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id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Arbitrary-precision arithmetic</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C3%BDpo%C4%8Dty_s_libovolnou_p%C5%99esnost%C3%AD" title="Výpočty s libovolnou přesností – Czech" lang="cs" hreflang="cs" data-title="Výpočty s libovolnou přesností" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Langzahlarithmetik" title="Langzahlarithmetik – German" lang="de" hreflang="de" data-title="Langzahlarithmetik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Precisi%C3%B3n_arbitraria" title="Precisión arbitraria – Spanish" lang="es" hreflang="es" data-title="Precisión arbitraria" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Arithm%C3%A9tique_multipr%C3%A9cision" title="Arithmétique multiprécision – French" lang="fr" hreflang="fr" data-title="Arithmétique multiprécision" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%BB%E6%84%8F%E7%B2%BE%E5%BA%A6%E6%BC%94%E7%AE%97" title="任意精度演算 – Japanese" lang="ja" hreflang="ja" data-title="任意精度演算" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Du%C5%BCe_liczby_ca%C5%82kowite" title="Duże liczby całkowite – Polish" lang="pl" hreflang="pl" data-title="Duże liczby całkowite" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Bignum" title="Bignum – Portuguese" lang="pt" hreflang="pt" data-title="Bignum" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%BB%D0%B8%D0%BD%D0%BD%D0%B0%D1%8F_%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Длинная арифметика – Russian" lang="ru" hreflang="ru" data-title="Длинная арифметика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bignum-aritmetik" title="Bignum-aritmetik – Swedish" lang="sv" hreflang="sv" data-title="Bignum-aritmetik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B3%D0%B0_%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Довга арифметика – Ukrainian" lang="uk" hreflang="uk" data-title="Довга арифметика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%AB%98%E7%B2%BE%E5%BA%A6%E8%AE%A1%E7%AE%97" title="高精度计算 – Chinese" lang="zh" hreflang="zh" data-title="高精度计算" data-language-autonym="中文" data-language-local-name="Chinese" 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a> <a href="/wiki/Computer_number_format" title="Computer number format">formats</a></th></tr><tr><th class="sidebar-heading"> <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a></th></tr><tr><td class="sidebar-content"> <ul><li>16-bit: <a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half</a> (binary16)</li> <li>32-bit: <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single</a> (binary32), <a href="/wiki/Decimal32_floating-point_format" title="Decimal32 floating-point format">decimal32</a></li> <li>64-bit: <a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double</a> (binary64), <a href="/wiki/Decimal64_floating-point_format" title="Decimal64 floating-point format">decimal64</a></li> <li>128-bit: <a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple</a> (binary128), <a href="/wiki/Decimal128_floating-point_format" title="Decimal128 floating-point format">decimal128</a></li> <li>256-bit: <a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple</a> (binary256)</li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Other</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16</a></li> <li><a href="/wiki/TensorFloat-32" title="TensorFloat-32">TensorFloat-32</a></li> <li><a href="/wiki/Microsoft_Binary_Format" title="Microsoft Binary Format">Microsoft Binary Format</a></li> <li><a href="/wiki/IBM_hexadecimal_floating-point" title="IBM hexadecimal floating-point">IBM floating-point architecture</a></li> <li><a href="/wiki/Power_Management_Bus#Linear11_Floating-Point_Format" title="Power Management Bus">PMBus Linear-11</a></li> <li><a href="/wiki/G.711" title="G.711">G.711 8-bit floats</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Alternatives</th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Arbitrary precision</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Tapered_floating_point" title="Tapered floating point">Tapered floating point</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Unum_(number_format)" title="Unum (number format)">Posit</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a 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:Floating-point" title="Template:Floating-point"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Floating-point" title="Template talk:Floating-point"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Floating-point" title="Special:EditPage/Template:Floating-point"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <b>arbitrary-precision arithmetic</b>, also called <b>bignum arithmetic</b>, <b>multiple-precision arithmetic</b>, or sometimes <b>infinite-precision arithmetic</b>, indicates that <a href="/wiki/Calculation" title="Calculation">calculations</a> are performed on numbers whose <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> of <a href="/wiki/Precision_(arithmetic)" class="mw-redirect" title="Precision (arithmetic)">precision</a> are potentially limited only by the available <a href="/wiki/Memory_(computers)" class="mw-redirect" title="Memory (computers)">memory</a> of the host system. This contrasts with the faster <a href="/wiki/Fixed-precision_arithmetic" class="mw-redirect" title="Fixed-precision arithmetic">fixed-precision arithmetic</a> found in most <a href="/wiki/Arithmetic_logic_unit" title="Arithmetic logic unit">arithmetic logic unit</a> (ALU) hardware, which typically offers between 8 and 64 <a href="/wiki/Bit" title="Bit">bits</a> of precision. </p><p>Several modern <a href="/wiki/Programming_language" title="Programming language">programming languages</a> have built-in support for bignums,<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><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><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><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> and others have libraries available for arbitrary-precision <a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">integer</a> and <a href="/wiki/Floating-point" class="mw-redirect" title="Floating-point">floating-point</a> math. Rather than storing values as a fixed number of bits related to the size of the <a href="/wiki/Processor_register" title="Processor register">processor register</a>, these implementations typically use variable-length <a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">arrays</a> of digits. </p><p>Arbitrary precision is used in applications where the speed of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> is not a limiting factor, or where <a href="/wiki/Floating_point_error_mitigation" class="mw-redirect" title="Floating point error mitigation">precise results</a> with very large numbers are required. It should not be confused with the <a href="/wiki/Symbolic_computation" class="mw-redirect" title="Symbolic computation">symbolic computation</a> provided by many <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>, which represent numbers by expressions such as <span class="texhtml"><i>π</i>·sin(2)</span>, and can thus <i>represent</i> any <a href="/wiki/Computable_number" title="Computable number">computable number</a> with infinite precision. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=1" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A common application is <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>, whose algorithms commonly employ arithmetic with integers having hundreds of digits.<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><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> Another is in situations where artificial limits and <a href="/wiki/Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">overflows</a> would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining optimal or near-optimal values for coefficients needed in formulae, for example the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {\frac {1}{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {\frac {1}{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2efb76ff22445ab918a112a0d4892d1c5d192f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\textstyle {\sqrt {\frac {1}{3}}}}"></span> that appears in <a href="/wiki/Gaussian_integration" class="mw-redirect" title="Gaussian integration">Gaussian integration</a>.<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><p>Arbitrary precision arithmetic is also used to compute fundamental <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constants</a> such as <a href="/wiki/Pi" title="Pi">π</a> to millions or more digits and to analyze the properties of the digit strings<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> or more generally to investigate the precise behaviour of functions such as the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> where certain questions are difficult to explore via analytical methods. Another example is in rendering <a href="/wiki/Fractal" title="Fractal">fractal</a> images with an extremely high magnification, such as those found in the <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a>. </p><p>Arbitrary-precision arithmetic can also be used to avoid <a href="/wiki/Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">overflow</a>, which is an inherent limitation of fixed-precision arithmetic. Similar to an automobile's <a