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<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"/><title>3.2 Numbers</title><link rel="stylesheet" type="text/css" href="../scribble.css" title="default"/><link rel="stylesheet" type="text/css" href="../racket.css" title="default"/><link rel="stylesheet" type="text/css" href="icons.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-style.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../doc-site.css" title="default"/><script type="text/javascript" src="../scribble-common.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../doc-site.js"></script><script type="text/javascript" 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align="right"></td><td><a href="doc-bibliography.html" class="tocviewlink" data-pltdoc="x">Bibliography</a></td></tr><tr><td align="right"></td><td><a href="doc-index.html" class="tocviewlink" data-pltdoc="x">Index</a></td></tr></table></div></div><div class="tocviewlist"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_1");">▼</a></td><td>3 </td><td><a href="datatypes.html" class="tocviewlink" data-pltdoc="x">Built-<wbr></wbr>In Datatypes</a></td></tr></table><div class="tocviewsublistbottom" style="display: block;" id="tocview_1"><table cellspacing="0" cellpadding="0"><tr><td align="right">3.1 </td><td><a href="booleans.html" class="tocviewlink" data-pltdoc="x">Booleans</a></td></tr><tr><td align="right">3.2 </td><td><a href="" class="tocviewselflink" data-pltdoc="x">Numbers</a></td></tr><tr><td align="right">3.3 </td><td><a href="characters.html" class="tocviewlink" data-pltdoc="x">Characters</a></td></tr><tr><td align="right">3.4 </td><td><a href="strings.html" class="tocviewlink" data-pltdoc="x">Strings (Unicode)</a></td></tr><tr><td align="right">3.5 </td><td><a href="bytestrings.html" class="tocviewlink" data-pltdoc="x">Bytes and Byte Strings</a></td></tr><tr><td align="right">3.6 </td><td><a href="symbols.html" class="tocviewlink" data-pltdoc="x">Symbols</a></td></tr><tr><td align="right">3.7 </td><td><a href="keywords.html" class="tocviewlink" data-pltdoc="x">Keywords</a></td></tr><tr><td align="right">3.8 </td><td><a href="pairs.html" class="tocviewlink" data-pltdoc="x">Pairs and Lists</a></td></tr><tr><td align="right">3.9 </td><td><a href="vectors.html" class="tocviewlink" data-pltdoc="x">Vectors</a></td></tr><tr><td align="right">3.10 </td><td><a href="hash-tables.html" class="tocviewlink" data-pltdoc="x">Hash Tables</a></td></tr><tr><td align="right">3.11 </td><td><a href="boxes.html" class="tocviewlink" data-pltdoc="x">Boxes</a></td></tr><tr><td align="right">3.12 </td><td><a href="void_undefined.html" class="tocviewlink" data-pltdoc="x">Void and Undefined</a></td></tr></table></div></div></div></div><div class="maincolumn"><div class="main"><div class="navsettop"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.1", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.1");">top</a></span><span class="navright"> <a href="booleans.html" title="backward to "3.1 Booleans"" data-pltdoc="x">← prev</a> <a href="datatypes.html" title="up to "3 Built-In Datatypes"" data-pltdoc="x">up</a> <a href="characters.html" title="forward to "3.3 Characters"" data-pltdoc="x">next →</a></span> </div><h4 x-source-module="(lib "scribblings/guide/guide.scrbl")" x-part-tag=""numbers"">3.2<tt> </tt><a name="(part._numbers)"></a>Numbers</h4><p>A Racket <a name="(tech._number)"></a><span style="font-style: italic">number</span> is either exact or inexact:</p><ul><li><p>An <span style="font-style: italic">exact</span> number is either</p><ul><li><p>an arbitrarily large or small integer, such as <span class="RktVal">5</span>,
<span class="RktVal">99999999999999999</span>, or <span class="RktVal"><span class="nobreak">-1</span>7</span>;</p></li><li><p>a rational that is exactly the ratio of two arbitrarily
small or large integers, such as <span class="RktVal">1/2</span>,
<span class="RktVal">99999999999999999/2</span>, or <span class="RktVal"><span class="nobreak">-3</span>/4</span>; or</p></li><li><p>a complex number with exact real and imaginary parts
(where the imaginary part is not zero), such as <span class="RktVal">1+2i</span> or
<span class="RktVal">1/2+3/4i</span>.</p></li></ul></li><li><p>An <span style="font-style: italic">inexact</span> number is either</p><ul><li><p>an IEEE floating-point representation of a number, such
as <span class="RktVal">2.0</span> or <span class="RktVal">3.14e+87</span>, where the IEEE
infinities and not-a-number are written
<span class="RktVal">+inf.0</span>, <span class="RktVal"><span class="nobreak">-i</span>nf.0</span>, and <span class="RktVal">+nan.0</span>
(or <span class="RktVal">-nan.0</span>); or</p></li><li><p>a complex number with real and imaginary parts that are
IEEE floating-point representations, such as
<span class="RktVal">2.0+3.0i</span> or <span class="RktVal"><span class="nobreak">-i</span>nf.0+nan.0i</span>; as a
special case, an inexact complex number can have an
exact zero real part with an inexact imaginary part.</p></li></ul></li></ul><p>Inexact numbers print with a decimal point or exponent specifier, and
exact numbers print as integers and fractions. The same conventions
apply for reading number constants, but <span class="RktInBG"><span class="hspace"></span><span class="RktIn">#e</span><span class="hspace"></span></span> or
<span class="RktInBG"><span class="hspace"></span><span class="RktIn">#i</span><span class="hspace"></span></span> can prefix a number to force its parsing as an exact
or inexact number. The prefixes <span class="RktInBG"><span class="hspace"></span><span class="RktIn">#b</span><span class="hspace"></span></span>, <span class="RktInBG"><span class="hspace"></span><span class="RktIn">#o</span><span class="hspace"></span></span>, and
<span class="RktInBG"><span class="hspace"></span><span class="RktIn">#x</span><span class="hspace"></span></span> specify binary, octal, and hexadecimal
interpretation of digits.</p><blockquote class="refpara"><blockquote class="refcolumn"><blockquote class="refcontent"><p><span class="imageleft"><img src="magnify.png" alt="+" width="24" height="24"/></span><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=reader.html%23%2528part._parse-number%2529&version=6.1" class="Sq" data-pltdoc="x">Reading Numbers</a> in <a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=index.html&version=6.1" class="Sq" data-pltdoc="x"><span style="font-weight: bold">The Racket Reference</span></a> documents the fine points of the syntax of numbers.</p></blockquote></blockquote></blockquote><p><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><p>Examples:</p></td></tr><tr><td><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktVal">0.5</span></td></tr><tr><td><p><span class="RktRes">0.5</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktVal">#e0.5</span></td></tr><tr><td><p><span class="RktRes">1/2</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktVal">#x03BB</span></td></tr><tr><td><p><span class="RktRes">955</span></p></td></tr></table></blockquote></td></tr></table></p><p>Computations that involve an inexact number produce inexact results,
