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252 lines
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<title>Reals and Rationals (Guile Reference Manual)</title>
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<div class="subsubsection-level-extent" id="Reals-and-Rationals">
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<p>
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Next: <a href="Complex-Numbers.html" accesskey="n" rel="next">Complex Numbers</a>, Previous: <a href="Integers.html" accesskey="p" rel="prev">Integers</a>, Up: <a href="Numbers.html" accesskey="u" rel="up">Numerical data types</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<hr>
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<h4 class="subsubsection" id="Real-and-Rational-Numbers"><span>6.6.2.3 Real and Rational Numbers<a class="copiable-link" href="#Real-and-Rational-Numbers"> ¶</a></span></h4>
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<a class="index-entry-id" id="index-Real-numbers"></a>
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<a class="index-entry-id" id="index-Rational-numbers"></a>
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<a class="index-entry-id" id="index-real_003f-2"></a>
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<a class="index-entry-id" id="index-rational_003f-2"></a>
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<p>Mathematically, the real numbers are the set of numbers that describe
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all possible points along a continuous, infinite, one-dimensional line.
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The rational numbers are the set of all numbers that can be written as
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fractions <var class="var">p</var>/<var class="var">q</var>, where <var class="var">p</var> and <var class="var">q</var> are integers.
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All rational numbers are also real, but there are real numbers that
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are not rational, for example <em class="math">the square root of 2</em>, and
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<em class="math">pi</em>.
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</p>
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<p>Guile can represent both exact and inexact rational numbers, but it
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cannot represent precise finite irrational numbers. Exact rationals are
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represented by storing the numerator and denominator as two exact
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integers. Inexact rationals are stored as floating point numbers using
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the C type <code class="code">double</code>.
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</p>
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<p>Exact rationals are written as a fraction of integers. There must be
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no whitespace around the slash:
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</p>
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<div class="example lisp">
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<pre class="lisp-preformatted">1/2
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-22/7
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</pre></div>
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<p>Even though the actual encoding of inexact rationals is in binary, it
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may be helpful to think of it as a decimal number with a limited
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number of significant figures and a decimal point somewhere, since
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this corresponds to the standard notation for non-whole numbers. For
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example:
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</p>
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<div class="example lisp">
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<pre class="lisp-preformatted">0.34
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-0.00000142857931198
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-5648394822220000000000.0
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4.0
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</pre></div>
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<p>The limited precision of Guile’s encoding means that any finite “real”
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number in Guile can be written in a rational form, by multiplying and
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then dividing by sufficient powers of 10 (or in fact, 2). For example,
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‘<samp class="samp">-0.00000142857931198</samp>’ is the same as −142857931198 divided
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by 100000000000000000. In Guile’s current incarnation, therefore, the
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<code class="code">rational?</code> and <code class="code">real?</code> predicates are equivalent for finite
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numbers.
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</p>
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<p>Dividing by an exact zero leads to a error message, as one might expect.
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However, dividing by an inexact zero does not produce an error.
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Instead, the result of the division is either plus or minus infinity,
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depending on the sign of the divided number and the sign of the zero
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divisor (some platforms support signed zeroes ‘<samp class="samp">-0.0</samp>’ and
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‘<samp class="samp">+0.0</samp>’; ‘<samp class="samp">0.0</samp>’ is the same as ‘<samp class="samp">+0.0</samp>’).
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</p>
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<p>Dividing zero by an inexact zero yields a <abbr class="acronym">NaN</abbr> (‘not a number’)
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value, although they are actually considered numbers by Scheme.
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Attempts to compare a <abbr class="acronym">NaN</abbr> value with any number (including
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itself) using <code class="code">=</code>, <code class="code"><</code>, <code class="code">></code>, <code class="code"><=</code> or <code class="code">>=</code>
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always returns <code class="code">#f</code>. Although a <abbr class="acronym">NaN</abbr> value is not
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<code class="code">=</code> to itself, it is both <code class="code">eqv?</code> and <code class="code">equal?</code> to itself
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and other <abbr class="acronym">NaN</abbr> values. However, the preferred way to test for
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them is by using <code class="code">nan?</code>.
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</p>
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<p>The real <abbr class="acronym">NaN</abbr> values and infinities are written ‘<samp class="samp">+nan.0</samp>’,
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‘<samp class="samp">+inf.0</samp>’ and ‘<samp class="samp">-inf.0</samp>’. This syntax is also recognized by
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<code class="code">read</code> as an extension to the usual Scheme syntax. These special
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values are considered by Scheme to be inexact real numbers but not
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rational. Note that non-real complex numbers may also contain
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infinities or <abbr class="acronym">NaN</abbr> values in their real or imaginary parts. To
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test a real number to see if it is infinite, a <abbr class="acronym">NaN</abbr> value, or
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neither, use <code class="code">inf?</code>, <code class="code">nan?</code>, or <code class="code">finite?</code>, respectively.
