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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>Some <a href="/wiki/Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a> provide a built-in (primitive) <b>rational <a href="/wiki/Data_type" title="Data type">data type</a></b> to represent <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> like 1/3 and −11/17 without <a href="/wiki/Rounding" title="Rounding">rounding</a>, and to do arithmetic on them. Examples are the <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">ratio</code> type of <a href="/wiki/Common_Lisp" title="Common Lisp">Common Lisp</a>, and analogous types provided by most languages for <a href="/wiki/Algebraic_computation" class="mw-redirect" title="Algebraic computation">algebraic computation</a>, such as <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> and <a href="/wiki/Maple_software" class="mw-redirect" title="Maple software">Maple</a>. Many languages that do not have a built-in rational type still provide it as a <a href="/wiki/Library_(computer_science)" class="mw-redirect" title="Library (computer science)">library</a>-defined type. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Representation">Representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=1" title="Edit section: Representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A variable or value of that type is usually represented as a <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> <i>m</i>/<i>n</i> where <i>m</i> and <i>n</i> are two <a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">integer</a> numbers, either with a fixed or <a href="/wiki/Arbitrary_precision_arithmetic" class="mw-redirect" title="Arbitrary precision arithmetic">arbitrary precision</a>. Depending on the language, the <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a> <i>n</i> <a href="/wiki/Division_by_zero" title="Division by zero">may be constrained to be non-zero</a>, and the two numbers may be kept in <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">reduced</a> form (without any common <a href="/wiki/Divisor" title="Divisor">divisors</a> except 1). </p><p>Languages that support a rational data type usually provide special syntax for building such values, and also extend the basic arithmetic operations ('+', '−', '×', '/', integer <a href="/wiki/Power_(mathematics)" class="mw-redirect" title="Power (mathematics)">powers</a>) and comparisons ('=', '<', '>', '≤') to act on them — either natively or through <a href="/wiki/Operator_overloading" title="Operator overloading">operator overloading</a> facilities provided by the language. These operations may be translated by the <a href="/wiki/Compiler" title="Compiler">compiler</a> into a sequence of integer <a href="/wiki/Machine_instruction" class="mw-redirect" title="Machine instruction">machine instructions</a>, or into <a href="/wiki/Library_(computer_science)" class="mw-redirect" title="Library (computer science)">library</a> calls. Support may also extend to other operations, such as formatting, rounding to an integer or <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> value, etc.. As in mathematics, those languages often interpret an integer value as equivalent to a rational value with a unit denominator. </p> <div class="mw-heading mw-heading2"><h2 id="Language_support">Language support</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=2" title="Edit section: Language support"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Built-in or core library</b>: </p> <ul><li><a href="/wiki/C%2B%2B" title="C++">C++</a> has included support for compile-time rational arithmetic in the form of the contents of <a href="/wiki/C%2B%2B_Standard_Library" title="C++ Standard Library">its standard library</a>'s <a rel="nofollow" class="external text" href="http://en.cppreference.com/w/cpp/header/ratio"><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr"><ratio></code> header</a> since <a href="/wiki/C%2B%2B11" title="C++11">its 2011 revision</a>.</li> <li><a href="/wiki/Clojure" title="Clojure">Clojure</a> can perform arithmetic on rational numbers and offers a literal form to represent them.</li> <li><a href="/wiki/Go_(programming_language)" title="Go (programming language)">Go</a> provides rational numbers in the standard library, in the <a rel="nofollow" class="external text" href="https://golang.org/pkg/math/big/#Rat"><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">math/big</code> package</a>.</li> <li><a href="/wiki/J_(programming_language)" title="J (programming language)">J</a> provides rational numbers in the base language. For example, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">1r3</code> is one-third. Rationals in J use <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">arbitrary precision</a> integers for both the numerator and denominator, allowing arbitrary precision non-integers. For instance, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">12683021339465478347804472r7322545784478161858100577</code> represents the square root of three to 50 decimal digits.