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docItemCol_VOVn"><div class="docItemContainer_Djhp"><article><nav class="theme-doc-breadcrumbs breadcrumbsContainer_Z_bl" aria-label="Breadcrumbs"><ul class="breadcrumbs"><li class="breadcrumbs__item"><a aria-label="Home page" class="breadcrumbs__link" href="/"><svg viewBox="0 0 24 24" class="breadcrumbHomeIcon_YNFT"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 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">Complex</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>Complex</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>
<table><thead><tr><th style="text-align:left">Symbol</th><th style="text-align:left">Description</th><th></th></tr></thead><tbody><tr><td style="text-align:left"><code>ImaginaryUnit</code></td><td style="text-align:left"><code class="language-math math-inline">\imaginaryI</code></td><td>The imaginary unit, solution of <code class="language-math math-inline">x^2+1=0</code></td></tr></tbody></table>
<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>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="real">Real<a href="#real" class="hash-link" aria-label="Direct link to Real" title="Direct link to Real" translate="no"></a></h3></nav>
<section id="Real" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Real</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\Re</span><span>(3+4</span><span class="command_yVKk">\imaginaryI</span><span>)</span><br></div></div><div class="display_sbg0">$$$\Re(3+4\imaginaryI)$$</div></div><p>Evaluate to the real part of a complex number.</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)">"Real"</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)">"Complex"</span><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><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>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="imaginary">Imaginary<a href="#imaginary" class="hash-link" aria-label="Direct link to Imaginary" title="Direct link to Imaginary" translate="no"></a></h3></nav>
<section id="Imaginary" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Imaginary</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\Im</span><span>(3+4</span><span class="command_yVKk">\imaginaryI</span><span>)</span><br></div></div><div class="display_sbg0">$$$\Im(3+4\imaginaryI)$$</div></div><p>Evaluate to the imaginary part of a complex number. If <em>z</em> is a real number, the
imaginary part is zero.</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)">"Imaginary"</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)">"Complex"</span><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><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><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)">"Imaginary"</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)">"Pi"</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>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="conjugate">Conjugate<a href="#conjugate" class="hash-link" aria-label="Direct link to Conjugate" title="Direct link to Conjugate" translate="no"></a></h3></nav>
<section id="Conjugate" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Conjugate</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>z^</span><span class="command_yVKk">\ast</span><br></div></div><div class="display_sbg0">$$$z^\ast$$</div></div><p>Evaluate to the complex conjugate of a complex number. The conjugates of complex
numbers give the mirror image of the complex number about the real axis.</p><code class="language-math math-display">z^\ast = \Re z - \imaginaryI \Im z</code><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)">"Conjugate"</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)">"Complex"</span><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><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)">// ➔ ["Complex", 3, -4]</span><br></div></code></pre></div></div></section>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="abs">Abs<a href="#abs" class="hash-link" aria-label="Direct link to Abs" title="Direct link to Abs" translate="no"></a></h3></nav>
<section id="Abs" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Abs</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>|z|</span><br></div></div><div class="display_sbg0">$$$|z|$$</div></div><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>abs</span><span class="punctuation_YpjP">}</span><span>(z)</span><br></div></div><div class="display_sbg0">$$$\operatorname{abs}(z)$$</div></div><p>Evaluate to the magnitude of a complex number.</p><p>The <strong>magnitude</strong> of a complex number is the distance from the origin to the
point representing the complex number in the complex plane.</p><code class="language-math math-display">|z| = \sqrt{(\Im z)^2 + (\Re z)^2}</code><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)">"Abs"</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)">"Complex"</span><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><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>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="arg">Arg<a href="#arg" class="hash-link" aria-label="Direct link to Arg" title="Direct link to Arg" translate="no"></a></h3></nav>
<section id="Arg" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Arg</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\arg</span><span>(z)</span><br></div></div><div class="display_sbg0">$$$\arg(z)$$</div></div><p>Evaluate to the argument of a complex number.</p><p>The <strong>argument</strong> of a complex number is the angle between the positive real axis
and the line joining the origin to the point representing the complex number in
