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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="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a></div>
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<div><h1>% -- a binary operator, usually used for remainder and reduction</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>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The usual meaning for this operator is remainder, or normal form with respect to a Gröbner basis.<p/>
For integers, the remainder is non-negative.<table class="examples"><tr><td><pre>i1 : 1232132141242345 % 1000000
o1 = 242345</pre>
</td></tr>
<tr><td><pre>i2 : (-4)%5
o2 = 1</pre>
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</table>
<p/>
In polynomial rings, the division algorithm is used.<table class="examples"><tr><td><pre>i3 : A = ZZ[a,b]
o3 = A
o3 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i4 : (3*a^3-a*b-4) % (5*a-b)
3 2
o4 = - 2a + a b - a*b - 4
o4 : A</pre>
</td></tr>
<tr><td><pre>i5 : pseudoRemainder(3*a^3-a*b-4, 5*a-b)
3 2
o5 = 3b - 25b - 500
o5 : A</pre>
</td></tr>
<tr><td><pre>i6 : B = QQ[a,b]
o6 = B
o6 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i7 : (3*a^3-a*b-4) % (5*a-b)
3 3 1 2
o7 = ---b - -b - 4
125 5
o7 : B</pre>
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</table>
In more complicated situations, Gröbner bases are usually needed. See <a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a>.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_remainder.html" title="matrix remainder">remainder</a> -- matrix remainder</span></li>
<li><span><a href="_remainder_sq.html" title="matrix quotient and remainder (opposite)">remainder'</a> -- matrix quotient and remainder (opposite)</span></li>
<li><span><a href="_pseudo__Remainder.html" title="compute the pseudo-remainder">pseudoRemainder</a> -- compute the pseudo-remainder</span></li>
<li><span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</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>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 % GroebnerBasis</li>
<li>ZZ % Ideal</li>
<li>ZZ % MonomialIdeal</li>
<li>ZZ % ZZ</li>
<li><span><tt>InexactNumber % RingElement</tt> (missing documentation<!-- tag: (%,InexactNumber,RingElement) -->)</span></li>
<li><span><a href="___Matrix_sp_pc_sp__Groebner__Basis.html" title="calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis">Matrix % GroebnerBasis</a> -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis</span></li>
<li><span>RingElement % GroebnerBasis, see <span><a href="___Matrix_sp_pc_sp__Groebner__Basis.html" title="calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis">Matrix % GroebnerBasis</a> -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis</span></span></li>
<li><span>Matrix % Ideal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % Matrix, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % Module, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % MonomialIdeal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % RingElement, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>RingElement % Ideal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>RingElement % Matrix, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>RingElement % MonomialIdeal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>RingElement % RingElement, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span><tt>RingElement % InexactNumber</tt> (missing documentation<!-- tag: (%,RingElement,InexactNumber) -->)</span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</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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