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a> provides rational numbers with the rational operator, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">//</code>. For example, <code class="mw-highlight mw-highlight-lang-julia mw-content-ltr" dir="ltr"><span class="mi">6</span><span class="o">//</span><span class="mi">9</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="o">//</span><span class="mi">3</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">typeof</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="o">//</span><span class="mi">9</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kt">Rational</span><span class="p">{</span><span class="kt">Int64</span><span class="p">}</span></code>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a> provides a <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Rational</code> type, which is really an alias for <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Ratio Integer</code> (<code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Ratio</code> being a polymorphic type implementing rational numbers for any <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Integral</code> type of numerators and denominators). The fraction is constructed using the % operator.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/OCaml" title="OCaml">OCaml</a>'s Num library implements arbitrary-precision rational numbers.</li> <li><a href="/wiki/Perl" title="Perl">Perl</a>: <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Math::BigRat</code> core module implements arbitrary-precision rational numbers. The <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">bigrat</code> pragma can be used to turn on transparent BigRat support.</li> <li><a href="/wiki/Raku_(programming_language)" title="Raku (programming language)">Raku</a>: use by default <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Rat</code><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> type (rational numbers with limited-precision). <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">FatRat</code><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> data type implements arbitrary-precision rational numbers.</li> <li><a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>: The standard library includes a <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Fraction</code> class in the module <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fractions</code>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>: native support using special syntax.</li> <li><a href="/wiki/Smalltalk" title="Smalltalk">Smalltalk</a> represents rational numbers using a <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Fraction</code> class in the form <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">p/q</code> where <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">p</code> and <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">q</code> are arbitrary size integers. Applying the arithmetic operations <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">*</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">+</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">-</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">/</code>, to fractions returns a <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">reduced fraction</a>.</li></ul> <p><b>With external libraries</b>: </p> <ul><li>For <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a> & <a href="/wiki/C%2B%2B" title="C++">C++</a> there are projects like the <a href="/wiki/GNU_Multiple_Precision_Arithmetic_Library" title="GNU Multiple Precision Arithmetic Library">GNU Multiple Precision Arithmetic Library</a>.</li> <li>The <a href="/wiki/Apache_Commons" title="Apache Commons">Apache Commons</a> Math library provides rational numbers for <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a> with its <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Fraction</code> class.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Common_Lisp">Common Lisp</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=3" title="Edit section: Common Lisp"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Common_Lisp" title="Common Lisp">Common Lisp</a> provides a numeric data type for arbitrarily sized rational numbers: <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">RATIO</span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-highlight mw-highlight-lang-lisp mw-content-ltr" dir="ltr"><pre><span></span><span class="w"> </span><span class="m">1/3</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="m">1/3</span> </pre></div> <p>The type of a rational number is <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">RATIO</code>: </p> <div class="mw-highlight mw-highlight-lang-lisp mw-content-ltr" dir="ltr"><pre><span></span><span class="w"> </span><span class="p">(</span><span class="nb">type-of</span><span class="w"> </span><span class="m">1/3</span><span class="p">)</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="nv">RATIO</span> </pre></div> <p>Dividing two integers may return a rational number and the multiplication of a rational number may return an integer number: </p> <div class="mw-highlight mw-highlight-lang-lisp mw-content-ltr" dir="ltr"><pre><span></span><span class="w"> </span><span class="p">(</span><span class="nb">/</span><span class="w"> </span><span class="mi">6</span><span class="w"> </span><span class="mi">8</span><span class="p">)</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="m">3/4</span> <span class="w"> </span><span class="p">(</span><span class="nb">*</span><span class="w"> </span><span class="m">3/4</span><span