href="/wiki/Odometer" title="Odometer">odometer</a> display which may change from 99999 to 00000, a fixed-precision integer may exhibit <i><a href="/wiki/Integer_overflow" title="Integer overflow">wraparound</a></i> if numbers grow too large to represent at the fixed level of precision. Some processors can instead deal with overflow by <i><a href="/wiki/Saturation_arithmetic" title="Saturation arithmetic">saturation</a>,</i> which means that if a result would be unrepresentable, it is replaced with the nearest representable value. (With 16-bit unsigned saturation, adding any positive amount to 65535 would yield 65535.) Some processors can generate an <a href="/wiki/Exception_handling" title="Exception handling">exception</a> if an arithmetic result exceeds the available precision. Where necessary, the exception can be caught and recovered from—for instance, the operation could be restarted in software using arbitrary-precision arithmetic. </p><p>In many cases, the task or the programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow. Such guarantees may be based on pragmatic limits: a school attendance program may have a task limit of 4,000 students. A programmer may design the computation so that intermediate results stay within specified precision boundaries. </p><p>Some programming languages such as <a href="/wiki/Lisp_(programming_language)" title="Lisp (programming language)">Lisp</a>, <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="/wiki/Perl" title="Perl">Perl</a>, <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>, <a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a> and <a href="/wiki/Raku_(programming_language)" title="Raku (programming language)">Raku</a> use, or have an option to use, arbitrary-precision numbers for <i>all</i> integer arithmetic. Although this reduces performance, it eliminates the possibility of incorrect results (or exceptions) due to simple overflow. It also makes it possible to guarantee that arithmetic results will be the same on all machines, regardless of any particular machine's <a href="/wiki/Word_(data_type)" class="mw-redirect" title="Word (data type)">word size</a>. The exclusive use of arbitrary-precision numbers in a programming language also simplifies the language, because <i>a number is a number</i> and there is no need for multiple types to represent different levels of precision. </p> <div class="mw-heading mw-heading2"><h2 id="Implementation_issues">Implementation issues</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=2" title="Edit section: Implementation issues"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Arbitrary-precision arithmetic is considerably slower than arithmetic using numbers that fit entirely within processor registers, since the latter are usually implemented in <a href="/wiki/Arithmetic_logic_unit" title="Arithmetic logic unit">hardware arithmetic</a> whereas the former must be implemented in software. Even if the <a href="/wiki/Computer" title="Computer">computer</a> lacks hardware for certain operations (such as integer division, or all floating-point operations) and software is provided instead, it will use number sizes closely related to the available hardware registers: one or two words only. There are exceptions, as certain <i><a href="/wiki/Variable_word_length_machine" class="mw-redirect" title="Variable word length machine">variable word length</a></i> machines of the 1950s and 1960s, notably the <a href="/wiki/IBM_1620" title="IBM 1620">IBM 1620</a>, <a href="/wiki/IBM_1401" title="IBM 1401">IBM 1401</a> and the <a href="/wiki/Honeywell_200" title="Honeywell 200">Honeywell 200</a> series, could manipulate numbers bound only by available storage, with an extra bit that delimited the value. </p><p>Numbers can be stored in a <a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">fixed-point</a> format, or in a <a href="/wiki/Floating-point" class="mw-redirect" title="Floating-point">floating-point</a> format as a <a href="/wiki/Significand" title="Significand">significand</a> multiplied by an arbitrary exponent. However, since division almost immediately introduces infinitely repeating sequences of digits (such as 4/7 in decimal, or 1/10 in binary), should this possibility arise then either the representation would be truncated at some satisfactory size or else rational numbers would be used: a large integer for the <a href="/wiki/Numerator" class="mw-redirect" title="Numerator">numerator</a> and for the <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a>. But even with the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> divided out, arithmetic with rational numbers can become unwieldy very quickly: 1/99 − 1/100 = 1/9900, and if 1/101 is then