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<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"/><meta name="viewport" content="width=device-width, initial-scale=0.8"/><title>4.3 Numbers</title><link rel="stylesheet" type="text/css" href="../scribble.css" title="default"/><link rel="stylesheet" type="text/css" href="extras.css" title="default"/><link rel="stylesheet" type="text/css" href="icons.css" title="default"/><link rel="stylesheet" type="text/css" href="../racket.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" src="../local-redirect/local-redirect.js"></script><script type="text/javascript" src="../local-redirect/local-user-redirect.js"></script><!--[if IE 6]><style type="text/css">.SIEHidden { overflow: hidden; }</style><![endif]--></head><body id="doc-racket-lang-org"><div class="tocset"><div class="tocview"><div class="tocviewlist tocviewlisttopspace"><div class="tocviewtitle"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_0");">►</a></td><td></td><td><a href="index.html" class="tocviewlink" data-pltdoc="x">The Racket Reference</a></td></tr></table></div><div class="tocviewsublisttop" style="display: none;" id="tocview_0"><table cellspacing="0" cellpadding="0"><tr><td align="right">1 </td><td><a href="model.html" class="tocviewlink" data-pltdoc="x">Language Model</a></td></tr><tr><td align="right">2 </td><td><a href="notation.html" class="tocviewlink" data-pltdoc="x">Notation for Documentation</a></td></tr><tr><td align="right">3 </td><td><a href="syntax.html" class="tocviewlink" data-pltdoc="x">Syntactic Forms</a></td></tr><tr><td align="right">4 </td><td><a href="data.html" class="tocviewselflink" data-pltdoc="x">Datatypes</a></td></tr><tr><td align="right">5 </td><td><a href="structures.html" class="tocviewlink" data-pltdoc="x">Structures</a></td></tr><tr><td align="right">6 </td><td><a href="mzlib_class.html" class="tocviewlink" data-pltdoc="x">Classes and Objects</a></td></tr><tr><td align="right">7 </td><td><a href="mzlib_unit.html" class="tocviewlink" data-pltdoc="x">Units</a></td></tr><tr><td align="right">8 </td><td><a href="contracts.html" class="tocviewlink" data-pltdoc="x">Contracts</a></td></tr><tr><td align="right">9 </td><td><a href="match.html" class="tocviewlink" data-pltdoc="x">Pattern Matching</a></td></tr><tr><td align="right">10 </td><td><a href="control.html" class="tocviewlink" data-pltdoc="x">Control Flow</a></td></tr><tr><td align="right">11 </td><td><a href="concurrency.html" class="tocviewlink" data-pltdoc="x">Concurrency and Parallelism</a></td></tr><tr><td align="right">12 </td><td><a href="Macros.html" class="tocviewlink" data-pltdoc="x">Macros</a></td></tr><tr><td align="right">13 </td><td><a href="input-and-output.html" class="tocviewlink" data-pltdoc="x">Input and Output</a></td></tr><tr><td align="right">14 </td><td><a href="security.html" class="tocviewlink" data-pltdoc="x">Reflection and Security</a></td></tr><tr><td align="right">15 </td><td><a href="os.html" class="tocviewlink" data-pltdoc="x">Operating System</a></td></tr><tr><td align="right">16 </td><td><a href="memory.html" class="tocviewlink" data-pltdoc="x">Memory Management</a></td></tr><tr><td align="right">17 </td><td><a href="unsafe.html" class="tocviewlink" data-pltdoc="x">Unsafe Operations</a></td></tr><tr><t
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<a name="(tech._real._number)"></a><span style="font-style: italic">real numbers</span>, and all of the real numbers that can be
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represented are also <a name="(tech._rational._number)"></a><span style="font-style: italic">rational numbers</span>, except for
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<a name="(idx._(gentag._111._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">+inf.0</span> (positive <a name="(idx._(gentag._112._(lib._scribblings/reference/reference..scrbl)))"></a>infinity),
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<a name="(idx._(gentag._113._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">+inf.f</span> (single-precision variant, when
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enabled via <span class="RktSym"><a href="Reading.html#%28def._%28%28quote._~23~25kernel%29._read-single-flonum%29%29" class="RktValLink" data-pltdoc="x">read-single-flonum</a></span>),
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<a name="(idx._(gentag._114._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal"><span class="nobreak">-i</span>nf.0</span> (negative infinity),