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Every real number in Scheme belongs to precisely one of those three
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classes.
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</p>
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<p>On platforms that follow <abbr class="acronym">IEEE</abbr> 754 for their floating point
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arithmetic, the ‘<samp class="samp">+inf.0</samp>’, ‘<samp class="samp">-inf.0</samp>’, and ‘<samp class="samp">+nan.0</samp>’ values
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are implemented using the corresponding <abbr class="acronym">IEEE</abbr> 754 values.
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They behave in arithmetic operations like <abbr class="acronym">IEEE</abbr> 754 describes
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it, i.e., <code class="code">(= +nan.0 +nan.0)</code> ⇒ <code class="code">#f</code>.
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</p>
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<dl class="first-deffn">
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<dt class="deffn" id="index-real_003f"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">real?</strong> <var class="def-var-arguments">obj</var><a class="copiable-link" href="#index-real_003f"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005freal_005fp"><span class="category-def">C Function: </span><span><strong class="def-name">scm_real_p</strong> <var class="def-var-arguments">(obj)</var><a class="copiable-link" href="#index-scm_005freal_005fp"> ¶</a></span></dt>
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<dd><p>Return <code class="code">#t</code> if <var class="var">obj</var> is a real number, else <code class="code">#f</code>. Note
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that the sets of integer and rational values form subsets of the set
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of real numbers, so the predicate will also be fulfilled if <var class="var">obj</var>
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is an integer number or a rational number.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-rational_003f"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">rational?</strong> <var class="def-var-arguments">x</var><a class="copiable-link" href="#index-rational_003f"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005frational_005fp"><span class="category-def">C Function: </span><span><strong class="def-name">scm_rational_p</strong> <var class="def-var-arguments">(x)</var><a class="copiable-link" href="#index-scm_005frational_005fp"> ¶</a></span></dt>
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<dd><p>Return <code class="code">#t</code> if <var class="var">x</var> is a rational number, <code class="code">#f</code> otherwise.
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Note that the set of integer values forms a subset of the set of
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rational numbers, i.e. the predicate will also be fulfilled if
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<var class="var">x</var> is an integer number.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-rationalize"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">rationalize</strong> <var class="def-var-arguments">x eps</var><a class="copiable-link" href="#index-rationalize"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005frationalize"><span class="category-def">C Function: </span><span><strong class="def-name">scm_rationalize</strong> <var class="def-var-arguments">(x, eps)</var><a class="copiable-link" href="#index-scm_005frationalize"> ¶</a></span></dt>
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<dd><p>Returns the <em class="emph">simplest</em> rational number differing
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from <var class="var">x</var> by no more than <var class="var">eps</var>.
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</p>
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<p>As required by <abbr class="acronym">R5RS</abbr>, <code class="code">rationalize</code> only returns an
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exact result when both its arguments are exact. Thus, you might need
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to use <code class="code">inexact->exact</code> on the arguments.
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</p>
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<div class="example lisp">
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<pre class="lisp-preformatted">(rationalize (inexact->exact 1.2) 1/100)
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⇒ 6/5
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</pre></div>
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</dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-inf_003f"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">inf?</strong> <var class="def-var-arguments">x</var><a class="copiable-link" href="#index-inf_003f"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005finf_005fp"><span class="category-def">C Function: </span><span><strong class="def-name">scm_inf_p</strong> <var class="def-var-arguments">(x)</var><a class="copiable-link" href="#index-scm_005finf_005fp"> ¶</a></span></dt>
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<dd><p>Return <code class="code">#t</code> if the real number <var class="var">x</var> is ‘<samp class="samp">+inf.0</samp>’ or
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‘<samp class="samp">-inf.0</samp>’. Otherwise return <code class="code">#f</code>.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-nan_003f"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">nan?</strong> <var class="def-var-arguments">x</var><a class="copiable-link" href="#index-nan_003f"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005fnan_005fp"><span class="category-def">C Function: </span><span><strong class="def-name">scm_nan_p</strong> <var class="def-var-arguments">(x)</var><a class="copiable-link" href="#index-scm_005fnan_005fp"> ¶</a></span></dt>
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<dd><p>Return <code class="code">#t</code> if the real number <var class="var">x</var> is ‘<samp class="samp">+nan.0</samp>’, or
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<code class="code">#f</code> otherwise.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-finite_003f"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">finite?</strong> <var class="def-var-arguments">x</var><a class="copiable-link" href="#index-finite_003f"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005ffinite_005fp"><span class="category-def">C Function: </span><span><strong class="def-name">scm_finite_p</strong> <var class="def-var-arguments">(x)</var><a class="copiable-link" href="#index-scm_005ffinite_005fp"> ¶</a></span></dt>
