" class="hash-link" aria-label="Direct link to Non-Finite Results" title="Direct link to Non-Finite Results" translate="no"></a></h3>
<p>Arithmetic can leave the finite numbers in two ways, and the engine keeps the
two apart.</p>
<p><strong>A pole</strong> — a finite non-zero value divided by zero — evaluates to
<code>ComplexInfinity</code> (<code class="language-math math-inline">\tilde\infty</code>), an infinity of unspecified direction. The
signed infinities <code>PositiveInfinity</code> and <code>NegativeInfinity</code> arise from
arithmetic on infinite operands.</p>
<p><strong>An indeterminate form</strong> evaluates to <code>NaN</code>: <code class="language-math math-inline">0/0</code>, <code class="language-math math-inline">\infty - \infty</code> and
<code class="language-math math-inline">\infty \times 0</code> have no value that the surrounding expression could rely on.</p>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "ComplexInfinity"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "PositiveInfinity"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0</span><br></div></code></pre></div></div>
<p>These results are outside the bare numeric types. <code>integer</code>, <code>rational</code>, <code>real</code>
and <code>complex</code> each denote a <strong>finite</strong> value, so an infinity matches neither
<code>real</code> nor <code>complex</code>; it matches <code>infinity</code>, and <code>NaN</code> matches <code>nan</code>. All three
are still <code>number</code>, which is the union of the finite numbers with the
infinities and <code>NaN</code>.</p>
<p><strong><code>NaN</code> propagates.</strong> Any numeric function of <code>NaN</code> is <code>NaN</code>, under plain
<code>evaluate()</code> and not only under <code>N()</code> — <code>NaN</code> is not an exact value, so there is
nothing to hold symbolically:</p>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NaN"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NaN"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Mod"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NaN"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GCD"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NaN"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><br></div></code></pre></div></div>
<p>An <strong>infinity</strong>, by contrast, <em>is</em> an exact value. A function of an infinity
with no closed form therefore stays symbolic rather than numericizing:
<code>["Sin", "PositiveInfinity"]</code> evaluates to itself. Heads whose result is
required to be finite reject an infinite argument outright — see
<a class="" href="/compute-engine/reference/number-theory/">Number Theory</a>.</p>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="sums-and-products">Sums and Products<a href="#sums-and-products" class="hash-link" aria-label="Direct link to Sums and Products" title="Direct link to Sums and Products" translate="no"></a></h3>
<section id="Sum" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Sum</b>(<em>xs</em>: collection<!-- -->)</p><p>Evaluate to a sum of all the elements in <em>collection</em>. If all the elements are
numbers, the result is a number. If some elements are symbolic, the result is
an <code>["Add"]</code> expression.</p><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\sum</span><span> x_</span><span class="punctuation_YpjP">{</span><span>i</span><span class="punctuation_YpjP">}</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$\sum x_{i}$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 23</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"y"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", 12, "x", "y"]</span><br></div></code></pre></div></div><p>A list of points (tuples) sums to a point, and a list of rows sums to the row
of the column sums. An element that is not a number, a point or a row (a
boolean, a dictionary) is an <code>incompatible-type</code> error, also when it is the
only element.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Tuple"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Tuple"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Tuple", 4, 6]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["List", 4, 6]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"True"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Error", ["ErrorCode", "'incompatible-type'", "'number'", "'boolean'"], "True"]</span><br></div></code></pre></div></div><p>Note this is equivalent to:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Reduce"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">]</span><br></div></code></pre></div></div><p class="signature_CWyf"><b>Sum</b>(<em>body</em>: function, ...<em>bounds</em>: tuple<!-- -->) -> <span> <!-- -->number</span></p><p>Evaluate to the sum of <code>body</code> for each value in <code>bounds</code>.</p><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\sum</span><span class="punctuation_YpjP">{</span><span>i=1</span><span class="punctuation_YpjP">}</span><span>^</span><span class="punctuation_YpjP">{</span><span>10</span><span class="punctuation_YpjP">}</span><span> i+1</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$\sum{i=1}^{10} i+1$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"i"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Tuple"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"i"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 65</span><br></div></code></pre></div></div><p>The bounds can also be written as a <code>\le</code> (or <code>\ge</code>) range in the subscript,
instead of the <code>i=1</code>/superscript form. The chained form gives both bounds; a
one-sided form supplies a single bound (the other defaults):</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token comment" style="color:var(--base-05)">// \sum_{1 \le i \le 10} i^2</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce.parse('\\sum_</span><span class="token punctuation" style="color:var(--base-06)">{</span><span class="token number" style="color:var(--base-09)">1</span><span class="token plain"> \\le i \\le </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">}</span><span class="token plain"> i^</span><span class="token number" style="color:var(--base-09)">2</span><span class="token plain">').evaluate()</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 385</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// \sum_{i \le 10} i (lower bound defaults to 1)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce.parse('\\sum_</span><span class="token punctuation" style="color:var(--base-06)">{</span><span class="token plain">i \\le </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">}</span><span class="token plain"> i').evaluate()</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 55</span><br></div></code></pre></div></div><p>Only the <code>\le</code>/<code>\ge</code> form implies a bound. A sum with one explicit bound in
the <code>i=1</code>/superscript form, such as <code>\sum_{k=1} k</code> or <code>\sum_{k}^{10} k</code>,
stays unevaluated: no default is chosen for the missing bound. The subscript
<code>k=1</code> is the lower bound, <code>["Limits", "k", 1, "Nothing"]</code>. For an infinite
sum, write the upper bound <code>\infty</code>.</p><div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'\\sum_{k=1} k'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Sum", "k", ["Limits", "k", 1, "Nothing"]] (unevaluated)</span><br></div></code></pre></div></div><p>The bounds can also be written flat after the index, as for <code>Integrate</code>:
the index, then the lower bound, then the upper bound.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 55</span><br></div></code></pre></div></div><p class="signature_CWyf"><b>Sum</b>(<em>body</em>: function, ...<em>bounds</em>: Element<!-- -->) -> <span> <!-- -->number</span></p><p>Evaluate to the sum of <code>body</code> for each value in an Element-based indexing set.</p><p>This form uses <code>["Element", index, collection]</code> to specify that the index variable
iterates over a finite collection (Set, List, or Range).</p><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\sum</span><span>_</span><span class="punctuation_YpjP">{</span><span>n </span><span class="command_yVKk">\in</span><span> </span><span class="command_yVKk">\{</span><span>1,2,3</span><span class="command_yVKk">\}</span><span class="punctuation_YpjP">}</span><span> n</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$\sum_{n \in \{1,2,3\}} n$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 6</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 14 (1² + 2² + 3² = 1 + 4 + 9)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Range"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 15 (1 + 2 + 3 + 4 + 5)</span><br></div></code></pre></div></div><p>The indexing set can be:</p><ul>
<li class=""><strong>Set</strong>: <code>["Set", 1, 2, 3]</code> - explicit finite set of values</li>
<li class=""><strong>List</strong>: <code>["List", 1, 2, 3]</code> - ordered list of values</li>
<li class=""><strong>List (2-element)</strong>: <code>["List", 1, 5]</code> - when a List has exactly 2 integer elements,
it is treated as a Range. This allows bracket notation like <code>\sum_{n \in [1,5]} n</code>
to iterate over all integers from 1 to 5 (evaluates to 15, not 6).</li>
<li class=""><strong>Range</strong>: <code>["Range", 1, 10]</code> - integer range from 1 to 10</li>
<li class=""><strong>Interval</strong>: <code>["Interval", 1, 10]</code> - enumerates integers in the interval.
Supports <code>Open</code> and <code>Closed</code> boundary markers:<!-- -->
<ul>
<li class=""><code>["Interval", 1, 5]</code> → iterates 1, 2, 3, 4, 5 (closed bounds)</li>
<li class=""><code>["Interval", ["Open", 0], 5]</code> → iterates 1, 2, 3, 4, 5 (excludes 0)</li>
<li class=""><code>["Interval", 1, ["Open", 6]]</code> → iterates 1, 2, 3, 4, 5 (excludes 6)</li>
</ul>
</li>
</ul><p><strong>Note:</strong> Evaluation requires a finite, enumerable domain with at most 1000 elements.
Sums over infinite sets (like <code>\sum_{n \in \mathbb{N}}</code>) remain symbolic.</p><h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="multiple-indexing-sets">Multiple Indexing Sets<a href="#multiple-indexing-sets" class="hash-link" aria-label="Direct link to Multiple Indexing Sets" title="Direct link to Multiple Indexing Sets" translate="no"></a></h4><p>Multiple Element expressions can be specified for multi-index sums:</p><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\sum</span><span>_</span><span class="punctuation_YpjP">{</span><span>n </span><span class="command_yVKk">\in</span><span> A</span><span class="punctuation_YpjP">}</span><span class="command_yVKk">\sum</span><span>_</span><span class="punctuation_YpjP">{</span><span>m </span><span class="command_yVKk">\in</span><span> B</span><span class="punctuation_YpjP">}</span><span> n </span><span class="command_yVKk">\cdot</span><span> m</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$\sum_{n \in A}\sum_{m \in B} n \cdot m$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"m"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"m"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 21 (1·3 + 1·4 + 2·3 + 2·4)</span><br></div></code></pre></div></div><p>Mixed indexing sets (Element + Limits) can be used together:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"m"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Limits"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"m"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 12 (n iterates {1,2}, m iterates 1 to 2)</span><br></div></code></pre></div></div><h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="condition-filtering">Condition Filtering<a href="#condition-filtering" class="hash-link" aria-label="Direct link to Condition Filtering" title="Direct link to Condition Filtering" translate="no"></a></h4><p>A condition can be attached to an Element expression to filter values from the set.