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<a name="(idx._(gentag._115._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">-inf.f</span> (single-precision variant, when
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enabled), <a name="(idx._(gentag._116._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">+nan.0</span> (<a name="(idx._(gentag._117._(lib._scribblings/reference/reference..scrbl)))"></a>not-a-number), and
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<a name="(idx._(gentag._118._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">+nan.f</span> (single-precision variant, when
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enabled). Among the rational numbers, some are <a name="(tech._integer)"></a><span style="font-style: italic">integers</span>,
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because <span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._round%29%29" class="RktValLink" data-pltdoc="x">round</a></span> applied to the number produces the same number.</p><blockquote class="refpara"><blockquote class="refcolumn"><blockquote class="refcontent"><p><span class="imageleft"><img src="magnify.png" alt="+" width="24" height="24"/></span>See <a href="reader.html#%28part._parse-number%29" data-pltdoc="x">Reading Numbers</a> for information on the
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syntax of number literals.</p></blockquote></blockquote></blockquote><p>Orthogonal to those categories, each number is also either an
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<a name="(tech._exact._number)"></a><span style="font-style: italic">exact number</span> or an <a name="(tech._inexact._number)"></a><span style="font-style: italic">inexact number</span>. Unless
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otherwise specified, computations that involve an inexact number
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produce inexact results. Certain operations on inexact numbers,
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however, produce an exact number, such as multiplying an inexact
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number with an exact <span class="RktVal">0</span>. Operations that mathematically produce
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irrational numbers for some rational arguments (e.g., <span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._sqrt%29%29" class="RktValLink" data-pltdoc="x">sqrt</a></span>) may
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produce inexact results even for exact arguments.</p><p>In the case of complex numbers, either the real and imaginary parts
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are both exact or inexact with the same precision, or the number has
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an exact zero real part and an inexact imaginary part; a complex
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number with an exact zero imaginary part is a real number.</p><p>Inexact real numbers are implemented as double-precision
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<a name="(idx._(gentag._119._(lib._scribblings/reference/reference..scrbl)))"></a>IEEE floating-point numbers, also known as
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<a name="(tech._flonum)"></a><span style="font-style: italic">flonums</span>, or as single-precision IEEE floating-point numbers,
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also known as <a name="(tech._single._flonum)"></a><span style="font-style: italic">single-flonums</span>. Single-flonums are
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supported only when <span class="RktPn">(</span><span class="RktSym"><a href="number-types.html#%28def._%28%28quote._~23~25kernel%29._single-flonum-available~3f%29%29" class="RktValLink" data-pltdoc="x">single-flonum-available?</a></span><span class="RktPn">)</span> reports
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<span class="RktVal">#t</span>. Although we write <span class="RktVal">+inf.f</span>,
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<span class="RktVal">-inf.f</span>, and <span class="RktVal">+nan.f</span> to mean
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single-flonums, those forms read as double-precision flonums by
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default, since <span class="RktSym"><a href="Reading.html#%28def._%28%28quote._~23~25kernel%29._read-single-flonum%29%29" class="RktValLink" data-pltdoc="x">read-single-flonum</a></span> is <span class="RktVal">#f</span> by default.