the complex plane.</p><code class="language-math math-display">\arg z = \tan^{-1} \frac{\Im z}{\Re z}</code><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)">"Arg"</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)">"Complex"</span><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><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)">// ➔ 0.9272952180016122</span><br></div></code></pre></div></div></section>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="absarg">AbsArg<a href="#absarg" class="hash-link" aria-label="Direct link to AbsArg" title="Direct link to AbsArg" translate="no"></a></h3></nav>
<section id="AbsArg" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AbsArg</b>(<em>z</em>)</p><p>Return a tuple of the magnitude and argument of a complex number.</p><p>This corresponds to the polar representation of a complex number.</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)">"AbsArg"</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)">"Complex"</span><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><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, 0.9272952180016122]</span><br></div></code></pre></div></div><div class="language-math math-display 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-math 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">3+4\imaginaryI = 5 (\cos 0.9272 + \imaginaryI \sin 0.9272) = 5</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\exponentialE^{0.9272}</span><br></div></code></pre></div></div></section>
<nav class="hidden"><h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="complexroots">ComplexRoots<a href="#complexroots" class="hash-link" aria-label="Direct link to ComplexRoots" title="Direct link to ComplexRoots" translate="no"></a></h3></nav>
<section id="ComplexRoots" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>ComplexRoots</b>(<em>z</em>, <em>n</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>ComplexRoot</span><span class="punctuation_YpjP">}</span><span>(1, 3)</span><br></div></div><div class="display_sbg0">$$$\operatorname{ComplexRoot}(1, 3)$$</div></div><p>Return a list of the n<sup>th</sup> roots of a number <em>z</em>.</p><p>The complex roots of a number are the solutions of the equation <code class="language-math math-inline">z^n = a</code>.</p><ul>
<li class="">Wikipedia: <a href="https://en.wikipedia.org/wiki/Nth_root" target="_blank" rel="noopener noreferrer" class="">n<sup>th</sup> root</a></li>
<li class="">Wikipedia: <a href="https://en.wikipedia.org/wiki/Root_of_unity" target="_blank" rel="noopener noreferrer" class="">Root of unity</a></li>
<li class="">Wolfram MathWorld: <a href="http://mathworld.wolfram.com/nthRoot.html" target="_blank" rel="noopener noreferrer" class="">Root of unity</a></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 comment" style="color:var(--base-05)">// The three complex roots of unity (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)">"ComplexRoots"</span><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 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, -1/2 + sqrt(3)/2, -1/2 - sqrt(3)/2]</span><br></div></code></pre></div></div></section></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-edit-meta-row row"><div class="col"></div><div class="col lastUpdated_VsjB"></div></div></footer></article><nav class="docusaurus-mt-lg pagination-nav" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/compute-engine/reference/colors/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Colors</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/compute-engine/reference/control-structures/"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Control Structures</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#constants" class="table-of-contents__link toc-highlight">Constants</a></li><li><a href="#functions" class="table-of-contents__link toc-highlight">Functions</a><ul><li><a href="#real" class="table-of-contents__link toc-highlight">Real</a></li><li><a href="#imaginary" class="table-of-contents__link toc-highlight">Imaginary</a></li><li><a href="#conjugate" class="table-of-contents__link toc-highlight">Conjugate</a></li><li><a href="#abs" class="table-of-contents__link toc-highlight">Abs</a></li><li><a href="#arg" class="table-of-contents__link toc-highlight">Arg</a></li><li><a href="#absarg" class="table-of-contents__link toc-highlight">AbsArg</a></li><li><a href="#complexroots" class="table-of-contents__link toc-highlight">ComplexRoots</a></li></ul></li></ul></div></div></div></div></main></div></div></div><div class="shared-footer"><footer class="theme-layout-footer footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="theme-layout-footer-column col footer__col"><div class="footer__title">SDKs</div><ul class="footer__items clean-list"><li class="footer__item"><a class="footer__link-item" href="/mathfield/">Mathfield</a></li><li class="footer__item"><a class="footer__link-item" href="/compute-engine/">Compute Engine</a></li><li class="footer__item"><a class="footer__link-item" href="/math-json/">MathJSON</a></li><li class="footer__item"><a href="https://epsil.dev" target="_blank" rel="noopener noreferrer" class="footer__link-item">Epsil<svg width="13.5" height="13.5" aria-label="(opens in new tab)" class="iconExternalLink_nPIU"><use href="#theme-svg-external-link"></use></svg></a></li></ul></div><div class="theme-layout-footer-column col footer__col"><div class="footer__title">Community</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/cortex-js" target="_blank" rel="noopener noreferrer" class="footer__link-item">
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