The condition is the optional 4th operand of the Element expression.</p><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\sum</span><span>_</span><span class="punctuation_YpjP">{</span><span>n </span><span class="command_yVKk">\in</span><span> S, n > 0</span><span class="punctuation_YpjP">}</span><span> n</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$\sum_{n \in S, n > 0} n$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token comment" style="color:var(--base-05)">// Sum only positive values from S</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Greater"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 6 (only 1 + 2 + 3)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// Sum values greater than or equal to 2</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sum"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GreaterEqual"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 9 (only 2 + 3 + 4)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// Product of negative values</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Less"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 8 (only (-2) × (-4))</span><br></div></code></pre></div></div><p>Supported condition operators: <code>Greater</code>, <code>GreaterEqual</code>, <code>Less</code>, <code>LessEqual</code>, <code>NotEqual</code>.</p><h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="simplification">Simplification<a href="#simplification" class="hash-link" aria-label="Direct link to Simplification" title="Direct link to Simplification" translate="no"></a></h4><p>When <code>simplify()</code> is called on a <code>Sum</code> expression with symbolic bounds, the following closed-form formulas are applied when applicable:</p><table><thead><tr><th style="text-align:left">Pattern</th><th style="text-align:left">Simplifies to</th><th style="text-align:left">Name</th></tr></thead><tbody><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=1}^{b} c</code></td><td style="text-align:left"><code class="language-math math-inline">b \cdot c</code></td><td style="text-align:left">Constant body</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=a}^{b} n</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{b(b+1) - a(a-1)}{2}</code></td><td style="text-align:left">Triangular number (general bounds)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=1}^{b} n^2</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{b(b+1)(2b+1)}{6}</code></td><td style="text-align:left">Sum of squares</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=1}^{b} n^3</code></td><td style="text-align:left"><code class="language-math math-inline">\left[\frac{b(b+1)}{2}\right]^2</code></td><td style="text-align:left">Sum of cubes</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=0}^{b} r^n</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{1-r^{b+1}}{1-r}</code></td><td style="text-align:left">Geometric series</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=0}^{b} (-1)^n</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{1+(-1)^b}{2}</code></td><td style="text-align:left">Alternating unit series</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=0}^{b} (-1)^n \cdot n</code></td><td style="text-align:left"><code class="language-math math-inline">(-1)^b \lfloor\frac{b+1}{2}\rfloor</code></td><td style="text-align:left">Alternating linear series</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=0}^{b} (a + dn)</code></td><td style="text-align:left"><code class="language-math math-inline">(b+1)\left(a + \frac{db}{2}\right)</code></td><td style="text-align:left">Arithmetic progression</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} \binom{n}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">2^n</code></td><td style="text-align:left">Sum of binomial coefficients</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} (-1)^k \binom{n}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">0</code></td><td style="text-align:left">Alternating binomial sum</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} k \binom{n}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">n \cdot 2^{n-1}</code></td><td style="text-align:left">Weighted binomial sum</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=1}^{n} \frac{1}{k(k+1)}</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{n}{n+1}</code></td><td style="text-align:left">Partial fractions (telescoping)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=2}^{n} \frac{1}{k(k-1)}</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{n-1}{n}</code></td><td style="text-align:left">Partial fractions (telescoping)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} k^2 \binom{n}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">n(n+1) \cdot 2^{n-2}</code></td><td style="text-align:left">Weighted squared binomial sum</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} k^3 \binom{n}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">n^2(n+3) \cdot 2^{n-3}</code></td><td style="text-align:left">Weighted cubed binomial sum</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} (-1)^k k \binom{n}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">0</code></td><td style="text-align:left">Alternating weighted binomial sum (n ≥ 2)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{n} \binom{n}{k}^2</code></td><td style="text-align:left"><code class="language-math math-inline">\binom{2n}{n}</code></td><td style="text-align:left">Sum of binomial squares</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=1}^{n} k(k+1)</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{n(n+1)(n+2)}{3}</code></td><td style="text-align:left">Sum of consecutive products</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=m}^{b} (a + dn)</code></td><td style="text-align:left"><code class="language-math math-inline">(b-m+1)\left(a + \frac{d(m+b)}{2}\right)</code></td><td style="text-align:left">Arithmetic progression (general bounds)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{n=1}^{b} c \cdot f(n)</code></td><td style="text-align:left"><code class="language-math math-inline">c \cdot \sum_{n=1}^{b} f(n)</code></td><td style="text-align:left">Factor out constant</td></tr></tbody></table><p>Edge cases:</p><ul>
<li class="">Empty range (upper < lower): returns <code>0</code></li>
<li class="">Single iteration (upper = lower): substitutes the bound value and returns the body</li>
<li class="">Nested sums: inner sums are simplified first, enabling cascading simplification</li>
</ul><h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="infinite-series">Infinite Series<a href="#infinite-series" class="hash-link" aria-label="Direct link to Infinite Series" title="Direct link to Infinite Series" translate="no"></a></h4><p>A <code>Sum</code> over an <strong>infinite</strong> upper bound is evaluated exactly when it matches
a known convergent family; otherwise it stays symbolic (and <code>N()</code> computes an
accelerated numerical approximation). The recognized families:</p><table><thead><tr><th style="text-align:left">Series</th><th style="text-align:left">Value</th><th style="text-align:left">Name</th></tr></thead><tbody><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=a}^{\infty} k^{-s}, \; s > 1</code></td><td style="text-align:left"><code class="language-math math-inline">\zeta(s) - \sum_{k<a} k^{-s}</code></td><td style="text-align:left">p-series (<code class="language-math math-inline">\zeta(2) = \pi^2/6</code>, …)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=n_0}^{\infty} c\,r^k, \; \|r\| < 1</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{c\,r^{n_0}}{1-r}</code></td><td style="text-align:left">Geometric series</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^s}</code></td><td style="text-align:left"><code class="language-math math-inline">\eta(s)</code></td><td style="text-align:left">Alternating p-series (<code class="language-math math-inline">\eta(1) = \ln 2</code>, <code class="language-math math-inline">\eta(2) = \pi^2/12</code>)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=1}^{\infty} \frac{1}{(2k-1)^s}, \; s > 1</code></td><td style="text-align:left"><code class="language-math math-inline">(1 - 2^{-s})\,\zeta(s)</code></td><td style="text-align:left">Odd p-series (<code class="language-math math-inline">\pi^2/8</code> for <code class="language-math math-inline">s=2</code>)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^s}</code></td><td style="text-align:left"><code class="language-math math-inline">\beta(s)</code></td><td style="text-align:left">Dirichlet beta (<code class="language-math math-inline">\pi/4</code>, Catalan's constant, <code class="language-math math-inline">\pi^3/32</code>, <code class="language-math math-inline">5\pi^5/1536</code> for <code class="language-math math-inline">s = 1, 2, 3, 5</code>)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=a}^{\infty} \frac{r^k}{k!}</code></td><td style="text-align:left"><code class="language-math math-inline">e^r - \sum_{k<a} \frac{r^k}{k!}</code></td><td style="text-align:left">Exponential series (symbolic <code class="language-math math-inline">r</code> allowed)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=1}^{\infty} k\,r^k, \; \|r\| < 1</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{r}{(1-r)^2}</code></td><td style="text-align:left">First-moment geometric</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\sum_{k=1}^{\infty} \frac{r^k}{k}, \; \|r\| < 1</code></td><td style="text-align:left"><code class="language-math math-inline">-\ln(1-r)</code></td><td style="text-align:left">Logarithmic series</td></tr></tbody></table><p>The body is compared with each pattern as a rational function of the index, so
an equivalent spelling is recognized: <code class="language-math math-inline">\sum_{k=1}^{\infty} \frac{k}{k^3}</code> is
<code class="language-math math-inline">\frac{\pi^2}{6}</code>, <code class="language-math math-inline">\sum_{k=1}^{\infty} \frac{1}{(k+1)^2}</code> is
<code class="language-math math-inline">\frac{\pi^2}{6} - 1</code>, and the telescoping sum
<code class="language-math math-inline">\sum_{k=1}^{\infty} \frac{1}{k^2+k}</code> is <code class="language-math math-inline">1</code>. The comparison runs only for a
body whose denominator, as written, is zero at no integer index of the domain.
A pole there keeps the sum symbolic.</p><p>For a <strong>symbolic</strong> ratio <code class="language-math math-inline">r</code>, the families with a convergence region return a
<code>When</code>-guarded value recording it — for example
<code class="language-math math-inline">\sum k x^k \to \frac{x}{(1-x)^2} \; \{\|x\| < 1\}</code>.</p><p>A series whose divergence the engine can certify evaluates to its infinity,
under <code>evaluate()</code> and <code>.N()</code>: <code class="language-math math-inline">\sum_{n=1}^{\infty} \frac{1}{n}</code> and
<code class="language-math math-inline">\sum_{k=1}^{\infty} k \cdot 2^k</code> give <code>PositiveInfinity</code>, and
<code class="language-math math-inline">\sum_{n=1}^{\infty} \frac{-3}{n}</code> gives <code>NegativeInfinity</code>. The certificate is
the limit comparison with the harmonic series. A series with no value
(<code class="language-math math-inline">\sum (-1)^n</code>) and a series the test cannot decide (<code class="language-math math-inline">\sum \frac{1}{n \ln n}</code>)
stay symbolic. <code>isEqual()</code> against an infinity answers <code>true</code> for a certified
divergent series and <code>undefined</code> for a series left symbolic.</p></section>
<section id="Product" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Product</b>(<em>xs</em>: collection<!-- -->)</p><p>Evaluate to a product of all the elements in <code>collection</code>.</p><p>If all the elements are numbers, the result is a number. Otherwise it is a <code>["Multiply"]</code> expression.</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\prod</span><span> x_</span><span class="punctuation_YpjP">{</span><span>i</span><span class="punctuation_YpjP">}</span><br></div></div><div class="display_sbg0">$$$\prod x_{i}$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 385</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", 55, "x"]</span><br></div></code></pre></div></div><p>Note this is equivalent to:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Reduce"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">]</span><br></div></code></pre></div></div><p class="signature_CWyf"><b>Product</b>(<em>f</em>: function, <em>bounds</em>:tuple<!-- -->)</p><p>Return the product of <code>body</code> for each value in <code>bounds</code>.</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\prod</span><span>_</span><span class="punctuation_YpjP">{</span><span>i=1</span><span class="punctuation_YpjP">}</span><span>^</span><span class="punctuation_YpjP">{</span><span>n</span><span class="punctuation_YpjP">}</span><span> f(i)</span><br></div></div><div class="display_sbg0">$$$\prod_{i=1}^{n} f(i)$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Tuple"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 39916800</span><br></div></code></pre></div></div><p>A product with only one bound (<code>\prod_{k=1} k</code>) stays unevaluated, as a sum
does. The bounds can also be written flat after the index:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 120</span><br></div></code></pre></div></div><p class="signature_CWyf"><b>Product</b>(<em>body</em>: function, ...<em>bounds</em>: Element<!-- -->) -> <span> <!-- -->number</span></p><p>Evaluate to the product of <code>body</code> for each value in an Element-based indexing set.</p><p>This form uses <code>["Element", index, collection]</code> to specify that the index variable