so that inexactness acts as a kind of taint on numbers. Beware,
however, that Racket offers no “inexact booleans,” so computations
that branch on the comparison of inexact numbers can nevertheless
produce exact results. The procedures <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exact-%7E3einexact%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">exact->inexact</a></span> and
<span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._inexact-%7E3eexact%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">inexact->exact</a></span> convert between the two
types of numbers.</p><p><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><p>Examples:</p></td></tr><tr><td><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">1/2</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">2.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.5</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528quote._%7E23%7E25kernel%2529._if%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">if</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="hspace"> </span><span class="RktVal">3.0</span><span class="hspace"> </span><span class="RktVal">2.999</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">2</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._inexact-%7E3eexact%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">inexact->exact</a></span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">3602879701896397/36028797018963968</span></p></td></tr></table></blockquote></td></tr></table></p><p>Inexact results are also produced by procedures such as <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._sqrt%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">sqrt</a></span>,
<span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">log</a></span>, and <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._sin%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">sin</a></span> when an exact result would require
representing real numbers that are not rational. Racket can represent
only rational numbers and complex numbers with rational parts.</p><p><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><p>Examples:</p></td></tr><tr><td><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._sin%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">sin</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">rational...</span></td></tr><tr><td><p><span class="RktRes">0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._sin%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">sin</a></span><span class="hspace"> </span><span class="RktVal">1/2</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">not rational...</span></td></tr><tr><td><p><span class="RktRes">0.479425538604203</span></p></td></tr></table></blockquote></td></tr></table></p><p>In terms of performance, computations with small integers are
typically the fastest, where “small” means that the number fits into
one bit less than the machine’s word-sized representation for signed
numbers. Computation with very large exact integers or with
non-integer exact numbers can be much more expensive than computation
with inexact numbers.</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">sigma</span><span class="hspace"> </span><span class="RktSym">f</span><span class="hspace"> </span><span class="RktSym">a</span><span class="hspace"> </span><span class="RktSym">b</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528quote._%7E23%7E25kernel%2529._if%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">if</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="hspace"> </span><span class="RktSym">a</span><span class="hspace"> </span><span class="RktSym">b</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktVal">0</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">f</span><span class="hspace"> </span><span class="RktSym">a</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">sigma</span><span class="hspace"> </span><span class="RktSym">f</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="hspace"> </span><span class="RktSym">a</span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym">b</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p> </p></td></tr><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=time.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fmore-scheme..rkt%2529._time%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">time</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._round%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">round</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">sigma</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._lambda%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">lambda</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktSym">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">2000</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktOut">cpu time: 484 real time: 123 gc time: 0</span></p></td></tr><tr><td><p><span class="RktRes">8</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=time.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fmore-scheme..rkt%2529._time%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">time</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._round%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">round</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">sigma</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._lambda%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">lambda</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="hspace"> </span><span class="RktSym">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">2000</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktOut">cpu time: 0 real time: 1 gc time: 0</span></p></td></tr><tr><td><p><span class="RktRes">8.0</span></p></td></tr></table></td></tr></table></blockquote><p>The number categories <a name="(tech._integer)"></a><span style="font-style: italic">integer</span>, <a name="(tech._rational)"></a><span style="font-style: italic">rational</span>,
