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<head><title>// -- a binary operator, usually used for quotient</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="___The_sp__Macaulay2_splanguage.html" title="">The Macaulay2 language</a> > <a href="_operators.html" title="">operators</a> > <a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a></div>
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<div><h1>// -- a binary operator, usually used for quotient</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>x // y</tt></div>
</dd></dl>
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</li>
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<div class="single"><h2>Description</h2>
<div>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 href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> and <a href="__sl.html" title="a binary operator, usually used for division">/</a>.<table class="examples"><tr><td><pre>i1 : 4/2
o1 = 2
o1 : QQ</pre>
</td></tr>
<tr><td><pre>i2 : 4//2
o2 = 2</pre>
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<table class="examples"><tr><td><pre>i3 : R = QQ[x];</pre>
</td></tr>
<tr><td><pre>i4 : (x^2-3)//(x-1)
o4 = x + 1
o4 : R</pre>
</td></tr>
<tr><td><pre>i5 : (x^2-3)%(x-1)
o5 = -2
o5 : R</pre>
</td></tr>
<tr><td><pre>i6 : (x^2-3)/(x-1)
2
x - 3
o6 = ------
x - 1
o6 : frac(R)</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></li>
<li><span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>//</tt> :</h2>
<ul><li>CC // CC</li>
<li>CC // QQ</li>
<li>CC // RR</li>
<li>CC // ZZ</li>
<li>InfiniteNumber // ZZ</li>
<li>Matrix // ZZ</li>
<li>Number // RingElement</li>
<li>QQ // QQ</li>
<li>QQ // ZZ</li>
<li>RingElement // Number</li>
<li>RR // QQ</li>
<li>RR // RR</li>
<li>RR // ZZ</li>
<li>ZZ // InfiniteNumber</li>
<li>ZZ // Matrix</li>
<li>ZZ // MonomialIdeal</li>
<li>ZZ // ZZ</li>
<li><span><tt>InexactNumber // RingElement</tt> (missing documentation<!-- tag: (//,InexactNumber,RingElement) -->)</span></li>
<li><span>Matrix // GroebnerBasis, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></li>
<li><span>Matrix // MonomialIdeal, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>Matrix // RingElement, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // GroebnerBasis, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // Matrix, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // MonomialIdeal, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // RingElement, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span><tt>RingElement // InexactNumber</tt> (missing documentation<!-- tag: (//,RingElement,InexactNumber) -->)</span></li>
<li><span>Thing // Command, see <span><a href="___Thing_sp_sl_sl_sp__Function.html" title="apply a function">Thing // Function</a> -- apply a function</span></span></li>
<li><span><a href="___Thing_sp_sl_sl_sp__Function.html" title="apply a function">Thing // Function</a> -- apply a function</span></li>
<li><span>Thing // SelfInitializingType, see <span><a href="___Thing_sp_sl_sl_sp__Function.html" title="apply a function">Thing // Function</a> -- apply a function</span></span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> is <span>a <a href="___Keyword.html">keyword</a></span>.</p>
<div><div><p>This operator may be used as a binary operator in an expression like <tt>x//y</tt>. The user may install <a href="_binary_spmethods.html" title="">binary methods</a> for handling such expressions with code such as</p>
<pre> X // Y := (x,y) -> ...</pre>
<p>where <tt>X</tt> is the class of <tt>x</tt> and <tt>Y</tt> is the class of <tt>y</tt>.</p>
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