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<!DOCTYPE html> <html lang="en"> <head> <title>/ -- a binary operator, usually used for division</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="index.html">Documentation </a> <br><a href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Macaulay2 documentation" href="index.html">Macaulay2Doc</a> » <a href="___The_sp__Macaulay2_splanguage.html">The Macaulay2 language</a> » <a href="_operators.html">operators</a> » <a title="a binary operator, usually used for division" href="__sl.html">/</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="__sl_sl.html">next</a> | <a href="__st.html">previous</a> | <a href="__sl_sl.html">forward</a> | <a href="__st.html">backward</a> | <a href="_operators.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>/ -- a binary operator, usually used for division</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">x / y</code></dd> </dl> </li> </ul> <div> <h2>Description</h2> This operator is currently used in several ways in Macaulay2. <ul> <li>division in a ring, yielding a fraction</li> <li>division in unevaluated expressions</li> <li>quotient rings, modules and sheaves</li> <li>applying a function or ring map to every element of a list or set</li> </ul> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : 2/3 2 o1 = - 3 o1 : QQ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : 2./3 o2 = .6666666666666666 o2 : RR (of precision 53)</code></pre> </td> </tr> </table> <h3>Intervals</h3> <p>If one of the inputs is an <a title="the class of all real intervals" href="___R__Ri.html">RRi</a>, the output is an interval containing all quotients of pairs in the inputs.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : 2/interval(1,3) o3 = [.6666666666666666,2] o3 : RRi (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : interval(-1,2)/interval(1,3) o4 = [-1,2] o4 : RRi (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : interval(1,2)/interval(1,2) o5 = [.5,2] o5 : RRi (of precision 53)</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> -- a binary operator, usually used for quotient</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">symbol /</span>:</h2> <ul> <li><kbd>CC / CC</kbd></li> <li><kbd>CC / QQ</kbd></li> <li><kbd>CC / RR</kbd></li> <li><kbd>CC / ZZ</kbd></li> <li><kbd>QQ / CC</kbd></li> <li><kbd>QQ / QQ</kbd></li> <li><kbd>QQ / RR</kbd></li> <li><kbd>QQ / RRi</kbd></li> <li><kbd>QQ / ZZ</kbd></li> <li><kbd>RR / CC</kbd></li> <li><kbd>RR / QQ</kbd></li> <li><kbd>RR / RR</kbd></li> <li><kbd>RR / RRi</kbd></li> <li><kbd>RR / ZZ</kbd></li> <li><kbd>RRi / QQ</kbd></li> <li><kbd>RRi / RR</kbd></li> <li><kbd>RRi / RRi</kbd></li> <li><kbd>RRi / ZZ</kbd></li> <li><kbd>ZZ / CC</kbd></li> <li><kbd>ZZ / QQ</kbd></li> <li><kbd>ZZ / RR</kbd></li> <li><kbd>ZZ / RRi</kbd></li> <li><kbd>ZZ / ZZ</kbd></li> <li><span><a title="apply a function to generators of an ideal" href="___Ideal_sp_sl_sp__Function.html">Ideal / Function</a> -- apply a function to generators of an ideal</span></li> <li><span><a title="quotient module" href="___Ideal_sp_sl_sp__Ideal.html">Ideal / Ideal</a> -- quotient module</span></li> <li><span><kbd>List / Number</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>List / RingElement</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></li> <li><span><kbd>Matrix / RingElement</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>Vector / Number</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>Vector / RingElement</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>Module / Ideal</kbd> -- see <span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></span></li> <li><span><kbd>Module / List</kbd> -- see <span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></span></li> <li><span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></li> <li><span><kbd>Module / RingElement</kbd> -- see <span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></span></li> <li><span><kbd>Module / Sequence</kbd> -- see <span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></span></li> <li><span><kbd>Module / Vector</kbd> -- see <span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></span></li> <li><span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></li> <li><span><kbd>Ring / List</kbd> -- see <span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></span></li> <li><span><kbd>Ring / Module</kbd> -- see <span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></span></li> <li><span><kbd>Ring / RingElement</kbd> -- see <span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></span></li> <li><span><kbd>Ring / Sequence</kbd> -- see <span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></span></li> <li><span><kbd>Ring / ZZ</kbd> -- see <span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></span></li> <li><span><a title="fraction" href="___Ring__Element_sp_sl_sp__Ring__Element.html">RingElement / RingElement</a> -- fraction</span></li> <li><span><kbd>List / Command</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>List / Function</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>List / SelfInitializingType</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>Set / Command</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>Set / Function</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>String / Command</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>String / Function</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>VirtualTally / Command</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>VirtualTally / Function</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>VisibleList / Command</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></li> <li><span><kbd>VisibleList / RingMap</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> <li><span><kbd>VisibleList / SelfInitializingType</kbd> -- see <span><a title="apply a function to elements of a list" href="___Visible__List_sp_sl_sp__Function.html">VisibleList / Function</a> -- apply a function to elements of a list</span></span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a binary operator, usually used for division" href="__sl.html">/</a> is <span>a <a title="the class of all keywords" href="___Keyword.html">keyword</a></span>.</p> <div> <p>This operator may be used as a binary operator in an expression like <span class="tt">x/y</span>. The user may install <a href="_binary_spmethods.html">binary methods</a> for handling such expressions with code such as</p> <pre> X / Y := (x,y) -> ...</pre> <p>where <span class="tt">X</span> is the class of <span class="tt">x</span> and <span class="tt">Y</span> is the class of <span class="tt">y</span>.</p> </div> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Macaulay2Doc/operators/division.m2:77:0</span>.</p> </div> </div> </div> </body> </html>
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