iterates over a finite collection (Set, List, or Range).</p><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\prod</span><span>_</span><span class="punctuation_YpjP">{</span><span>k </span><span class="command_yVKk">\in</span><span> </span><span class="command_yVKk">\{</span><span>1,2,3,4</span><span class="command_yVKk">\}</span><span class="punctuation_YpjP">}</span><span> k</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$\prod_{k \in \{1,2,3,4\}} k$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 24 (4!)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Product"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Set"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 36 (1² × 2² × 3² = 1 × 4 × 9)</span><br></div></code></pre></div></div><p>See the <code>Sum</code> documentation above for details on supported collection types.</p><h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="simplification-1">Simplification<a href="#simplification-1" class="hash-link" aria-label="Direct link to Simplification" title="Direct link to Simplification" translate="no"></a></h4><p>When <code>simplify()</code> is called on a <code>Product</code> expression with symbolic bounds, the following closed-form formulas are applied when applicable:</p><table><thead><tr><th style="text-align:left">Pattern</th><th style="text-align:left">Simplifies to</th><th style="text-align:left">Name</th></tr></thead><tbody><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{n=1}^{b} c</code></td><td style="text-align:left"><code class="language-math math-inline">c^b</code></td><td style="text-align:left">Constant body</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{n=1}^{b} n</code></td><td style="text-align:left"><code class="language-math math-inline">b!</code></td><td style="text-align:left">Factorial</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{n=1}^{b} (n+c)</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{(b+c)!}{c!}</code></td><td style="text-align:left">Shifted factorial</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{n=1}^{b} (2n-1)</code></td><td style="text-align:left"><code class="language-math math-inline">(2b-1)!!</code></td><td style="text-align:left">Odd double factorial</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{n=1}^{b} 2n</code></td><td style="text-align:left"><code class="language-math math-inline">2^b \cdot b!</code></td><td style="text-align:left">Even double factorial</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=0}^{n-1} (x+k)</code></td><td style="text-align:left"><code class="language-math math-inline">(x)_n</code></td><td style="text-align:left">Rising factorial (Pochhammer)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=0}^{n-1} (x-k)</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{x!}{(x-n)!}</code></td><td style="text-align:left">Falling factorial</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=1}^{n} \frac{k+1}{k}</code></td><td style="text-align:left"><code class="language-math math-inline">n+1</code></td><td style="text-align:left">Telescoping product</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=2}^{n} (1 - \frac{1}{k^2})</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{n+1}{2n}</code></td><td style="text-align:left">Wallis-like product</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{n=1}^{b} c \cdot f(n)</code></td><td style="text-align:left"><code class="language-math math-inline">c^b \cdot \prod_{n=1}^{b} f(n)</code></td><td style="text-align:left">Factor out constant</td></tr></tbody></table><p>Edge cases:</p><ul>
<li class="">Empty range (upper < lower): returns <code>1</code></li>
<li class="">Single iteration (upper = lower): substitutes the bound value and returns the body</li>
</ul><p>The Wallis-like product is also given its closed form by <code>evaluate()</code>, for any
positive integer lower bound <code class="language-math math-inline">a</code>: <code class="language-math math-inline">\prod_{k=a}^{n} \left(1 - \frac{1}{k^2}\right)</code>
evaluates to <code class="language-math math-inline">\frac{(a-1)(n+1)}{a \cdot n}</code>, so <code class="language-math math-inline">\prod_{k=2}^{n}</code> is
<code class="language-math math-inline">\frac{n+1}{2n}</code>, <code class="language-math math-inline">\prod_{k=3}^{n}</code> is <code class="language-math math-inline">\frac{2(n+1)}{3n}</code>, and
<code class="language-math math-inline">\prod_{k=1}^{n}</code> is <code class="language-math math-inline">0</code>. The body is compared as a rational function, so
<code class="language-math math-inline">\frac{k^2-1}{k^2}</code> and <code class="language-math math-inline">\frac{(k-1)(k+1)}{k^2}</code> are recognized too.</p><h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="infinite-products">Infinite Products<a href="#infinite-products" class="hash-link" aria-label="Direct link to Infinite Products" title="Direct link to Infinite Products" translate="no"></a></h4><p>A <code>Product</code> over an <strong>infinite</strong> upper bound is evaluated exactly when its body
is one of these known convergent families, or the reciprocal of one. The body
is compared with each pattern algebraically, so any spelling of the same
rational function is recognized: <code class="language-math math-inline">\prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1}</code> is
<code class="language-math math-inline">\frac{\pi}{2}</code> (the reciprocal of the Wallis product),
<code class="language-math math-inline">2\prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1}</code> is <code class="language-math math-inline">\pi</code>, and
<code class="language-math math-inline">\prod_{k=1}^{\infty} \frac{k^2}{k^2+1}</code> is <code class="language-math math-inline">\frac{\pi}{\sinh \pi}</code>. A body
that matches no entry stays symbolic. A body with a float literal gives a
float, and a body whose numerator and denominator share a factor stays
symbolic.</p><table><thead><tr><th style="text-align:left">Product</th><th style="text-align:left">Value</th><th style="text-align:left">Name</th></tr></thead><tbody><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=a}^{\infty} \left(1 - \frac{1}{k^2}\right), \; a \ge 2</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{a-1}{a}</code></td><td style="text-align:left">Telescoping</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=1}^{\infty} \left(1 - \frac{1}{(2k)^2}\right)</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{2}{\pi}</code></td><td style="text-align:left">Wallis product</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=1}^{\infty} \left(1 - \frac{1}{(2k+1)^2}\right)</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{\pi}{4}</code></td><td style="text-align:left">Odd Wallis analog</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">\prod_{k=1}^{\infty} \left(1 + \frac{1}{k^2}\right)</code></td><td style="text-align:left"><code class="language-math math-inline">\frac{\sinh \pi}{\pi}</code></td><td style="text-align:left">From the <code class="language-math math-inline">\sin</code> product formula</td></tr></tbody></table></section>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="interpreting-elliptical-notation">Interpreting Elliptical Notation<a href="#interpreting-elliptical-notation" class="hash-link" aria-label="Direct link to Interpreting Elliptical Notation" title="Direct link to Interpreting Elliptical Notation" translate="no"></a></h3>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="interpret">Interpret<a href="#interpret" class="hash-link" aria-label="Direct link to Interpret" title="Direct link to Interpret" translate="no"></a></h3></nav>
<section id="Interpret" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Interpret</b>(<em>expr</em>)</p><p>Give formal meaning to an <strong>elliptical</strong> sum or product — one written with an
ellipsis, such as <code class="language-math math-inline">1 + 2 + \dots + n</code> — by turning it into a <code>Sum</code> or
<code>Product</code>. A sum or product containing an ellipsis is otherwise an inert
notational object (see <a class="" href="/compute-engine/guides/latex-syntax/#elliptical-notation">Elliptical Notation</a>),
returned unchanged by <code>evaluate()</code>.</p><p>Interpretation is an explicit <strong>opt-in</strong>: a plain <code>evaluate()</code> never guesses.
The recognizer handles arithmetic progressions, polynomial general terms (via
finite differences), geometric patterns, and linear recurrences (via
Berlekamp–Massey and <code>RSolve</code>). The gate is strict by design — an ambiguous or
unproven pattern is returned unchanged rather than interpreted incorrectly.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Interpret"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ContinuationPlaceholder"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Sum", "k", ["Limits", "k", 1, "n"]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Interpret"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ContinuationPlaceholder"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">100</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ a Sum that evaluates to 5050</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Interpret"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ContinuationPlaceholder"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"n"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Product", ["Multiply", 2, "k"], ["Limits", "k", 1, "n"]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Interpret"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">8</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ContinuationPlaceholder"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">55</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Sum", ["Fibonacci", "k"], ["Limits", "k", 1, 10]] (evaluates to 143)</span><br></div></code></pre></div></div><div class="readMore_KWn8"><a href="/compute-engine/guides/sequences/#interpreting-elliptical-notation"><p>Read more about <strong>interpreting elliptical notation</strong> and the recognizer families <svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div></section>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="transcendental-functions">Transcendental Functions<a href="#transcendental-functions" class="hash-link" aria-label="Direct link to Transcendental Functions" title="Direct link to Transcendental Functions" translate="no"></a></h3>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Notation</th><th style="text-align:left"></th></tr></thead><tbody><tr><td style="text-align:left"><code>Exp</code></td><td style="text-align:left"><code class="language-math math-inline">\exponentialE^{x}</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q168698" target="_blank" rel="noopener noreferrer" class="">Exponential function</a>. Applied to a matrix, broadcasts element-wise — it is <strong>not</strong> the matrix exponential <code class="language-math math-inline">e^M</code></td></tr><tr><td style="text-align:left"><code>Ln</code></td><td style="text-align:left"><code class="language-math math-inline">\ln(x)</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q11197" target="_blank" rel="noopener noreferrer" class="">Logarithm function</a>, the natural logarithm, the inverse of <code>Exp</code></td></tr><tr><td style="text-align:left"><code>Log</code></td><td style="text-align:left"><code class="language-math math-inline">\log_b(x)</code></td><td style="text-align:left"><code>["Log", <v>, <b>]</code><br> Logarithm of base <em>b</em>, default 10</td></tr><tr><td style="text-align:left"><code>Lb</code></td><td style="text-align:left"><code class="language-math math-inline">\log_2(x)</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q581168" target="_blank" rel="noopener noreferrer" class="">Binary logarithm function</a>, the base-2 logarithm</td></tr><tr><td style="text-align:left"><code>Lg</code></td><td style="text-align:left"><code class="language-math math-inline">\log_{10}(x)</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q966582" target="_blank" rel="noopener noreferrer" class="">Common logarithm</a>, the base-10 logarithm</td></tr><tr><td style="text-align:left"><code>LogOnePlus</code></td><td style="text-align:left"><code class="language-math math-inline">\ln(x+1)</code></td><td style="text-align:left"></td></tr></tbody></table></div>
<div class="readMore_KWn8"><a href="/compute-engine/reference/trigonometry/"><p>See also <strong>Trigonometry</strong> for trigonometric functions <svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div>
<div class="readMore_KWn8"><a href="/compute-engine/reference/complex/"><p>See also <strong>Complex</strong> for complex functions <svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div>
<div class="readMore_KWn8"><a href="/compute-engine/reference/statistics/"><p>See also <strong>Statistics</strong> for statistics functions and functions on lists<svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="rounding">Rounding<a href="#rounding" class="hash-link" aria-label="Direct link to Rounding" title="Direct link to Rounding" translate="no"></a></h3>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Notation</th><th style="text-align:left"></th></tr></thead><tbody><tr><td style="text-align:left"><code>Abs</code></td><td style="text-align:left"><code class="language-math math-inline">\|x\| </code></td><td style="text-align:left">Absolute value, <a href="https://www.wikidata.org/wiki/Q3317982" target="_blank" rel="noopener noreferrer" class="">magnitude</a></td></tr><tr><td style="text-align:left"><code>Ceil</code></td><td style="text-align:left"><code class="language-math math-inline">\lceil x \rceil </code></td><td style="text-align:left">Rounds a number up to the next largest integer</td></tr><tr><td style="text-align:left"><code>Floor</code></td><td style="text-align:left"><code class="language-math math-inline">\lfloor x \rfloor</code></td><td style="text-align:left">Round a number to the greatest integer less than the input value</td></tr><tr><td style="text-align:left"><code>Chop</code></td><td style="text-align:left"></td><td style="text-align:left">Replace real numbers that are very close to 0 (less than <code class="language-math math-inline">10^{-10}</code>) with 0</td></tr><tr><td style="text-align:left"><code>Round</code></td><td style="text-align:left"></td><td style="text-align:left">Rounds a number to the nearest integer. <code>Round(x, step)</code> rounds to the nearest multiple of the step</td></tr></tbody></table></div>
<p><code>Floor</code>, <code>Ceil</code>, <code>Truncate</code> and <code>Round</code> take an optional second argument, a
step: the value is rounded to a multiple of the step instead of an integer.
