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<p><a id="X87003045878E74DF" java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<div java.lang.StringIndexOutOfBoundsException: Range [64, 10) out of bounds for length 118
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X7A76497986DA921F">17.1 <span class="Heading">Rationals: Global Variables</span></a>
</span>
<div class="ContSSBlock">
<="< >span class"n;&<>a ="chap17_mj>1711Rationals/>/
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X826E2AA88679B3DF">17.2 <span class="Heading">Elementary Operations for Rationals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17_mj.html#X7ED018F5794935F7">17.2-1 IsRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17_mj.html#X7BD6E170840F045D">17.2-2 IsPosRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17_mj.html#X81179AC87AC951A8">17.2-3 IsNegRat</a></span>
<span class="ContSS"><br /><span class="nocss"> &java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 1
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17_mj.html#X81F6B5877A81E727">17.2-5 DenominatorRat</a></span>
<span class="ContSS"><br /><span class="nocss">  <*     -----  ----    -----
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17_mj.html#X7C8F8693825C28A4">17.2-7 Random</a></span>
</div></div>
</div>

<h3>17 <span class="Heading">Rational Numbers</span></h3>

<p>The <em>rationals</em> form a very important field. On the one hand it is the quotient field of the integers (see chapter <a href="chap14_mj.html#X853DF11B80068ED5"><span class="RefLink">14</span></a>). On the other hand it is the prime field of the fields of characteristic zero (see chapter <*     SO           SO>        -11643-1992 Plane1,GB2312 --165

<p>The former comment suggests the representation actually used. A rational      SS3          ESCO     CNS11643-1992 Planes 3-7

<p>Because the underlying integer arithmetic can compute with arbitrary size integers, the rational arithmetic is always exact, even for rationals whose numerators and denominators have thousands of digits.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">2/3;</span>
2/3
<span class="GAPprompt">gap></span> <span class="GAPinput">66/123;  # numerator and denominator are made relatively prime</span>
22/41
<span class="GAPprompt">gap></span> <span class="GAPinput">17/-13;  # the numerator carries the sign;</span>
-17/13
<span class="GAPprompt">gap></span> <span class="GAPinput">121/11;  # rationals with denominator 1 (when canceled) are integers</span>
11
</pre></div>

<p><a id="X7A76497986DA921F" *      SOdesignator  : ESC"" ""finalchar_for_SO

<h4>17.1 <span class="Heading">Rationals: Global Variables</span></h4>

<p><a id="X7B6029D18570C08A" name="X7B6029D18570C08A"></a></p>

<h5>17.1-1 Rationals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Rationals</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsRationals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<pjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<p><code class="func">IsRationals</code> returns <code class="keyw">true</code> for a prime field that consists of cyclotomic numbers –for example the <strong class="pkg">GAP      $ E         bytes  SO  


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( Rationals ); 2/3 in Rationals;</span>
infinity
true
</pre></div>

<p><a id="X826E2AA88679B3DF" name="X826E2AA88679B3DF"></a></p>

<h4>17.2 <span class="Heading">Elementary Operations for Rationals</span></h4>

<p><a id="X7ED018F5794935F7" name="X7ED018F5794935F7"></a></p>

<h5>17.2-1 IsRat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">      $)G   SO 
<p>Every rational number lies in the category <code class="func">IsRat</code>, which is a subcategory of <code class="func">IsCyc</code> (<a href="chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></*        11643plane,until another


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRat( 2/3 );</span>
true
<span class=*
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRat( 11 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRat( IsRat );  # `IsRat' is a function, not a rational</span>
   ESC$*H      Indicates the two bytes immediately following
</pre></div>

<p><a id="X7BD6E170840F045D" name="X7BD6E170840F045D"></a></p>

<h5>17.2-2 IsPosRat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPosRat</java.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 52
<java.lang.StringIndexOutOfBoundsException: Range [33, 2) out of bounds for length 94

<p><a id="X81179AC87AC951A8" name="X81179AC87AC951A8"></a></p>

<h5>17.2-3 IsNegRat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsNegRat</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Every negative rational number lies in the category <code class="func">IsNegRat</code>.</p>

<p><a id="X7D830E7482E7F528" name="X7D830E7482E7F528"></a></p>

<h5>17.2-4 NumeratorRat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NumeratorRat</code>( <var class="Arg">rat</var> )</td><td class="tdright">(&       $+       Indicatesimmediate  bytes
<p><code class="func">NumeratorRat</code> returns the numerator of the rational <var class="Arg">rat</var>. Because the numerator holds the sign<       11643--3 untilanotherSS3designation


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumeratorRat( 2/3 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput"># numerator and denominator are made relatively prime:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumeratorRat( 66/123 );</span>
22
<span class="GAPprompt">gap></span> <span class="GAPinput">NumeratorRat( 17/-13 );  # numerator holds the sign of the rational</span>
-17
<span class="GAPprompt">gap></span> <span class="GAPinput">NumeratorRat( 11 );      # integers are rationals with denominator 1</span>
11
</pre></div>

<p><a id="X81F6B5877A81E727" name="X81F6B5877A81E727"></a></p>

<h5>17.2-5 DenominatorRat</h5>

<java.lang.StringIndexOutOfBoundsException: Range [16, 4) out of bounds for length 223
<>codeclass=func"DenominatorRat</code> returns the denominator of the rational <var class="Arg">rat</var>. Because the numerator holds the sign of the rational the denominator is always a positive integer. Integers are rationals with the denominator 1, thus <code class="func">DenominatorRat</code> returns 1 for integers.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DenominatorRat( 2/3 );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput"># numerator and denominator are made relatively prime:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DenominatorRat( 66/123 );</span>
41
<span class="GAPprompt">gap></span> <span class="GAPinput"># the denominator holds the sign of the rational:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DenominatorRat( 17/-13 );</span>
13
<span class="GAPprompt">gap></span> <span class="GAPinput">DenominatorRat( 11 ); # integers are rationals with denominator 1</span>
1
</pre></div>

<p><a id="X7EB4C646806A2BDE" name="X7EB4C646806A2BDE"></a></p>

<h5>17.2-6 Rat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Rat</code>( <var class="Arg">elm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">Rat</code> returns a rational number <var class="Arg">rat</var> whose meaning depends on the type of <var class="Arg">elm</var>.</p>

<p>If <var class="Arg">elm</var> is a string consisting of digits <code class="code">'0'</code>, <code class="code">'1'</code>, <span


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Rat( "1/2" );  Rat( "35/14" );  Rat*ESC $+K        immediatetwobytesjava.lang.StringIndexOutOfBoundsException: Range [70, 66) out of bounds for length 70
1/2
5/2
-35/27
314159/100000
</pre></div>

<p><a id="X7C8F8693825C28A4" name="X7C8F8693825C28A4"></a></p>

<h5>17.2-7 Random</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func",java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 51
<p><code class="func">Random</code> for rationals returns pseudo random rationals which are the quotient of two random integers. See the description of <code class="func">Random</code> (<a href="chap14_mj.html#X8185784B7E228DEA"><span class="RefLink">14


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