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<dd><p>Return <code class="code">#t</code> if the real number <var class="var">x</var> is neither infinite nor a
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NaN, <code class="code">#f</code> otherwise.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-nan"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">nan</strong><a class="copiable-link" href="#index-nan"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005fnan"><span class="category-def">C Function: </span><span><strong class="def-name">scm_nan</strong> <var class="def-var-arguments">()</var><a class="copiable-link" href="#index-scm_005fnan"> ¶</a></span></dt>
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<dd><p>Return ‘<samp class="samp">+nan.0</samp>’, a <abbr class="acronym">NaN</abbr> value.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-inf"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">inf</strong><a class="copiable-link" href="#index-inf"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005finf"><span class="category-def">C Function: </span><span><strong class="def-name">scm_inf</strong> <var class="def-var-arguments">()</var><a class="copiable-link" href="#index-scm_005finf"> ¶</a></span></dt>
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<dd><p>Return ‘<samp class="samp">+inf.0</samp>’, positive infinity.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-numerator"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">numerator</strong> <var class="def-var-arguments">x</var><a class="copiable-link" href="#index-numerator"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005fnumerator"><span class="category-def">C Function: </span><span><strong class="def-name">scm_numerator</strong> <var class="def-var-arguments">(x)</var><a class="copiable-link" href="#index-scm_005fnumerator"> ¶</a></span></dt>
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<dd><p>Return the numerator of the rational number <var class="var">x</var>.
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</p></dd></dl>
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<dl class="first-deffn">
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<dt class="deffn" id="index-denominator"><span class="category-def">Scheme Procedure: </span><span><strong class="def-name">denominator</strong> <var class="def-var-arguments">x</var><a class="copiable-link" href="#index-denominator"> ¶</a></span></dt>
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<dt class="deffnx def-cmd-deffn" id="index-scm_005fdenominator"><span class="category-def">C Function: </span><span><strong class="def-name">scm_denominator</strong> <var class="def-var-arguments">(x)</var><a class="copiable-link" href="#index-scm_005fdenominator"> ¶</a></span></dt>
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<dd><p>Return the denominator of the rational number <var class="var">x</var>.
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</p></dd></dl>
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<dl class="first-deftypefn">
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<dt class="deftypefn" id="index-scm_005fis_005freal"><span class="category-def">C Function: </span><span><code class="def-type">int</code> <strong class="def-name">scm_is_real</strong> <code class="def-code-arguments">(SCM val)</code><a class="copiable-link" href="#index-scm_005fis_005freal"> ¶</a></span></dt>
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<dt class="deftypefnx def-cmd-deftypefn" id="index-scm_005fis_005frational"><span class="category-def">C Function: </span><span><code class="def-type">int</code> <strong class="def-name">scm_is_rational</strong> <code class="def-code-arguments">(SCM val)</code><a class="copiable-link" href="#index-scm_005fis_005frational"> ¶</a></span></dt>
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<dd><p>Equivalent to <code class="code">scm_is_true (scm_real_p (val))</code> and
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<code class="code">scm_is_true (scm_rational_p (val))</code>, respectively.
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</p></dd></dl>
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<dl class="first-deftypefn">
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<dt class="deftypefn" id="index-scm_005fto_005fdouble"><span class="category-def">C Function: </span><span><code class="def-type">double</code> <strong class="def-name">scm_to_double</strong> <code class="def-code-arguments">(SCM val)</code><a class="copiable-link" href="#index-scm_005fto_005fdouble"> ¶</a></span></dt>
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<dd><p>Returns the number closest to <var class="var">val</var> that is representable as a
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<code class="code">double</code>. Returns infinity for a <var class="var">val</var> that is too large in
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magnitude. The argument <var class="var">val</var> must be a real number.
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</p></dd></dl>
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<dl class="first-deftypefn">
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<dt class="deftypefn" id="index-scm_005ffrom_005fdouble"><span class="category-def">C Function: </span><span><code class="def-type">SCM</code> <strong class="def-name">scm_from_double</strong> <code class="def-code-arguments">(double val)</code><a class="copiable-link" href="#index-scm_005ffrom_005fdouble"> ¶</a></span></dt>
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<dd><p>Return the <code class="code">SCM</code> value that represents <var class="var">val</var>. The returned
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value is inexact according to the predicate <code class="code">inexact?</code>, but it
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will be exactly equal to <var class="var">val</var>.
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</p></dd></dl>
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</div>
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<hr>
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<p>
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Next: <a href="Complex-Numbers.html">Complex Numbers</a>, Previous: <a href="Integers.html">Integers</a>, Up: <a href="Numbers.html">Numerical data types</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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