<code>Floor(x, step)</code> is the greatest multiple of the step that is at most <code>x</code>,
whatever the sign of the step, <code>Round(x, step)</code> is the nearest multiple, and
a zero step gives <code>Indeterminate</code>. To round to <code>n</code> decimal places, use the
step <code>10^-n</code>: <code>Round(3.14159, 1/100)</code> is <code>157/50</code>.</p>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Floor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">226</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 220</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Ceil"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">226</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 230</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Floor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2.7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 5/2</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Round"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3.14159</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">100</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 157/50</span><br></div></code></pre></div></div>
<p>A value exactly halfway between two integers (a tie) is rounded with the rule
of <code>ce.roundingTies</code>. The default rule rounds a tie away from zero:
<code>Round(2.5)</code> is <code>3</code> and <code>Round(-2.5)</code> is <code>-3</code>. See
<a class="" href="/compute-engine/guides/numeric-evaluation/#rounding-ties">Rounding Ties</a>.</p>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="other-relational-operators">Other Relational Operators<a href="#other-relational-operators" class="hash-link" aria-label="Direct link to Other Relational Operators" title="Direct link to Other Relational Operators" translate="no"></a></h3>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="congruent">Congruent<a href="#congruent" class="hash-link" aria-label="Direct link to Congruent" title="Direct link to Congruent" translate="no"></a></h3></nav>
<section id="Congruent" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Congruent</b>(<em>a</em>, <em>b</em>, <em>modulus</em>)</p><p>Evaluate to <code>True</code> if <code>a</code> is congruent to <code>b</code> modulo <code>modulus</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Congruent"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">26</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ True</span><br></div></code></pre></div></div><div class="latexWrapperColumn_tWwQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>26 </span><span class="command_yVKk">\equiv</span><span> 11 </span><span class="command_yVKk">\pmod</span><span class="punctuation_YpjP">{</span><span>5</span><span class="punctuation_YpjP">}</span><br></div></div><div class="latexWrapperColumnDisplay_W6wd">$$$26 \equiv 11 \pmod{5}$$</div></div></section>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="other-functions">Other Functions<a href="#other-functions" class="hash-link" aria-label="Direct link to Other Functions" title="Direct link to Other Functions" translate="no"></a></h3>
<section id="Clamp" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Clamp</b>(<em>value</em>, <em>lower</em>, <em>upper</em>)</p><ul>
<li class="">If <code>value</code> is less than <code>lower</code>, evaluate to <code>lower</code></li>
<li class="">If <code>value</code> is greater than <code>upper</code>, evaluate to <code>upper</code></li>
<li class="">Otherwise, evaluate to <code>value</code></li>
</ul><p>All three arguments are required: <code>["Clamp", 0.42]</code> is an arity error. There
is no one-argument form with default bounds (decided 2026-09-21).</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Clamp"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4.2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Clamp"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"+Infinity"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Clamp"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">100</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 11</span><br></div></code></pre></div></div></section>
<section id="Smoothstep" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Smoothstep</b>(<em>e0</em>, <em>e1</em>, <em>x</em>)</p><p>Smooth Hermite interpolation from 0 to 1 as <code>x</code> goes from <code>e0</code> to <code>e1</code>:
<code>t² (3 − 2t)</code>, with <code>t = Clamp((x − e0) / (e1 − e0), 0, 1)</code>. This is the
formula of the GLSL and WGSL built-in <code>smoothstep</code>, with the operands in the
same order. The result is exact for exact operands, and it broadcasts over
collection arguments.</p><ul>
<li class="">When <code>e0 = e1</code>, the result is the step function: 0 when <code>x < e0</code>, else 1.</li>
<li class="">When <code>e0 > e1</code>, the formula applies as written, and the result goes from 1
to 0.</li>
<li class="">Every compile target gives these results. The GLSL and WGSL built-in
<code>smoothstep</code> has no defined result when <code>e0 ≥ e1</code>, so the shader targets
write the formula out instead of calling it.</li>
</ul><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Smoothstep"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 7/27</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Smoothstep"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Smoothstep"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0.5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0</span><br></div></code></pre></div></div></section>
<section id="Max" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Max</b>(<em>x1</em>, <em>x2</em>, ...<!-- -->)</p><p class="signature_CWyf"><b>Max</b>(<em>list</em>)</p><p>If all the arguments are numbers, evaluate to the largest of them. The
infinities take part in the comparison like any other value, so
<code>["Max", 1, "PositiveInfinity"]</code> is <code>PositiveInfinity</code>.</p><p>If any argument is <code>NaN</code>, the whole expression is <code>NaN</code>. An undefined value
absorbs the comparison, and it does so even when other arguments are still
unknown — there is no ordering that could rule it out.</p><p>Otherwise, simplify the expression by removing values that are smaller than or
equal to the largest known number.</p><p>A dictionary is not a collection of numbers. <code>Max</code>, <code>Min</code>, <code>Supremum</code> and
<code>Infimum</code> of a dictionary that has an entry give the <code>incompatible-type</code> error
that <code>Sum</code> gives, and the error names the first entry. An empty dictionary
contributes no value, as an empty list does.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Max"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 5</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Max"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "PositiveInfinity"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Max"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7.1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Max", 7.1, "x"]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Max"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7.1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NaN"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><br></div></code></pre></div></div></section>
<section id="Min" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Max</b>(<em>x1</em>, <em>x2</em>, ...<!-- -->)</p><p class="signature_CWyf"><b>Max</b>(<em>list</em>)</p><p>If all the arguments are numbers, evaluate to the smallest of them. As with
<code>Max</code>, the infinities take part in the comparison, and a <code>NaN</code> argument makes
the whole expression <code>NaN</code>.</p><p>Otherwise, simplify the expression by removing values that are greater than or
equal to the smallest known number.</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\min</span><span>(0, 7.1, 3) = 0</span><br></div></div><div class="display_sbg0">$$$\min(0, 7.1, 3) = 0$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Min"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ -1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Min"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NegativeInfinity"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NegativeInfinity"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Min"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7.1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Min", 0, "x"]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Min"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7.1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"NaN"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "NaN"</span><br></div></code></pre></div></div></section>
<section id="Mod" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Mod</b>(<em>a</em>, <em>b</em>)</p><p>Evaluate to the Euclidian division (modulus) of <code>a</code> by <code>b</code>.</p><p>When <code>a</code> and <code>b</code> are positive integers, this is equivalent to the <code>%</code> operator in
JavaScript, and returns the remainder of the division of <code>a</code> by <code>b</code>.</p><p>However, when <code>a</code> and <code>b</code> are not positive integers, the result is different.</p><p>The result is always the same sign as <code>b</code>, or 0.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Mod"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 2</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Mod"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 3</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Mod"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ -3</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Mod"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ -2</span><br></div></code></pre></div></div><p><code>Mod</code> of exact operands is exact. A radical dividend is reduced without
rounding, and <code>N()</code> gives the float.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Mod"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", -1, ["Sqrt", 2]] (√2 − 1; N() gives 0.4142…)</span><br></div></code></pre></div></div></section>
<section id="Rational" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Rational</b>(<em>n</em>)</p><p>Evaluate to a rational approximating the value of the number <code>n</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0.42</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Rational", 21, 50]</span><br></div></code></pre></div></div><br><p class="signature_CWyf"><b>Rational</b>(<em>numerator</em>, <em>denominator</em>)</p><p>Represent a rational number equal to <code>numerator</code>over <code>denominator</code>.</p></section>
<section id="Numerator" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Numerator</b>(<em>expr</em>)</p><p>Return the numerator of <code>expr</code>.</p><p>Note that <code>expr</code> may be a non-canonical form.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Numerator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 4</span><br></div></code></pre></div></div></section>
<section id="Denominator" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Denominator</b>(<em>expr</em>)</p><p>Return the denominator of <code>expr</code>.</p><p>Note that <code>expr</code> may be a non-canonical form.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Denominator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 5</span><br></div></code></pre></div></div></section>
<section id="NumeratorDenominator" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>NumeratorDenominator</b>(<em>expr</em>)</p><p>Return the numerator and denominator of <code>expr</code> as a sequence.</p><p>Note that <code>expr</code> may be a non-canonical form.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"NumeratorDenominator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Sequence", 4, 5]</span><br></div></code></pre></div></div><p>The sequence can be used with another function, for example GCD to
check if the fraction is in its canonical form:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GCD"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"NumeratorDenominator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GCD"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"NumeratorDenominator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Rational"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">8</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 2</span><br></div></code></pre></div></div></section>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="random-numbers">Random Numbers<a href="#random-numbers" class="hash-link" aria-label="Direct link to Random Numbers" title="Direct link to Random Numbers" translate="no"></a></h3>
<p>The random operators take a <strong>domain</strong> to draw from, never a seed. Seeding is
done by wrapping the work in a <code>WithRandomSeed</code> block, which makes every draw
inside it reproducible.</p>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Result</th></tr></thead><tbody><tr><td style="text-align:left"><code>WithRandomSeed</code></td><td style="text-align:left">evaluate an expression with a seeded block</td></tr><tr><td style="text-align:left"><code>Random</code></td><td style="text-align:left">one draw</td></tr><tr><td style="text-align:left"><code>RandomChoice</code></td><td style="text-align:left"><code>k</code> draws, with replacement</td></tr><tr><td style="text-align:left"><code>RandomSample</code></td><td style="text-align:left"><code>k</code> elements, without replacement</td></tr><tr><td style="text-align:left"><code>RandomShuffle</code></td><td style="text-align:left">a permutation</td></tr></tbody></table></div>
<p><strong>One draw, or <code>k</code> draws:</strong></p>
<div class="symbols-table first-column-header" style="--first-col-width:22ch"><table><thead><tr><th style="text-align:left"></th><th style="text-align:left">one draw</th><th style="text-align:left"><code>k</code> draws</th></tr></thead><tbody><tr><td style="text-align:left">with replacement</td><td style="text-align:left"><code>Random(xs)</code></td><td style="text-align:left"><code>RandomChoice(xs, k)</code></td></tr><tr><td style="text-align:left">without replacement</td><td style="text-align:left"><code>Random(xs)</code></td><td style="text-align:left"><code>RandomSample(xs, k)</code></td></tr><tr><td style="text-align:left">all of them</td><td style="text-align:left">—</td><td style="text-align:left"><code>RandomShuffle(xs)</code></td></tr></tbody></table></div>
<p><code>Random(xs)</code> and <code>RandomChoice(xs, 1)</code> have the same distribution: with a single
draw, "with" and "without" replacement coincide. The wrapper is the only
difference — <code>RandomChoice</code> returns a list.</p>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="withrandomseed">WithRandomSeed<a href="#withrandomseed" class="hash-link" aria-label="Direct link to WithRandomSeed" title="Direct link to WithRandomSeed" translate="no"></a></h3></nav>
<section id="WithRandomSeed" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>WithRandomSeed</b>(<em>seed</em>: real | string, <em>body</em>: any<!-- -->)</p><p>Evaluate <em>body</em> with a random seed frame installed. Every draw inside the frame
is deterministic, and the whole block replays identically on re-evaluation,
while repeated draws <strong>within</strong> the frame still differ.</p><p>The seed must be a <strong>finite</strong> real number: <code>real</code> excludes <code class="language-math math-inline">\pm\infty</code> and
<code>NaN</code>, so an infinite or undefined seed is rejected with an <code>out-of-range</code>
error rather than silently hashed. (Earlier releases spelled this parameter
type <code>finite_real</code>. That name is retired; bare <code>real</code> now carries the same
finiteness guarantee.)</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"PositiveInfinity"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Error", ["ErrorCode", "'out-of-range'",</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// "'a finite real number or a string'", "'+oo'"]]</span><br></div></code></pre></div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">42</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.7367300395263549 — the same value on every evaluation</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">42</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Delimiter"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sequence"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (0.7367300395263549, 0.4498528692283148) — two different draws,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// and the same pair every time</span><br></div></code></pre></div></div><p>Outside a frame, draws are <strong>live</strong>: <code>["Random"]</code> on its own is
non-deterministic, which is what an animation or a ticker needs.</p><p>Frames nest and the innermost wins. Counters are per frame, so a nested frame
never disturbs its parent's later draws:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Delimiter"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sequence"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ the first and third draws are the frame-1 stream's 1st and 2nd values;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// the inner frame does not shift them</span><br></div></code></pre></div></div><p>Scoping is <strong>dynamic</strong>: the frame is active through calls to user-defined
functions, not just lexically inside <em>body</em>. The seed is evaluated once on entry
to the frame, not once per draw, and it may be a string
(<code>["WithRandomSeed", "'cell-a7'", …]</code>).</p><p>Seed a <strong>row or cell</strong>, not a whole document: with one document-wide frame,
inserting a draw near the top shifts every later value below it.</p><p><strong>A frame around a lazy collection does not make it replayable.</strong> Only
evaluation consumes draw indices, and a lazy view is not evaluated until its
elements are read — by which time the frame has exited, so the draws escape it.