class="w"> </span><span class="mi">16</span><span class="p">)</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="mi">12</span> </pre></div> <p>The <a href="/wiki/Numerator" class="mw-redirect" title="Numerator">numerator</a> and <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a> may be obtained using the homonymous functions, that reduce a rational to canonical form and compute the numerator or denominator of that form respectively:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-highlight mw-highlight-lang-lisp mw-content-ltr" dir="ltr"><pre><span></span><span class="w"> </span><span class="p">(</span><span class="nb">numerator</span><span class="w"> </span><span class="m">12/16</span><span class="p">)</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="mi">3</span> <span class="w"> </span><span class="p">(</span><span class="nb">denominator</span><span class="w"> </span><span class="m">12/16</span><span class="p">)</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="mi">4</span> </pre></div> <p>Computing with large integers returning a large rational number: </p> <div class="mw-highlight mw-highlight-lang-lisp mw-content-ltr" dir="ltr"><pre><span></span><span class="w"> </span><span class="p">(</span><span class="nb">/</span><span class="w"> </span><span class="p">(</span><span class="nb">1-</span><span class="w"> </span><span class="p">(</span><span class="nb">expt</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="mi">200</span><span class="p">))</span><span class="w"> </span><span class="p">(</span><span class="nb">1-</span><span class="w"> </span><span class="p">(</span><span class="nb">expt</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="mi">43</span><span class="p">)))</span> <span class="w"> </span><span class="err">⇒</span><span class="w"> </span><span class="m">1606938044258990275541962092341162602522202993782792835301375/8796093022207</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Clojure">Clojure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=4" title="Edit section: Clojure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>(print (+ 1/10 2/10)) ⇒ 3/10 </pre> <div class="mw-heading mw-heading3"><h3 id="Julia">Julia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=5" title="Edit section: Julia"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-highlight mw-highlight-lang-jlcon mw-content-ltr" dir="ltr"><pre><span></span><span class="gp">julia></span><span class="w"> </span><span class="mi">1</span><span class="o">//</span><span class="mi">10</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="o">//</span><span class="mi">10</span> <span class="go">3//10</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Haskell">Haskell</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=6" title="Edit section: Haskell"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In module Data.Ratio </p> <pre>(1 % 10) + (2 % 10) ⇒ 3 % 10 </pre> <div class="mw-heading mw-heading3"><h3 id="Racket_(PLT_Scheme)"><span id="Racket_.28PLT_Scheme.29"></span>Racket (PLT Scheme)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=7" title="Edit section: Racket (PLT Scheme)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-highlight mw-highlight-lang-racket mw-content-ltr" dir="ltr"><pre><span></span><span class="nb">></span><span class="w"> </span><span class="p">(</span><span class="nb">+</span><span class="w"> </span><span class="m">1/10</span><span class="w"> </span><span class="m">2/10</span><span class="p">)</span> <span class="m">3/10</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Raku">Raku</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=8" title="Edit section: Raku"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Raku_(programming_language)" title="Raku (programming language)">Raku</a> provides <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Rat</code> type by default. </p> <div class="mw-highlight mw-highlight-lang-pl6 mw-content-ltr" dir="ltr"><pre><span></span><span class="k">my</span> <span class="nv">$v</span> = <span class="mf">0.2</span>; <span class="nb">say</span> <span class="s">"{$v} is {$v.^name} and has numerator {$v.numerator} and denominator {$v.denominator}"</span>; <span class="c1"># ⇒ 0.2 is Rat and has numerator 1 and denominator 5</span> </pre></div> <div class="mw-highlight mw-highlight-lang-pl6 mw-content-ltr" dir="ltr"><pre><span></span><span class="nb">say</span> <span class="mf">0.1</span> + <span class="mf">0.2</span> <span class="c1"># ⇒ 0.3</span> </pre></div> <div class="mw-highlight mw-highlight-lang-pl6 mw-content-ltr" dir="ltr"><pre><span></span><span class="nb">say</span> (<span class="mf">0.1</span> + <span class="mf">0.2</span> - <span class="mf">0.3</span>).