added, the result is 10001/999900. </p><p>The size of arbitrary-precision numbers is limited in practice by the total storage available, and computation time. </p><p>Numerous <a href="/wiki/Algorithms" class="mw-redirect" title="Algorithms">algorithms</a> have been developed to efficiently perform arithmetic operations on numbers stored with arbitrary precision. In particular, supposing that <span class="texhtml"><i>N</i></span> digits are employed, algorithms have been designed to minimize the asymptotic <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity</a> for large <span class="texhtml"><i>N</i></span>. </p><p>The simplest algorithms are for <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, where one simply adds or subtracts the digits in sequence, carrying as necessary, which yields an <span class="texhtml">O(<i>N</i>)</span> algorithm (see <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>). </p><p><a href="/wiki/Comparison_(computer_programming)" class="mw-redirect" title="Comparison (computer programming)">Comparison</a> is also very simple. Compare the high-order digits (or machine words) until a difference is found. Comparing the rest of the digits/words is not necessary. The worst case is <span class="texhtml"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }"></span>(<i>N</i>)</span>, but it may complete much faster with operands of similar magnitude. </p><p>For <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, the most straightforward algorithms used for multiplying numbers by hand (as taught in primary school) require <span class="texhtml"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }"></span>(<i>N</i><sup>2</sup>)</span> operations, but <a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">multiplication algorithms</a> that achieve <span class="texhtml">O(<i>N</i> log(<i>N</i>) log(log(<i>N</i>)))</span> complexity have been devised, such as the <a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen algorithm</a>, based on <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transforms</a>, and there are also algorithms with slightly worse complexity but with sometimes superior real-world performance for smaller <span class="texhtml"><i>N</i></span>. The <a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba</a> multiplication is such an algorithm. </p><p>For <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, see <a href="/wiki/Division_algorithm" title="Division algorithm">division algorithm</a>. </p><p>For a list of algorithms along with complexity estimates, see <a href="/wiki/Computational_complexity_of_mathematical_operations" title="Computational complexity of mathematical operations">computational complexity of mathematical operations</a>. </p><p>For examples in <a href="/wiki/X86" title="X86">x86</a> assembly, see <a href="#External_links">external links</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Pre-set_precision">Pre-set precision</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=3" title="Edit section: Pre-set precision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In some languages such as <a href="/wiki/REXX" class="mw-redirect" title="REXX">REXX</a> and <a href="/wiki/Object_REXX" title="Object REXX">ooRexx</a>, the precision of all calculations must be set before doing a calculation. Other languages, such as <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> and <a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>, extend the precision automatically to prevent overflow. </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=4" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The calculation of <a href="/wiki/Factorial" title="Factorial">factorials</a> can easily produce very large numbers. This is not a problem for their usage in many formulas (such as <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>) because they appear along with other terms, so that—given careful attention to the order of evaluation—intermediate calculation values are not troublesome. If approximate values of factorial numbers are desired, <a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a> gives good results using floating-point arithmetic. The largest representable value for a fixed-size integer variable may be exceeded even for relatively small arguments as shown in the table below. Even floating-point numbers are soon outranged, so it may help to recast the calculations in terms of the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of the number. </p><p>But if exact values for large factorials are desired, then special software is required, as in the <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a> that follows, which implements the classic algorithm to calculate 1, 1×2, 1×2×3, 1×2×3×4, etc. the successive factorial numbers. </p> <pre>constants: Limit = 1000 <i>% Sufficient digits.