The expression <em>looks</em> framed and the failure is silent:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">12345</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Comprehension"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Range"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// the elements draw LIVE (outside the frame), not the seeded stream:</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// this instance: [0.9120…, 0.1273…, 0.4710…, …] ✗ NOT the framed values</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// re-created (a fresh parse or evaluation): different values again</span><br></div></code></pre></div></div><p>Reading the elements of the <strong>same instance</strong> twice does give the same values
— a lazy collection's elements are cached per instance, so one instance is
one draw set (two readers of the same list agree). But that is read
coherence, not replay: the values are live draws, and a re-created instance
draws fresh ones.</p><p>Materialize inside the frame, and it replays. <code>ListFrom</code> is the general form —
it works over any finite collection body — and indexing or a reducer will do it
too:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">12345</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"ListFrom"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Comprehension"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Element"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"k"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Range"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ the same six values on every evaluation ✓</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// (identical to ["RandomChoice", ["Interval", 0, 1], 6] in the same frame)</span><br></div></code></pre></div></div><p>Note that <code>["Repeat", ["Random"], n]</code> is eager and replayable but draws
<strong>once</strong>, yielding n copies of a single value — it is not a uniform batch.</p><div class="readMore_KWn8"><a href="https://github.com/cortex-js/compute-engine/blob/main/docs/RANDOMNESS-MODEL.md"><p>The generator, the seed folding, the interpreted/compiled/GPU parity contract,
and how many draws each operator consumes are specified in the
<strong>Randomness Model</strong> note<svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div></section>
<section id="Random" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Random</b>()</p><p class="signature_CWyf"><b>Random</b>(<em>domain</em>: collection<!-- -->)</p><p><code>Random</code> is <strong>not</strong> a pure function: it carries the <code>random</code> effect and
evaluates to a different value on each evaluation. Wrapping it in
<code>WithRandomSeed</code> discharges that effect — the block as a whole is pure and
replays identically.</p><ul>
<li class=""><code>["Random"]</code> evaluates to a real number in the interval <code class="language-math math-inline">[0, 1)</code>.</li>
<li class=""><code>["Random", ["Interval", a, b]]</code> evaluates to a real number in <code class="language-math math-inline">[a, b)</code>.
Open/closed endpoint markers are ignored: a floating-point draw cannot respect
an open endpoint.</li>
<li class=""><code>["Random", ["Range", …]]</code> evaluates to an element of the range, which is
<strong>inclusive</strong> at both ends and normalizes reversed bounds
(<code>["Range", 7, 2]</code> descends).</li>
<li class=""><code>["Random", xs]</code> evaluates to an element of the finite collection <code>xs</code>.</li>
</ul><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.6233… (for example)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Range"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 4 (for example) — a die roll</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Interval"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ -0.318… (for example)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Random"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">10</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">18</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 10 (for example)</span><br></div></code></pre></div></div><p>The domain is never materialized: <code>["Random", ["Range", 1, 1000000]]</code> indexes
the range, it does not build it.</p><p><code>Random</code> draws along the <strong>first axis</strong> of a non-flat domain, so
<code>["Random", ["List", ["List", 1, 2], ["List", 3, 4]]]</code> returns a <em>row</em>.
A <code>Tuple</code> is treated as an ordinary finite indexed collection and yields one of
its elements — deliberately unlike <code>Join</code>, where a tuple operand is a single
atomic element.</p><p>An unbounded or empty <code>Interval</code>, an infinite <code>Range</code> or collection, and an
empty collection are all errors.</p></section>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="randomchoice">RandomChoice<a href="#randomchoice" class="hash-link" aria-label="Direct link to RandomChoice" title="Direct link to RandomChoice" translate="no"></a></h3></nav>
<section id="RandomChoice" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>RandomChoice</b>(<em>domain</em>: collection, <em>k</em>: number<!-- -->)</p><p>Evaluate to a list of <code>k</code> independent draws from <em>domain</em>, <strong>with
replacement</strong>. Because each draw is independent, <code>k</code> may exceed the size of the
domain — that is what replacement means.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"RandomChoice"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Range"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [4, 1, 4] (for example) — repeats are expected</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"WithRandomSeed"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">7</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"RandomChoice"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Interval"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ a reproducible list of 4 reals in [0, 1)</span><br></div></code></pre></div></div><p><code>k</code> is typed <code>number</code>, not <code>integer</code>, so a computed count does not have to be
rounded by the caller; it is rounded on evaluation (half toward <code class="language-math math-inline">+\infty</code>:
<code>2.5</code> → <code>3</code>, <code>-2.5</code> → <code>-2</code>). A negative or out-of-range count is an error, and
<code>k = 0</code> evaluates to the empty list.</p><p>Only the <code>k</code> drawn elements are materialized — the source domain is not, so
<code>["RandomChoice", ["Range", 1, 1000000000], 5]</code> is cheap.</p></section>
<section id="RandomSample" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>RandomSample</b>(<em>xs</em>: indexed_collection, <em>k</em>: number<!-- -->)</p><p>Evaluate to a list of <code>k</code> elements drawn from <em>xs</em> <strong>without replacement</strong>.</p><p>"Without replacement" is over <strong>positions</strong>, not values. On a multiset, repeated
values are expected: <code>["RandomSample", ["List", 1, 1, 2], 2]</code> returns <code>[1, 1]</code>
in about a quarter of trials. It does not return distinct <em>values</em>.</p><p>Unlike <code>RandomChoice</code>, <code>k</code> may not exceed the size of <code>xs</code> — there are only that
many positions to draw. <code>k</code> equal to the size returns a permutation.</p><div class="readMore_KWn8"><a href="/compute-engine/reference/statistics/#randomsample"><p>See <strong>Statistics</strong> for the full description of <code>RandomSample</code><svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div></section>
<section id="RandomShuffle" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>RandomShuffle</b>(<em>xs</em>: indexed_collection<!-- -->)</p><p>Evaluate to a random permutation of the elements of <em>xs</em>.</p><div class="readMore_KWn8"><a href="/compute-engine/reference/collections/#randomshuffle"><p>See <strong>Collections</strong> for the full description of <code>RandomShuffle</code><svg class="icon_qj0s chevronRightBold_bbMj"><use xlink:href="/icons.svg#chevron-right-bold"></use></svg></p></a></div></section>
<h4 class="anchor anchorTargetStickyNavbar_Vzrq" id="migrating-from-the-previous-random-operators">Migrating from the previous random operators<a href="#migrating-from-the-previous-random-operators" class="hash-link" aria-label="Direct link to Migrating from the previous random operators" title="Direct link to Migrating from the previous random operators" translate="no"></a></h4>
<p><code>RandomInteger</code>, <code>RandomList</code>, <code>RandomSeed</code>, <code>Sample</code> and <code>Shuffle</code> have been
removed, along with the <code>ce.randomSeed</code> property and every per-operator <code>seed</code>
argument. Each removed head throws an <code>operator-removed</code> error naming its
replacement for one release, so a stored document fails loudly rather than
silently stopping to randomize.</p>
<div class="symbols-table first-column-header" style="--first-col-width:26ch"><table><thead><tr><th style="text-align:left">Was</th><th style="text-align:left">Now</th></tr></thead><tbody><tr><td style="text-align:left"><code>Random(seed)</code></td><td style="text-align:left"><code>WithRandomSeed(seed, Random())</code></td></tr><tr><td style="text-align:left"><code>Random(n)</code>, <code class="language-math math-inline">n > 0</code></td><td style="text-align:left"><code>Random(Range(0, n-1))</code></td></tr><tr><td style="text-align:left"><code>Random(m, n)</code>, <code class="language-math math-inline">m < n</code></td><td style="text-align:left"><code>Random(Range(m, n-1))</code></td></tr><tr><td style="text-align:left"><code>RandomInteger(a, b)</code></td><td style="text-align:left"><code>Random(Range(a, b))</code></td></tr><tr><td style="text-align:left"><code>RandomList(n)</code></td><td style="text-align:left"><code>RandomChoice(Interval(0, 1), n)</code></td></tr><tr><td style="text-align:left"><code>RandomList(n, seed)</code></td><td style="text-align:left"><code>WithRandomSeed(seed, RandomChoice(Interval(0,1), n))</code></td></tr><tr><td style="text-align:left"><code>RandomSeed(s)</code></td><td style="text-align:left"><code>WithRandomSeed(s, …)</code> around the work</td></tr><tr><td style="text-align:left"><code>Sample(xs, k)</code></td><td style="text-align:left"><code>RandomSample(xs, k)</code></td></tr><tr><td style="text-align:left"><code>Shuffle(xs)</code></td><td style="text-align:left"><code>RandomShuffle(xs)</code></td></tr><tr><td style="text-align:left"><code>Shuffle(xs, seed)</code></td><td style="text-align:left"><code>WithRandomSeed(seed, RandomShuffle(xs))</code></td></tr><tr><td style="text-align:left"><code>ce.randomSeed = s</code></td><td style="text-align:left"><code>WithRandomSeed(s, …)</code></td></tr></tbody></table></div>
<p>Note the <code>-1</code> on the <code>Random(n)</code> and <code>Random(m, n)</code> rows: the old bounds were
upper-<strong>exclusive</strong> and <code>Range</code> is inclusive. The formulas hold only in the