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When single-flonums are supported, inexact numbers are still
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represented as flonums by default, and single precision is used only
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when a computation starts with single-flonums.</p><p>Inexact numbers can be coerced to exact form, except for the inexact
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numbers <span class="RktVal">+inf.0</span>, <span class="RktVal">+inf.f</span>,
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<span class="RktVal"><span class="nobreak">-i</span>nf.0</span>, <span class="RktVal">-inf.f</span>, <span class="RktVal">+nan.0</span>, and <span class="RktVal">+nan.f</span>, which
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have no exact form. <a name="(idx._(gentag._120._(lib._scribblings/reference/reference..scrbl)))"></a>Dividing a
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number by exact zero raises an exception; dividing a non-zero number
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other than <span class="RktVal">+nan.0</span> or <span class="RktVal">+nan.f</span> by an inexact zero returns <span class="RktVal">+inf.0</span>,
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<span class="RktVal">+inf.f</span>, <span class="RktVal"><span class="nobreak">-i</span>nf.0</span>
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or <span class="RktVal">-inf.f</span>, depending on the sign and precision of the dividend. The
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<span class="RktVal">+nan.0</span> value is not <span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._~3d%29%29" class="RktValLink" data-pltdoc="x">=</a></span> to itself, but <span class="RktVal">+nan.0</span>
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is <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span> to itself, and <span class="RktVal">+nan.f</span> is similarly <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span> but
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not <span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._~3d%29%29" class="RktValLink" data-pltdoc="x">=</a></span> to itself. Conversely, <span class="RktPn">(</span><span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._~3d%29%29" class="RktValLink" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0.0</span><span class="stt"> </span><span class="RktVal"><span class="nobreak">-0</span>.0</span><span class="RktPn">)</span> is
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<span class="RktVal">#t</span>, but <span class="RktPn">(</span><span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span><span class="stt"> </span><span class="RktVal">0.0</span><span class="stt"> </span><span class="RktVal"><span class="nobreak">-0</span>.0</span><span class="RktPn">)</span> is <span class="RktVal">#f</span>, and the
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same for <span class="RktVal">0.0f0</span> and <span class="RktVal">-0.0f0</span> (which are single-precision variants). The datum
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<a name="(idx._(gentag._121._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">-nan.0</span> refers to the same constant as <span class="RktVal">+nan.0</span>,
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and <a name="(idx._(gentag._122._(lib._scribblings/reference/reference..scrbl)))"></a><span class="RktVal">-nan.f</span> is the same as <span class="RktVal">+nan.f</span>.</p><p>Calculations with infinities produce results consistent with IEEE
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double- or single-precision floating point where IEEE specifies the result; in
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cases where IEEE provides no specification,
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the result corresponds to the limit approaching
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infinity, or <span class="RktVal">+nan.0</span> or <span class="RktVal">+nan.f</span> if no such limit exists.</p><p>The precision and size of exact numbers is limited only by available
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memory (and the precision of operations that can produce irrational
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numbers). In particular, adding, multiplying, subtracting, and
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dividing exact numbers always produces an exact result.</p><p>A <a name="(tech._fixnum)"></a><span style="font-style: italic">fixnum</span> is an exact integer whose two’s complement
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representation fits into 30 or 31 bits (depending on the Racket variant)
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on a 32-bit platform or 61 or 63 bits (depending on the Racket variant) on a
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64-bit platform. No allocation is required when computing
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with fixnums. See also the <a href="fixnums.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">racket/fixnum</span></a> module, below.</p><p>Two fixnums that are <span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._~3d%29%29" class="RktValLink" data-pltdoc="x">=</a></span> are also the same
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according to <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eq~3f%29%29" class="RktValLink" data-pltdoc="x">eq?</a></span>. Otherwise, the result of <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eq~3f%29%29" class="RktValLink" data-pltdoc="x">eq?</a></span>
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applied to two numbers is undefined, except that numbers produced
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by the default reader in <span class="RktSym"><a href="Reading.html#%28def._%28%28quote._~23~25kernel%29._read-syntax%29%29" class="RktValLink" data-pltdoc="x">read-syntax</a></span> mode are <a href="reader.html#%28tech._interned%29" class="techoutside" data-pltdoc="x"><span class="techinside">interned</span></a> and therefore <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eq~3f%29%29" class="RktValLink" data-pltdoc="x">eq?</a></span>
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when they are <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span>.</p><p>Two real numbers are <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span> when they are both inexact with the same precision or both
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exact, and when they are <span class="RktSym"><a href="generic-numbers.html#%28def._%28%28quote._~23~25kernel%29._~3d%29%29" class="RktValLink" data-pltdoc="x">=</a></span> (except for <span class="RktVal">+nan.0</span>, <span class="RktVal">+nan.f</span>,
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<span class="RktVal">0.0</span>, <span class="RktVal">+0.0f0</span>, <span class="RktVal"><span class="nobreak">-0</span>.0</span>, and <span class="RktVal">-0.0f0</span>, as noted above).