</i> Base = 10 <i>% The base of the simulated arithmetic.</i> FactorialLimit = 365 <i>% Target number to solve, 365!</i> tdigit: Array[0:9] of character = ["0","1","2","3","4","5","6","7","8","9"] variables: digit: Array[1:Limit] of 0..9 <i>% The big number.</i> carry, d: Integer <i>% Assistants during multiplication.</i> last: Integer <i>% Index into the big number's digits.</i> text: Array[1:Limit] of character <i>% Scratchpad for the output.</i> digit[*] := 0 <i>% Clear the whole array.</i> last := 1 <i>% The big number starts as a single-digit,</i> digit[1] := 1 <i>% its only digit is 1.</i> <b>for</b> n := 1 <b>to</b> FactorialLimit: <i>% Step through producing 1!, 2!, 3!, 4!, etc. </i> carry := 0 <i>% Start a multiply by n.</i> <b>for</b> i := 1 <b>to</b> last: <i>% Step along every digit.</i> d := digit[i] * n + carry <i>% Multiply a single digit.</i> digit[i] := d <b>mod</b> Base <i>% Keep the low-order digit of the result.</i> carry := d <b>div</b> Base <i>% Carry over to the next digit.</i> <b>while</b> carry > 0: <i>% Store the remaining carry in the big number.</i> <b>if</b> last >= Limit: error("overflow") last := last + 1 <i>% One more digit.</i> digit[last] := carry <b>mod</b> Base carry := carry <b>div</b> Base <i>% Strip the last digit off the carry.</i> text[*] := " " <i>% Now prepare the output.</i> <b>for</b> i := 1 <b>to</b> last: <i>% Translate from binary to text.</i> text[Limit - i + 1] := tdigit[digit[i]] <i>% Reversing the order.</i> <b>print</b> text[Limit - last + 1:Limit], " = ", n, "!" </pre> <p>With the example in view, a number of details can be discussed. The most important is the choice of the representation of the big number. In this case, only integer values are required for digits, so an array of fixed-width integers is adequate. It is convenient to have successive elements of the array represent higher powers of the base. </p><p>The second most important decision is in the choice of the base of arithmetic, here ten. There are many considerations. The scratchpad variable <span class="texhtml mvar" style="font-style:italic;">d</span> must be able to hold the result of a single-digit multiply <i>plus the carry</i> from the prior digit's multiply. In base ten, a sixteen-bit integer is certainly adequate as it allows up to 32767. However, this example cheats, in that the value of <span class="texhtml mvar" style="font-style:italic;">n</span> is not itself limited to a single digit. This has the consequence that the method will fail for <span class="texhtml"><i>n</i> > 3200</span> or so. In a more general implementation, <span class="texhtml mvar" style="font-style:italic;">n</span> would also use a multi-digit representation. A second consequence of the shortcut is that after the multi-digit multiply has been completed, the last value of <i>carry</i> may need to be carried into multiple higher-order digits, not just one. </p><p>There is also the issue of printing the result in base ten, for human consideration. Because the base is already ten, the result could be shown simply by printing the successive digits of array <i>digit</i>, but they would appear with the highest-order digit last (so that 123 would appear as "321"). The whole array could be printed in reverse order, but that would present the number with leading zeroes ("00000...000123") which may not be appreciated, so this implementation builds the representation in a space-padded text variable and then prints that. The first few results (with spacing every fifth digit and annotation added here) are: </p> <table class="wikitable" style="text-align: right; white-space: nowrap; line-height: 80%"> <tbody><tr> <th colspan="2" style="text-align: center">Factorial numbers </th> <th colspan="2" style="text-align: center">Reach of computer integers </th></tr> <tr> <td>1 =</td> <td>1! </td></tr> <tr> <td>2 =</td> <td>2! </td></tr> <tr> <td>6 =</td> <td>3! </td></tr> <tr> <td>24 =</td> <td>4! </td></tr> <tr> <td>120 =</td> <td>5! </td> <td>8-bit</td> <td style="text-align: left">255 </td></tr> <tr> <td>720 =</td> <td>6! </td></tr> <tr> <td>5040 =</td> <td>7! </td></tr> <tr> <td>40320 =</td> <td>8! </td> <td>16-bit</td> <td style="text-align: left">65535 </td></tr> <tr> <td>3 62880 =</td> <td>9! </td></tr> <tr> <td>36 28800 =</td> <td>10! </td></tr> <tr> <td>399 16800 =</td> <td>11! </td></tr> <tr> <td>4790 01600 =</td> <td>12! </td> <td>32-bit</td> <td style="text-align: left">42949 67295 </td></tr> <tr> <td>62270 20800 =</td> <td>13! </td></tr> <tr> <td>8 71782 91200 =</td> <td>14! </td></tr> <tr> <td>130 76743 68000 =</td> <td>15! </td></tr> <tr> <td>2092 