<span class="nb">fmt</span>(<span class="s">"%.17f"</span>) <span class="c1"># ⇒ 0.00000000000000000</span> </pre></div> <div class="mw-highlight mw-highlight-lang-pl6 mw-content-ltr" dir="ltr"><pre><span></span><span class="nb">say</span> <span class="mi">1</span> / (<span class="mf">0.1</span> + <span class="mf">0.2</span> - <span class="mf">0.3</span>) <span class="c1"># ⇒ Attempt to divide by zero when coercing Rational to Str</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Ruby">Ruby</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=9" title="Edit section: Ruby"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using special syntax in 2.1 or newer: </p> <div class="mw-highlight mw-highlight-lang-irb mw-content-ltr" dir="ltr"><pre><span></span><span class="gp">irb(main):001:0> </span><span class="nb">puts</span><span class="w"> </span><span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="n">r</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="o">/</span><span class="mi">10</span><span class="n">r</span> <span class="go">3/10</span> <span class="go">=> nil</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_data_type&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.jsoftware.com/jwiki/Vocabulary/NumericPrecisions">"Vocabulary/NumericPrecisions - J Wiki"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Vocabulary%2FNumericPrecisions+-+J+Wiki&rft_id=http%3A%2F%2Fwww.jsoftware.com%2Fjwiki%2FVocabulary%2FNumericPrecisions&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+data+type" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120715070938/http://docs.julialang.org/en/latest/manual/complex-and-rational-numbers/">"Complex and Rational Numbers — Julia Language development documentation"</a>. <i>docs.julialang.org</i>. Archived from <a rel="nofollow" class="external text" href="http://docs.julialang.org/en/latest/manual/complex-and-rational-numbers/#rational-numbers">the original</a> on 2012-07-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.julialang.org&rft.atitle=Complex+and+Rational+Numbers+%E2%80%94+Julia+Language+development+documentation&rft_id=http%3A%2F%2Fdocs.julialang.org%2Fen%2Flatest%2Fmanual%2Fcomplex-and-rational-numbers%2F%23rational-numbers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+data+type" class="Z3988"></span> </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.haskell.org/onlinereport/ratio.html">"The Haskell 98 Library Report: Rational Numbers"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Haskell+98+Library+Report%3A+Rational+Numbers&rft_id=https%3A%2F%2Fwww.haskell.org%2Fonlinereport%2Fratio.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+data+type" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.raku.org/type/Rat">"Class Rat"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Class+Rat&rft_id=https%3A%2F%2Fdocs.raku.org%2Ftype%2FRat&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+data+type" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.raku.org/type/FatRat">"Class FatRat"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Class+FatRat&rft_id=https%3A%2F%2Fdocs.raku.org%2Ftype%2FFatRat&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+data+type" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.python.org/3/library/fractions.html">"Fractions — Rational numbers — Python 3.10.0 documentation"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fractions+%E2%80%94+Rational+numbers+%E2%80%94+Python+3.10.0+documentation&rft_id=https%3A%2F%2Fdocs.python.org%2F3%2Flibrary%2Ffractions.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+data+type" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.lispworks.com/documentation/lw50/CLHS/Body/t_ratio.htm#ratio">Common Lisp HyperSpec: RATIO</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.lispworks.com/documentation/lw50/CLHS/Body/f_numera.htm">Function NUMERATOR, DENOMINATOR</a> at the <a href="/wiki/Common_Lisp_HyperSpec" title="Common Lisp HyperSpec">Common Lisp HyperSpec</a></span> </li> </ol></div> <ul><li><a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>, <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>, vol. 2. 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Data_types" title="Template:Data types"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Data_types" title="Template talk:Data types"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Data_types" title="Special:EditPage/Template:Data types"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_types" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_type" title="Data type">Data types</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Units_of_information" title="Units of information">Uninterpreted</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit" title="Bit">Bit</a></li> <li><a href="/wiki/Byte" title="Byte">Byte</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">Trit</a></li> <li><a href="/wiki/Ternary_numeral_system#Tryte" title="Ternary numeral system">Tryte</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li> <li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary-precision or bignum</a></li> <li><a href="/wiki/Complex_data_type" title="Complex data type">Complex</a></li> <li><a href="/wiki/Decimal_data_type" title="Decimal data type">Decimal</a></li> <li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed point</a></li> <li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating point</a> <ul><li>Reduced precision <ul><li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half precision</a></li> <li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16</a></li></ul></li> <li><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single precision</a></li> <li><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double precision</a></li> <li><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple precision</a></li> <li><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple precision</a></li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a> <ul><li><a