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<!DOCTYPE html> <html lang="en"> <head> <title>// -- a binary operator, usually used for quotient</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 quotient" href="__sl_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="__pc.html">next</a> | <a href="__sl.html">previous</a> | <a href="__pc.html">forward</a> | <a href="__sl.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 quotient</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> For ring elements in the integers, polynomial rings, and other rings, there are two types of division: exact division, whose result is often in a larger field, such as the rationals or a function field, and division with remainder, whose result is in the same ring. In Macaulay2, '/' denotes the first kind of division, while '//' denotes the latter kind. The following example shows the difference between <a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> and <a title="a binary operator, usually used for division" href="__sl.html">/</a>. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : 4/2 o1 = 2 o1 : QQ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : 4//2 o2 = 2</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : R = QQ[x];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : (x^2-3)//(x-1) o4 = x + 1 o4 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : (x^2-3)%(x-1) o5 = -2 o5 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : (x^2-3)/(x-1) 2 x - 3 o6 = ------ x - 1 o6 : frac R</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="a binary operator, usually used for division" href="__sl.html">/</a> -- a binary operator, usually used for division</span></li> <li><span><a title="a binary operator, usually used for remainder and reduction" href="__pc.html">%</a> -- a binary operator, usually used for remainder and reduction</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>List // Number</kbd></li> <li><kbd>List // RingElement</kbd></li> <li><kbd>Number // InfiniteNumber</kbd></li> <li><kbd>Number // RingElement</kbd></li> <li><kbd>QQ // QQ</kbd></li> <li><kbd>QQ // ZZ</kbd></li> <li><kbd>RingElement // Number</kbd></li> <li><kbd>RingElement // RingElement</kbd></li> <li><kbd>RR // QQ</kbd></li> <li><kbd>RR // RR</kbd></li> <li><kbd>RR // ZZ</kbd></li> <li><kbd>ZZ // MonomialIdeal</kbd></li> <li><kbd>ZZ // ZZ</kbd></li> <li><span><kbd>Matrix // GroebnerBasis</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>Matrix // Matrix</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>Matrix // MonomialIdeal</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>Matrix // Number</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>Matrix // RingElement</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>Number // Matrix</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>RingElement // GroebnerBasis</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>RingElement // Matrix</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>RingElement // MonomialIdeal</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>Thing // Command</kbd> -- see <span><a title="apply a function" href="___Visible__List_sp_sl_sl_sp__Function.html">VisibleList // Function</a> -- apply a function</span></span></li> <li><span><kbd>Thing // Function</kbd> -- see <span><a title="apply a function" href="___Visible__List_sp_sl_sl_sp__Function.html">VisibleList // Function</a> -- apply a function</span></span></li> <li><span><kbd>Thing // SelfInitializingType</kbd> -- see <span><a title="apply a function" href="___Visible__List_sp_sl_sl_sp__Function.html">VisibleList // Function</a> -- apply a function</span></span></li> <li><span><kbd>VisibleList // Command</kbd> -- see <span><a title="apply a function" href="___Visible__List_sp_sl_sl_sp__Function.html">VisibleList // Function</a> -- apply a function</span></span></li> <li><span><a title="apply a function" href="___Visible__List_sp_sl_sl_sp__Function.html">VisibleList // Function</a> -- apply a function</span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a binary operator, usually used for quotient" href="__sl_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.m2:120:0</span>.</p> </div> </div> </div> </body> </html>