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Two complex numbers are <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span> when their real and imaginary parts are <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span>.
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Two numbers are <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._equal~3f%29%29" class="RktValLink" data-pltdoc="x">equal?</a></span> when they are <span class="RktSym"><a href="Equality.html#%28def._%28%28quote._~23~25kernel%29._eqv~3f%29%29" class="RktValLink" data-pltdoc="x">eqv?</a></span>.</p><p>See <a href="reader.html#%28part._parse-number%29" data-pltdoc="x">Reading Numbers</a>
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for information on <span class="RktSym"><a href="Reading.html#%28def._%28%28quote._~23~25kernel%29._read%29%29" class="RktValLink" data-pltdoc="x">read</a></span>ing
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numbers and <a href="printing.html#%28part._print-number%29" data-pltdoc="x">Printing Numbers</a>
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for information on <span class="RktSym"><a href="Writing.html#%28def._%28%28quote._~23~25kernel%29._print%29%29" class="RktValLink" data-pltdoc="x">print</a></span>ing numbers.</p><table cellspacing="0" cellpadding="0"><tr><td><p><span class="hspace"> </span><a href="number-types.html" class="toclink" data-pltdoc="x">4.3.1<span class="hspace"> </span>Number Types</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html" class="toclink" data-pltdoc="x">4.3.2<span class="hspace"> </span>Generic Numerics</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Arithmetic%29" class="toclink" data-pltdoc="x">4.3.2.1<span class="hspace"> </span>Arithmetic</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Number_.Comparison%29" class="toclink" data-pltdoc="x">4.3.2.2<span class="hspace"> </span>Number Comparison</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Powers_and_.Roots%29" class="toclink" data-pltdoc="x">4.3.2.3<span class="hspace"> </span>Powers and Roots</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Trigonometric_.Functions%29" class="toclink" data-pltdoc="x">4.3.2.4<span class="hspace"> </span>Trigonometric Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Complex_.Numbers%29" class="toclink" data-pltdoc="x">4.3.2.5<span class="hspace"> </span>Complex Numbers</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Bitwise_.Operations%29" class="toclink" data-pltdoc="x">4.3.2.6<span class="hspace"> </span>Bitwise Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Random_.Numbers%29" class="toclink" data-pltdoc="x">4.3.2.7<span class="hspace"> </span>Random Numbers</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Other_.Randomness_.Utilities%29" class="toclink" data-pltdoc="x">4.3.2.8<span class="hspace"> </span>Other Randomness Utilities</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Number--.String_.Conversions%29" class="toclink" data-pltdoc="x">4.3.2.9<span class="hspace"> </span>Number–String Conversions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="generic-numbers.html#%28part._.Extra_.Constants_and_.Functions%29" class="toclink" data-pltdoc="x">4.3.2.10<span class="hspace"> </span>Extra Constants and Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonums.html" class="toclink" data-pltdoc="x">4.3.3<span class="hspace"> </span>Flonums</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonums.html#%28part._.Flonum_.Arithmetic%29" class="toclink" data-pltdoc="x">4.3.3.1<span class="hspace"> </span>Flonum Arithmetic</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonums.html#%28part._flvectors%29" class="toclink" data-pltdoc="x">4.3.3.2<span class="hspace"> </span>Flonum Vectors</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="fixnums.html" class="toclink" data-pltdoc="x">4.3.4<span class="hspace"> </span>Fixnums</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="fixnums.html#%28part._.Fixnum_.Arithmetic%29" class="t
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