href="/wiki/Long_double" title="Long double">Long double</a></li></ul></li></ul></li> <li><a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer</a> <ul><li><a href="/wiki/Signedness" title="Signedness">signedness</a></li></ul></li> <li><a href="/wiki/Interval_arithmetic#Implementations" title="Interval arithmetic">Interval</a></li> <li><a class="mw-selflink selflink">Rational</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">Pointer</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Memory_address" title="Memory address">Address</a> <ul><li><a href="/wiki/Physical_address" title="Physical address">physical</a></li> <li><a href="/wiki/Virtual_address_space" title="Virtual address space">virtual</a></li></ul></li> <li><a href="/wiki/Reference_(computer_science)" title="Reference (computer science)">Reference</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plain_text" title="Plain text">Text</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Character_(computing)" title="Character (computing)">Character</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a> <ul><li><a href="/wiki/Null-terminated_string" title="Null-terminated string">null-terminated</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_data_type" title="Composite data type">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_data_type" title="Algebraic data type">Algebraic data type</a> <ul><li><a href="/wiki/Generalized_algebraic_data_type" title="Generalized algebraic data type">generalized</a></li></ul></li> <li><a href="/wiki/Array_data_type" class="mw-redirect" title="Array data type">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">Class</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent</a></li> <li><a href="/wiki/Intuitionistic_type_theory#Equality_type" title="Intuitionistic type theory">Equality</a></li> <li><a href="/wiki/Inductive_type" title="Inductive type">Inductive</a></li> <li><a href="/wiki/Intersection_type" title="Intersection type">Intersection</a></li> <li><a href="/wiki/List_(abstract_data_type)" title="List (abstract data type)">List</a></li> <li><a href="/wiki/Object_(computer_science)" title="Object (computer science)">Object</a> <ul><li><a href="/wiki/Metaobject" title="Metaobject">metaobject</a></li></ul></li> <li><a href="/wiki/Option_type" title="Option type">Option type</a></li> <li><a href="/wiki/Product_type" title="Product type">Product</a></li> <li><a href="/wiki/Record_(computer_science)" title="Record (computer science)">Record or Struct</a></li> <li><a href="/wiki/Refinement_type" title="Refinement type">Refinement</a></li> <li><a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">Set</a></li> <li><a href="/wiki/Union_type" title="Union type">Union</a> <ul><li><a href="/wiki/Tagged_union" title="Tagged union">tagged</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean</a></li> <li><a href="/wiki/Bottom_type" title="Bottom type">Bottom type</a></li> <li><a href="/wiki/Container_(abstract_data_type)" title="Container (abstract data type)">Collection</a></li> <li><a href="/wiki/Enumerated_type" title="Enumerated type">Enumerated type</a></li> <li><a href="/wiki/Exception_handling" title="Exception handling">Exception</a></li> <li><a href="/wiki/Function_type" title="Function type">Function type</a></li> <li><a href="/wiki/Opaque_data_type" title="Opaque data type">Opaque data type</a></li> <li><a href="/wiki/Recursive_data_type" title="Recursive data type">Recursive data type</a></li> <li><a href="/wiki/Semaphore_(programming)" title="Semaphore (programming)">Semaphore</a></li> <li><a href="/wiki/Stream_(computing)" title="Stream (computing)">Stream</a></li> <li><a href="/wiki/Strongly_typed_identifier" title="Strongly typed identifier">Strongly typed identifier</a></li> <li><a href="/wiki/Top_type" title="Top type">Top type</a></li> <li><a href="/wiki/Type_class" title="Type class">Type class</a></li> <li><a href="/wiki/Empty_type" title="Empty type">Empty type</a></li> <li><a href="/wiki/Unit_type" title="Unit type">Unit type</a></li> <li><a href="/wiki/Void_type" title="Void type">Void</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_data_type" title="Abstract data type">Abstract data type</a></li> <li><a href="/wiki/Boxing_(computer_science)" class="mw-redirect" title="Boxing (computer science)">Boxing</a></li> <li><a href="/wiki/Data_structure" title="Data structure">Data structure</a></li> <li><a href="/wiki/Generic_programming" title="Generic programming">Generic</a></li> <li><a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">Kind</a> <ul><li><a href="/wiki/Metaclass" title="Metaclass">metaclass</a></li></ul></li> <li><a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">Parametric polymorphism</a></li> <li><a href="/wiki/Primitive_data_type" title="Primitive data type">Primitive data type</a></li> <li><a href="/wiki/Interface_(object-oriented_programming)" title="Interface (object-oriented programming)">Interface</a></li> <li><a href="/wiki/Subtyping" title="Subtyping">Subtyping</a></li> <li><a href="/wiki/Type_constructor" title="Type constructor">Type constructor</a></li> <li><a href="/wiki/Type_conversion" title="Type conversion">Type conversion</a></li> <li><a href="/wiki/Type_system" title="Type system">Type system</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type 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