ranges stated — degenerate old calls (<code class="language-math math-inline">n \leq 0</code>, <code class="language-math math-inline">m \geq n</code>) need rewriting by
hand, because <code>Range</code> normalizes reversed bounds into a descending range instead
of producing an empty one.</p>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="relational-operators">Relational Operators<a href="#relational-operators" class="hash-link" aria-label="Direct link to Relational Operators" title="Direct link to Relational Operators" translate="no"></a></h2>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Notation</th><th style="text-align:left"></th></tr></thead><tbody><tr><td style="text-align:left"><code>Equal</code></td><td style="text-align:left"><code class="language-math math-inline">x = y</code></td><td style="text-align:left">Mathematical relationship asserting that two quantities have the same value. Stays unevaluated when the answer is not known. <code>NaN</code> and a pole (<code>1/0</code>) are equal to nothing, including themselves</td></tr><tr><td style="text-align:left"><code>NotEqual</code></td><td style="text-align:left"><code class="language-math math-inline">x \ne y</code></td><td style="text-align:left"></td></tr><tr><td style="text-align:left"><code>Greater</code></td><td style="text-align:left"><code class="language-math math-inline">x \gt y</code></td><td style="text-align:left"></td></tr><tr><td style="text-align:left"><code>GreaterEqual</code></td><td style="text-align:left"><code class="language-math math-inline">x \geq y</code></td><td style="text-align:left"></td></tr><tr><td style="text-align:left"><code>Less</code></td><td style="text-align:left"><code class="language-math math-inline">x \lt y</code></td><td style="text-align:left"></td></tr><tr><td style="text-align:left"><code>LessEqual</code></td><td style="text-align:left"><code class="language-math math-inline">x \leq y</code></td><td style="text-align:left"></td></tr><tr><td style="text-align:left"><code>IdenticallyEqual</code></td><td style="text-align:left"><code class="language-math math-inline">x \equiv y</code></td><td style="text-align:left">Assert an identity: the two expressions have the same value for <strong>every</strong> value of their free variables</td></tr></tbody></table><p>See below for additonal relational operators: <code>Congruent</code>, etc...</p></div>
<p>An ordered comparison (<code>Less</code>, <code>LessEqual</code>, <code>Greater</code>, <code>GreaterEqual</code>) with an
operand that has no order answers <code>False</code>: <code>NaN</code>, a pole (<code>1/0</code>, which
evaluates to <code class="language-math math-inline">\tilde\infty</code>) and a non-real complex number (<code>i</code>, <code>\ln(-1)</code>).
So a piecewise whose condition compares such a value takes its next branch:
<code>\{1/0 > 0: 1, 0\}</code> is <code>0</code>. <code>NotEqual</code> is not affected (<code>1/0 \ne 0</code> is
<code>True</code>), and a comparison with a free symbol stays unevaluated. Signed
infinities are ordered as usual (<code>0 < \infty</code> is <code>True</code>).</p>
<p>The three tiers of equality — <code>Same</code> (syntactic), <code>Equal</code> (same value) and
<code>IdenticallyEqual</code> (same value for every value of the free variables) — are
compared in
<a class="" href="/compute-engine/guides/symbolic-computing/#comparing-expressions">Comparing Expressions</a>.</p>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="polynomial-arithmetic">Polynomial Arithmetic<a href="#polynomial-arithmetic" class="hash-link" aria-label="Direct link to Polynomial Arithmetic" title="Direct link to Polynomial Arithmetic" translate="no"></a></h2>
<p>These functions operate on polynomial expressions.</p>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Description</th></tr></thead><tbody><tr><td style="text-align:left"><code>Expand</code></td><td style="text-align:left">Expand products and positive integer powers</td></tr><tr><td style="text-align:left"><code>ExpandAll</code></td><td style="text-align:left">Recursively expand products and positive integer powers</td></tr><tr><td style="text-align:left"><code>Factor</code></td><td style="text-align:left">Factor an expression into irreducible factors</td></tr><tr><td style="text-align:left"><code>AlgebraicGenerator</code></td><td style="text-align:left">A root of an irreducible polynomial, to select a number field</td></tr><tr><td style="text-align:left"><code>Together</code></td><td style="text-align:left">Combine rational expressions into a single fraction</td></tr><tr><td style="text-align:left"><code>Distribute</code></td><td style="text-align:left">Distribute multiplication over addition</td></tr><tr><td style="text-align:left"><code>PolynomialDegree</code></td><td style="text-align:left">Return the degree of a polynomial</td></tr><tr><td style="text-align:left"><code>CoefficientList</code></td><td style="text-align:left">Return the list of coefficients of a polynomial</td></tr><tr><td style="text-align:left"><code>Polynomial</code></td><td style="text-align:left">Construct a polynomial from a coefficient list</td></tr><tr><td style="text-align:left"><code>PolynomialQuotient</code></td><td style="text-align:left">Return the quotient of polynomial division</td></tr><tr><td style="text-align:left"><code>PolynomialRemainder</code></td><td style="text-align:left">Return the remainder of polynomial division</td></tr><tr><td style="text-align:left"><code>GCD</code></td><td style="text-align:left">Greatest common divisor of integers or polynomials</td></tr><tr><td style="text-align:left"><code>PolynomialGCD</code></td><td style="text-align:left">Return the greatest common divisor of two polynomials</td></tr><tr><td style="text-align:left"><code>Cancel</code></td><td style="text-align:left">Cancel common polynomial factors in a rational expression</td></tr><tr><td style="text-align:left"><code>PartialFraction</code></td><td style="text-align:left">Decompose a rational expression into partial fractions</td></tr><tr><td style="text-align:left"><code>Apart</code></td><td style="text-align:left">Alias for <code>PartialFraction</code></td></tr><tr><td style="text-align:left"><code>PolynomialRoots</code></td><td style="text-align:left">Return the roots of a polynomial</td></tr><tr><td style="text-align:left"><code>Discriminant</code></td><td style="text-align:left">Return the discriminant of a polynomial</td></tr></tbody></table></div>
<section id="Expand" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Expand</b>(<em>expr</em>)</p><p>Expand out products and positive integer powers in <code>expr</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Expand"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", ["Power", "x", 2], ["Multiply", 2, "x"], 1]</span><br></div></code></pre></div></div></section>
<section id="ExpandAll" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>ExpandAll</b>(<em>expr</em>)</p><p>Recursively expand out products and positive integer powers in <code>expr</code> and all subexpressions.</p></section>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="factor">Factor<a href="#factor" class="hash-link" aria-label="Direct link to Factor" title="Direct link to Factor" translate="no"></a></h3></nav>
<section id="Factor" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Factor</b>(<em>expr</em>)</p><p class="signature_CWyf"><b>Factor</b>(<em>expr</em>, <em>var</em>)</p><p class="signature_CWyf"><b>Factor</b>(<em>expr</em>, <em>var</em>, <em>generator</em>)</p><p>Factor a polynomial expression into a product of irreducible factors.</p><p>Supports:</p><ul>
<li class=""><strong>Univariate polynomials with rational coefficients</strong>: a complete
factorization into factors that are irreducible over the rationals, within
the work limits of the engine, also when no root is rational:
<code class="language-math math-inline">x^4 + 4 \to (x^2-2x+2)(x^2+2x+2)</code></li>
<li class=""><strong>Content extraction</strong>: <code class="language-math math-inline">6x^2 + 12x + 6 \to 6(x+1)^2</code> (extracts GCD of integer coefficients first)</li>
<li class=""><strong>Common factor extraction</strong>: <code class="language-math math-inline">2x + 4 \to 2(x+2)</code></li>
<li class=""><strong>Other expressions</strong>: structural methods, such as perfect square trinomials
<code class="language-math math-inline">a^2 \pm 2ab + b^2 \to (a \pm b)^2</code> and differences of squares
<code class="language-math math-inline">a^2 - b^2 \to (a-b)(a+b)</code></li>
</ul><p>The optional <code>var</code> parameter specifies which variable to factor over.</p><p>The optional <code>generator</code> parameter selects the coefficient field: see
<a href="#factoring-over-a-number-field" class="">Factoring over a number field</a> below.</p><p><strong>Perfect square trinomial:</strong></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Power", ["Add", "x", 1], 2] // (x+1)²</span><br></div></code></pre></div></div><p><strong>Quadratic with rational roots:</strong></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", ["Add", "x", 2], ["Add", "x", 3]] // (x+2)(x+3)</span><br></div></code></pre></div></div><p><strong>Difference of squares:</strong></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", ["Add", "x", -2], ["Add", "x", 2]] // (x-2)(x+2)</span><br></div></code></pre></div></div><p>A difference of squares is factored recursively, so a higher-degree input such
as <code class="language-math math-inline">x^6 - 1</code> is reduced all the way to its irreducible (cyclotomic) factors
rather than stopping at the first split:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (x-1)(x+1)(x²+x+1)(x²-x+1)</span><br></div></code></pre></div></div><p>The factors are always polynomials in <code>var</code>: <code>Factor</code> never introduces <code>Sqrt</code>
of the variable or <code>Abs</code> (e.g. <code class="language-math math-inline">x^3 - 1 \to (x-1)(x^2+x+1)</code>, not the
branch-dependent <code class="language-math math-inline">(x\sqrt{x}-1)(x\sqrt{x}+1)</code>). Without a <code>generator</code>, the
coefficients of the factors are rational.</p><p><strong>With coefficients:</strong></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">12</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">9</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Power", ["Add", ["Multiply", 2, "x"], 3], 2] // (2x+3)²</span><br></div></code></pre></div></div><p><strong>Cubic with rational roots:</strong></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Negate"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">11</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", ["Add", "x", -1], ["Add", "x", -2], ["Add", "x", -3]] // (x-1)(x-2)(x-3)</span><br></div></code></pre></div></div><p><strong>Several variables, and terms that are not polynomials:</strong> without <code>var</code>,
<code>Factor</code> takes out a factor common to every term of a sum, and it treats a
term that is not a polynomial, such as <code class="language-math math-inline">\sin(t)</code>, as a variable of its own.