<a name="(tech._real)"></a><span style="font-style: italic">real</span> (always rational), and <a name="(tech._complex)"></a><span style="font-style: italic">complex</span> are defined in
the usual way, and are recognized by the procedures <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._integer%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">integer?</a></span>,
<span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._rational%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">rational?</a></span>, <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._real%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">real?</a></span>, and <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._complex%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">complex?</a></span>, in addition
to the generic <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._number%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">number?</a></span>. A few mathematical procedures accept
only real numbers, but most implement standard extensions to complex
numbers.</p><p><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><p>Examples:</p></td></tr><tr><td><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._integer%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">integer?</a></span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._complex%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">complex?</a></span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._integer%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">integer?</a></span><span class="hspace"> </span><span class="RktVal">5.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._integer%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">integer?</a></span><span class="hspace"> </span><span class="RktVal">1+2i</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#f</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._complex%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">complex?</a></span><span class="hspace"> </span><span class="RktVal">1+2i</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._complex%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">complex?</a></span><span class="hspace"> </span><span class="RktVal">1.0+2.0i</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._abs%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">abs</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-5</span></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">5</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._abs%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">abs</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-5</span>+2i</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">abs: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: real?</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: -5+2i</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._sin%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">sin</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-5</span>+2i</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">3.6076607742131563+1.0288031496599335i</span></p></td></tr></table></blockquote></td></tr></table></p><p>The <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">=</a></span> procedure compares numbers for numerical equality. If
it is given both inexact and exact numbers to compare, it essentially
converts the inexact numbers to exact before comparing. The
<span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=booleans.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._eqv%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">eqv?</a></span> (and therefore <span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=booleans.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._equal%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">equal?</a></span>) procedure, in contrast,
compares numbers considering both exactness and numerical equality.</p><p><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><p>Examples:</p></td></tr><tr><td><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=booleans.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._eqv%7E3f%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">eqv?</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#f</span></p></td></tr></table></blockquote></td></tr></table></p><p>Beware of comparisons involving inexact numbers, which by their nature
can have surprising behavior. Even apparently simple inexact numbers
may not mean what you think they mean; for example, while a base-2
IEEE floating-point number can represent <span class="RktVal">1/2</span> exactly, it
can only approximate <span class="RktVal">1/10</span>:</p><p><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><p>Examples:</p></td></tr><tr><td><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="hspace"> </span><span class="RktVal">1/2</span><span class="hspace"> </span><span class="RktVal">0.5</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#t</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="hspace"> </span><span class="RktVal">1/10</span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">#f</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._inexact-%7E3eexact%2529%2529&version=6.1" class="RktValLink Sq" data-pltdoc="x">inexact->exact</a></span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">3602879701896397/36028797018963968</span></p></td></tr></table></blockquote></td></tr></table></p><blockquote class="refpara"><blockquote class="refcolumn"><blockquote class="refcontent"><p><span class="imageleft"><img src="magnify.png" alt="+" width="24" height="24"/></span><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=numbers.html&version=6.1" class="Sq" data-pltdoc="x">Numbers</a> in <a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=index.html&version=6.1" class="Sq" data-pltdoc="x"><span style="font-weight: bold">The Racket Reference</span></a> provides more on numbers and number procedures.</p></blockquote></blockquote></blockquote><div class="navsetbottom"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.1", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.1");">top</a></span><span class="navright"> <a href="booleans.html" title="backward to "3.1 Booleans"" data-pltdoc="x">← prev</a> <a href="datatypes.html" title="up to "3 Built-In Datatypes"" data-pltdoc="x">up</a> <a href="characters.html" title="forward to "3.3 Characters"" data-pltdoc="x">next →</a></span> </div></div></div><div id="contextindicator"> </div></body></html>
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