The like terms of the result are collected, as when the sum is evaluated.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"y"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", "a", ["Add", "x", "y"]] // a(x+y)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sin"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"t"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"y"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sin"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"t"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", ["Add", "x", "y"], ["Sin", "t"]] // (x+y)sin(t)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sin"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"t"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sin"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"t"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", 2, ["Sin", "t"]] // 2sin(t)</span><br></div></code></pre></div></div><p>A call of a function that is not declared, such as <code class="language-math math-inline">f(t)</code>, is not taken out
as a common factor, because its value could be a matrix.</p><p><strong>Automatic use in sqrt simplification:</strong></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Abs", ["Add", "x", 1]] // |x+1| (auto-factors before applying sqrt rule)</span><br></div></code></pre></div></div></section>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="factoring-over-a-number-field">Factoring over a number field<a href="#factoring-over-a-number-field" class="hash-link" aria-label="Direct link to Factoring over a number field" title="Direct link to Factoring over a number field" translate="no"></a></h3>
<p>With a third argument, <code>["Factor", expr, var, generator]</code> factors <code>expr</code> over
the field <code class="language-math math-inline">\mathbb{Q}(\alpha)</code>: the rationals extended by the algebraic
number <code class="language-math math-inline">\alpha</code> that <code>generator</code> selects. A polynomial that is irreducible
over the rationals can split over such a field.</p>
<p>The generator can be:</p>
<ul>
<li class="">an exact square root or other root of a rational number, such as
<code>["Sqrt", 2]</code>, <code>["Sqrt", -3]</code> or <code>["Root", 2, 3]</code>;</li>
<li class=""><code>"ImaginaryUnit"</code>;</li>
<li class="">an <a href="#algebraicgenerator" class=""><code>AlgebraicGenerator</code></a>, for any other algebraic
number, given by its defining polynomial.</li>
</ul>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (x-√2)(x+√2)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ImaginaryUnit"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (x-i)(x+i)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (x²-√2x+1)(x²+√2x+1)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Root"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (x²+∛2x+∛2²)(x-∛2)</span><br></div></code></pre></div></div>
<p>In LaTeX:</p>
<div class="language-latex codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-latex codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token function selector" style="color:var(--base-09)">\operatorname</span><span class="token punctuation" style="color:var(--base-06)">{</span><span class="token plain">Factor</span><span class="token punctuation" style="color:var(--base-06)">}</span><span class="token plain">(x^2-2, x, </span><span class="token function selector" style="color:var(--base-09)">\sqrt</span><span class="token punctuation" style="color:var(--base-06)">{</span><span class="token plain">2</span><span class="token punctuation" style="color:var(--base-06)">}</span><span class="token plain">)</span><br></div></code></pre></div></div>
<p>The result is exact, also with <code>.N()</code>. The coefficients of <code>expr</code> must be
exact rational numbers, the generator, and products and sums of these. The
variable must be a symbol.</p>
<p>A request stays unevaluated when the engine cannot answer it: a coefficient it
cannot read (another symbol, a float, an unrelated radical), a generator that
is not supported, or a request too large for the work limits. The engine never
factors over the rationals instead, and <strong>an unevaluated result is not a proof
that the polynomial is irreducible</strong>.</p>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"y"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Factor", ["Subtract", ["Power", "x", 2], "y"], "x", ["Sqrt", 2]]</span><br></div></code></pre></div></div>
<p>A polynomial that is irreducible over the field is returned unchanged:</p>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Sqrt"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ x²-3</span><br></div></code></pre></div></div>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="algebraicgenerator">AlgebraicGenerator<a href="#algebraicgenerator" class="hash-link" aria-label="Direct link to AlgebraicGenerator" title="Direct link to AlgebraicGenerator" translate="no"></a></h3></nav>
<section id="AlgebraicGenerator" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AlgebraicGenerator</b>(<em>coefficients</em>)</p><p>A root <code class="language-math math-inline">\alpha</code> of the polynomial
<code class="language-math math-inline">c_0 + c_1 t + \cdots + c_n t^n</code>, where <code>coefficients</code> is the list
<code class="language-math math-inline">[c_0, c_1, \ldots, c_n]</code> in <strong>ascending</strong> order of degree. Use it as the
third argument of <a href="#factor" class=""><code>Factor</code></a> to factor over a field that a single
radical does not describe, such as <code class="language-math math-inline">\mathbb{Q}(\sqrt{2}+\sqrt{3})</code>, or the
field of a root of <code class="language-math math-inline">t^5 - t - 1</code>, which has no expression in radicals.</p><p>The coefficients must be exact rational numbers, and the polynomial must be
irreducible over the rationals, of degree 1 to 128. Otherwise the result is an
<code>invalid-algebraic-generator</code> error.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token comment" style="color:var(--base-05)">// √2 + √3 is a root of t⁴ - 10t² + 1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-10</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AlgebraicGenerator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-10</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ four linear factors in α = AlgebraicGenerator([1, 0, -10, 0, 1])</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AlgebraicGenerator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Error", "'invalid-algebraic-generator'"] // t² - 1 is reducible</span><br></div></code></pre></div></div><p>The generator does not select <strong>which</strong> root of the polynomial it is: the
polynomial <code class="language-math math-inline">t^2 - 2</code> has two roots, <code class="language-math math-inline">\sqrt{2}</code> and <code class="language-math math-inline">-\sqrt{2}</code>,
and <code>["AlgebraicGenerator", ["List", -2, 0, 1]]</code> is "one of them". Every root
gives the same field, so the factors that <code>Factor</code> finds have the same degrees
and multiplicities for each choice. The result is expressed in terms of the
generator itself:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factor"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AlgebraicGenerator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ (x - α)(x + α), where α is the AlgebraicGenerator expression</span><br></div></code></pre></div></div><p>For this reason, an <code>AlgebraicGenerator</code> has no numeric value: <code>.N()</code> leaves it
unchanged, and it cannot be compiled. To work with a specific value, such as
the positive square root of 2, use <code>["Sqrt", 2]</code> instead.</p><p>A power <code class="language-math math-inline">\alpha^k</code> with <code class="language-math math-inline">n \le k \le 128</code>, where <code class="language-math math-inline">n</code> is the degree
of the polynomial, is reduced modulo the polynomial, as <code class="language-math math-inline">(\sqrt{2})^2</code> is
<code>2</code>. This holds for every root, so it does not choose one:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token comment" style="color:var(--base-05)">// α is a root of t² - 2</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AlgebraicGenerator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 2</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// β is a root of t³ - t - 1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AlgebraicGenerator"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ β + 1</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Expand"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> α</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> α</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ x² - 2</span><br></div></code></pre></div></div><p>A negative power, such as <code class="language-math math-inline">\alpha^{-1}</code>, is not reduced.</p></section>
<section id="Together" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Together</b>(<em>expr</em>)</p><p>Combine rational expressions into a single fraction with a common denominator.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Together"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"y"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Divide", ["Add", "x", "y"], ["Multiply", "x", "y"]]</span><br></div></code></pre></div></div><p>The like terms of the result are collected, as when the sum is evaluated, also
when there is nothing to combine:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Together"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", 2, "a"]</span><br></div></code></pre></div></div></section>
<section id="Distribute" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Distribute</b>(<em>expr</em>)</p><p>Distribute multiplication over addition in <code>expr</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Distribute"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"b"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"c"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", ["Multiply", "a", "b"], ["Multiply", "a", "c"]]</span><br></div></code></pre></div></div></section>
<p>The polynomial operators below (<code>PolynomialDegree</code>, <code>CoefficientList</code>,
<code>PolynomialQuotient</code>, <code>PolynomialRemainder</code>, <code>PolynomialGCD</code>, <code>Resultant</code>,
<code>Cancel</code>, <code>PartialFraction</code>, <code>Apart</code>, <code>PolynomialRoots</code> and <code>Discriminant</code>)
receive their operands unevaluated. A symbol that holds a polynomial is
replaced by its value, as <code>Expand</code> and <code>Factor</code> do, and the default variable is
read from that value: with <code>p := x^3 - 2x^2 - 4</code> and <code>q := x - 3</code>,
<code>["PolynomialQuotient", "p", "q", "x"]</code> is <code>x^2 + x + 3</code> and
<code>["PolynomialRemainder", "p", "q", "x"]</code> is <code>5</code>. The variable named in the call
is never substituted, even when it has a value: with <code>x := 5</code>,
<code>["PolynomialDegree", ["Add", ["Power", "x", 2], 1], "x"]</code> is <code>2</code>.</p>
<section id="PolynomialDegree" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>PolynomialDegree</b>(<em>poly</em>)</p><p class="signature_CWyf"><b>PolynomialDegree</b>(<em>poly</em>, <em>var</em>)</p><p>Return the degree of the polynomial <code>poly</code> with respect to the variable <code>var</code>.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of <code>poly</code>, or
to <code>x</code> when there are several free variables and one of them is <code>x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"PolynomialDegree"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 3</span><br></div></code></pre></div></div></section>
<section id="CoefficientList" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>CoefficientList</b>(<em>poly</em>)</p><p class="signature_CWyf"><b>CoefficientList</b>(<em>poly</em>, <em>var</em>)</p><p>Return the list of coefficients of the polynomial <code>poly</code> with respect to the variable <code>var</code>, ordered from highest to lowest degree.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of <code>poly</code>, or
to <code>x</code> when there are several free variables and one of them is <code>x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"CoefficientList"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["List", 1, 0, 2, 1]</span><br></div></code></pre></div></div><p>The result represents the polynomial <code class="language-math math-inline">1 \cdot x^3 + 0 \cdot x^2 + 2 \cdot x + 1</code>. The first element is the leading coefficient, and the last element is the constant term.</p><p>This function is also available as a method on expressions:</p><div class="language-javascript codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-javascript codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token keyword" style="color:var(--base-0e)">const</span><span class="token plain"> coeffs </span><span class="token operator" style="color:var(--base-06)">=</span><span class="token plain"> ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x^3 + 2x + 1'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialCoefficients</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [1, 0, 2, 1] (as Expression[])</span><br></div></code></pre></div></div><p>When the variable is omitted, it is auto-detected if the expression has exactly one unknown:</p><div class="language-javascript codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-javascript codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x^2 + 5'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialCoefficients</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [1, 0, 5]</span><br></div></code></pre></div></div><p>Returns <code>undefined</code> if the expression is not a polynomial in the given variable. This subsumes an <code>isPolynomial</code> check and degree computation:</p><div class="language-javascript codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-javascript codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token keyword" style="color:var(--base-0e)">const</span><span class="token plain"> isPolynomial </span><span class="token operator" style="color:var(--base-06)">=</span><span class="token plain"> expr</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialCoefficients</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token plain"> </span><span class="token operator" style="color:var(--base-06)">!==</span><span class="token plain"> </span><span class="token keyword" style="color:var(--base-0e)">undefined</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token keyword" style="color:var(--base-0e)">const</span><span class="token plain"> degree </span><span class="token operator" style="color:var(--base-06)">=</span><span class="token plain"> expr</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialCoefficients</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token operator" style="color:var(--base-06)">?.</span><span class="token plain">length </span><span class="token operator" style="color:var(--base-06)">-</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">;</span><br></div></code></pre></div></div><p><strong>Multivariate validation:</strong> Pass an array of variables to validate that the expression is polynomial in all of them. The coefficients are decomposed by the first variable:</p><div class="language-javascript codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-javascript codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x^2*y + 3x + y^2'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialCoefficients</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'y'</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [y, 3, y²] (polynomial in both x and y, decomposed by x)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'sin(x)*y + 1'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialCoefficients</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'y'</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ undefined (not polynomial in x)</span><br></div></code></pre></div></div><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="polynomialroots"><code>polynomialRoots()</code><a href="#polynomialroots" class="hash-link" aria-label="Direct link to polynomialroots" title="Direct link to polynomialroots" translate="no"></a></h3><p>The <code>polynomialRoots()</code> method returns the roots of a polynomial expression:</p><div class="language-javascript codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-javascript codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x^2 - 5x + 6'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialRoots</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [2, 3]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x^3 - 6x^2 + 11x - 6'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialRoots</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [1, 2, 3]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x^2 + 1'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialRoots</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ [] (no real roots)</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'sin(x)'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">polynomialRoots</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ undefined (not a polynomial)</span><br></div></code></pre></div></div><p>Returns <code>undefined</code> if the expression is not a polynomial. Returns an empty array if no roots can be found (e.g., irreducible over the rationals). Supports auto-detection of the variable when omitted.</p></section>
<section id="Polynomial" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Polynomial</b>(<em>coefficients</em>, <em>var</em>)</p><p>Construct a polynomial expression from a list of coefficients (highest degree first) and a variable. This is the inverse of <code>CoefficientList</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Polynomial"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", ["Power", "x", 3], ["Multiply", 2, "x"], 1]</span><br></div></code></pre></div></div><p>This constructs the polynomial <code class="language-math math-inline">x^3 + 2x + 1</code> from the coefficient list <code class="language-math math-inline">[1, 0, 2, 1]</code>.</p><p><strong>Round-trip with CoefficientList:</strong></p><p><code class="language-math math-inline">\operatorname{Polynomial}(\operatorname{CoefficientList}(p, x), x) = p</code></p></section>
<section id="PolynomialQuotient" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>PolynomialQuotient</b>(<em>dividend</em>, <em>divisor</em>)</p><p class="signature_CWyf"><b>PolynomialQuotient</b>(<em>dividend</em>, <em>divisor</em>, <em>var</em>)</p><p>Return the quotient of the polynomial division of <code>dividend</code> by <code>divisor</code> with respect to the variable <code>var</code>.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of the
operands, or to <code>x</code> when there are several free variables and one of them is
<code>x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"PolynomialQuotient"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", ["Power", "x", 2], "x", 1]</span><br></div></code></pre></div></div><p>This represents <code class="language-math math-inline">\frac{x^3 - 1}{x - 1} = x^2 + x + 1</code>.</p></section>
<section id="PolynomialRemainder" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>PolynomialRemainder</b>(<em>dividend</em>, <em>divisor</em>)</p><p class="signature_CWyf"><b>PolynomialRemainder</b>(<em>dividend</em>, <em>divisor</em>, <em>var</em>)</p><p>Return the remainder of the polynomial division of <code>dividend</code> by <code>divisor</code> with respect to the variable <code>var</code>.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of the
operands, or to <code>x</code> when there are several free variables and one of them is
<code>x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"PolynomialRemainder"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ -2</span><br></div></code></pre></div></div></section>
<section id="GCD" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>GCD</b>(<em>x-1</em>, <em>x-2</em>, ...<!-- -->)</p><p>Return the greatest common divisor of the arguments.</p><p>With integer arguments, <code>GCD</code> returns their integer greatest common divisor:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GCD"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">60</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">12</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">18</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 6</span><br></div></code></pre></div></div><p><code>GCD</code> also computes the greatest common divisor of <strong>polynomials</strong>. When the
arguments are univariate polynomials in the same variable that share a
non-trivial common factor, the variable is inferred and a (monic) polynomial
GCD is returned:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GCD"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">4</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", "x", 1]</span><br></div></code></pre></div></div><p>This represents <code class="language-math math-inline">\gcd(x^2 + 3x + 2, x^2 + 4x + 3) = x + 1</code>.</p><p>When the polynomial GCD would be a constant, the expression is left
unevaluated, so that a bare symbol keeps its integer-GCD reading — for example
<code>["GCD", "x", 6]</code> (where <code>x</code> may stand for an unknown integer) stays symbolic.
Use <code>PolynomialGCD</code> with an explicit variable when you want the coprime
→ <code>1</code> result, or to control which variable the GCD is taken over.</p></section>
<section id="PolynomialGCD" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>PolynomialGCD</b>(<em>a</em>, <em>b</em>)</p><p class="signature_CWyf"><b>PolynomialGCD</b>(<em>a</em>, <em>b</em>, <em>var</em>)</p><p>Return the greatest common divisor of two polynomials <code>a</code> and <code>b</code> with respect to the variable <code>var</code>.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of the
operands, or to <code>x</code> when there are several free variables and one of them is
<code>x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"PolynomialGCD"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Subtract", "x", 1]</span><br></div></code></pre></div></div><p>This represents <code class="language-math math-inline">\gcd(x^2 - 1, x - 1) = x - 1</code>.</p></section>
<section id="Cancel" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Cancel</b>(<em>expr</em>)</p><p class="signature_CWyf"><b>Cancel</b>(<em>expr</em>, <em>var</em>)</p><p>Cancel common polynomial factors in the numerator and denominator of the rational expression <code>expr</code> with respect to the variable <code>var</code>.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of <code>expr</code>, or
to <code>x</code> when there are several free variables and one of them is <code>x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Cancel"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Subtract"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", "x", 1]</span><br></div></code></pre></div></div><p>This represents <code class="language-math math-inline">\frac{x^2 - 1}{x - 1} = x + 1</code> after canceling the common factor <code class="language-math math-inline">(x - 1)</code>.</p><p>The like terms of the result are collected, as when the sum is evaluated. With
no <code>var</code> and no single free variable, nothing is canceled, but the like terms
are still collected:</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Cancel"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"a"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Multiply", 2, "a"]</span><br></div></code></pre></div></div></section>
<section id="PartialFraction" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>PartialFraction</b>(<em>expr</em>)</p><p class="signature_CWyf"><b>PartialFraction</b>(<em>expr</em>, <em>var</em>)</p><p>Decompose a rational expression into a sum of simpler fractions (partial fractions) with respect to the variable <code>var</code>.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of <code>expr</code>, or
to <code>x</code> when there are several free variables and one of them is <code>x</code> — so a
pipeline such as <code>expr |> Apart</code> works without naming the variable.</p><p>Supports:</p><ul>
<li class=""><strong>Distinct linear factors</strong>: <code class="language-math math-inline">\frac{1}{(x+1)(x+2)} \to \frac{1}{x+1} - \frac{1}{x+2}</code></li>
<li class=""><strong>Repeated linear factors</strong>: <code class="language-math math-inline">\frac{3x+5}{(x+1)^2} \to \frac{3}{x+1} + \frac{2}{(x+1)^2}</code></li>
<li class=""><strong>Irreducible quadratic factors</strong>: <code class="language-math math-inline">\frac{1}{(x+1)(x^2+1)} \to \frac{1}{2(x+1)} + \frac{-x+1}{2(x^2+1)}</code></li>
<li class=""><strong>Improper fractions</strong>: Performs polynomial division first, then decomposes the proper remainder.</li>
</ul><p>Returns the expression unchanged if it is not a rational expression in <code>var</code>, or if the denominator cannot be factored.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"PartialFraction"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Divide"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Add", ["Divide", 1, ["Add", "x", 1]], ["Divide", -1, ["Add", "x", 2]]]</span><br></div></code></pre></div></div></section>
<section id="Apart" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Apart</b>(<em>expr</em>)</p><p class="signature_CWyf"><b>Apart</b>(<em>expr</em>, <em>var</em>)</p><p>Alias for <code>PartialFraction</code>. Decompose a rational expression into partial fractions.</p></section>
<section id="PolynomialRoots" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>PolynomialRoots</b>(<em>poly</em>)</p><p class="signature_CWyf"><b>PolynomialRoots</b>(<em>poly</em>, <em>var</em>)</p><p>Return the roots of the polynomial <code>poly</code> with respect to the variable <code>var</code> as a set.</p><p>When <code>var</code> is omitted, it defaults to the single free variable of <code>poly</code>, or
to <code>x</code> when there are several free variables and one of them is <code>x</code>.</p><p>Returns <code>undefined</code> if the expression is not a polynomial or no roots can be found. Supports polynomials up to degree 4 with rational roots, and degree 2 with irrational roots.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"PolynomialRoots"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Set", 2, 3]</span><br></div></code></pre></div></div></section>
<section id="Discriminant" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Discriminant</b>(<em>poly</em>)</p><p class="signature_CWyf"><b>Discriminant</b>(<em>poly</em>, <em>var</em>)</p><p>Return the discriminant of the polynomial <code>poly</code> with respect to the variable <code>var</code>. Supports degree 2, 3, and 4 polynomials. Works with symbolic coefficients.</p><ul>
<li class=""><strong>Degree 2</strong> (<code class="language-math math-inline">ax^2 + bx + c</code>): returns <code class="language-math math-inline">b^2 - 4ac</code></li>
<li class=""><strong>Degree 3</strong> (<code class="language-math math-inline">ax^3 + bx^2 + cx + d</code>): returns <code class="language-math math-inline">b^2c^2 - 4ac^3 - 4b^3d + 18abcd - 27a^2d^2</code></li>
<li class=""><strong>Degree 4</strong>: returns the full 16-term discriminant formula</li>
</ul><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Discriminant"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Add"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Power"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Multiply"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">-5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">6</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"x"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1</span><br></div></code></pre></div></div><p>A discriminant of 0 indicates repeated roots. 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