27898 88000 =</td> <td>16! </td></tr> <tr> <td>35568 74280 96000 =</td> <td>17! </td></tr> <tr> <td>6 40237 37057 28000 =</td> <td>18! </td></tr> <tr> <td>121 64510 04088 32000 =</td> <td>19! </td></tr> <tr> <td>2432 90200 81766 40000 =</td> <td>20! </td> <td>64-bit</td> <td style="text-align: left">18446 74407 37095 51615 </td></tr> <tr> <td>51090 94217 17094 40000 =</td> <td>21! </td></tr> <tr> <td>11 24000 72777 76076 80000 =</td> <td>22! </td></tr> <tr> <td>258 52016 73888 49766 40000 =</td> <td>23! </td></tr> <tr> <td>6204 48401 73323 94393 60000 =</td> <td>24! </td></tr> <tr> <td>1 55112 10043 33098 59840 00000 =</td> <td>25! </td></tr> <tr> <td>40 32914 61126 60563 55840 00000 =</td> <td>26! </td></tr> <tr> <td>1088 88694 50418 35216 07680 00000 =</td> <td>27! </td></tr> <tr> <td>30488 83446 11713 86050 15040 00000 =</td> <td>28! </td></tr> <tr> <td>8 84176 19937 39701 95454 36160 00000 =</td> <td>29! </td></tr> <tr> <td>265 25285 98121 91058 63630 84800 00000 =</td> <td>30! </td></tr> <tr> <td>8222 83865 41779 22817 72556 28800 00000 =</td> <td>31! </td></tr> <tr> <td>2 63130 83693 36935 30167 21801 21600 00000 =</td> <td>32! </td></tr> <tr> <td>86 83317 61881 18864 95518 19440 12800 00000 =</td> <td>33! </td></tr> <tr> <td>2952 32799 03960 41408 47618 60964 35200 00000 =</td> <td>34! </td> <td>128-bit</td> <td style="text-align: left">3402 82366 92093 84634 63374 60743 17682 11455 </td></tr> <tr> <td>1 03331 47966 38614 49296 66651 33752 32000 00000 =</td> <td>35! </td></tr></tbody></table> <p>This implementation could make more effective use of the computer's built in arithmetic. A simple escalation would be to use base 100 (with corresponding changes to the translation process for output), or, with sufficiently wide computer variables (such as 32-bit integers) we could use larger bases, such as 10,000. Working in a power-of-2 base closer to the computer's built-in integer operations offers advantages, although conversion to a decimal base for output becomes more difficult. On typical modern computers, additions and multiplications take constant time independent of the values of the operands (so long as the operands fit in single machine words), so there are large gains in packing as much of a bignumber as possible into each element of the digit array. The computer may also offer facilities for splitting a product into a digit and carry without requiring the two operations of <i>mod</i> and <i>div</i> as in the example, and nearly all arithmetic units provide a <i><a href="/wiki/Carry_flag" title="Carry flag">carry flag</a></i> which can be exploited in multiple-precision addition and subtraction. This sort of detail is the grist of machine-code programmers, and a suitable assembly-language bignumber routine can run faster than the result of the compilation of a high-level language, which does not provide direct access to such facilities but instead maps the high-level statements to its model of the target machine using an optimizing compiler. </p><p>For a single-digit multiply the working variables must be able to hold the value (base−1)<sup>2</sup> + carry, where the maximum value of the carry is (base−1). Similarly, the variables used to index the digit array are themselves limited in width. A simple way to extend the indices would be to deal with the bignumber's digits in blocks of some convenient size so that the addressing would be via (block <i>i</i>, digit <i>j</i>) where <i>i</i> and <i>j</i> would be small integers, or, one could escalate to employing bignumber techniques for the indexing variables. Ultimately, machine storage capacity and execution time impose limits on the problem size. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=5" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>IBM's first business computer, the <a href="/wiki/IBM_702" title="IBM 702">IBM 702</a> (a <a href="/wiki/Vacuum-tube" class="mw-redirect" title="Vacuum-tube">vacuum-tube</a> machine) of the mid-1950s, implemented integer arithmetic <i>entirely in hardware</i> on digit strings of any length from 1 to 511 digits. The earliest widespread software implementation of arbitrary-precision arithmetic was probably that in <a href="/wiki/Maclisp" title="Maclisp">Maclisp</a>. Later, around 1980, the <a href="/wiki/Operating_system" title="Operating system">operating systems</a> <a href="/wiki/VAX/VMS" class="mw-redirect" title="VAX/VMS">VAX/VMS</a> and <a href="/wiki/VM/CMS" class="mw-redirect" title="VM/CMS">VM/CMS</a> offered bignum facilities as a collection of <a href="/wiki/Literal_string" class="mw-redirect" title="Literal string">string</a> <a href="/wiki/Subprogram" class="mw-redirect" title="Subprogram">functions</a> in the one case and in the languages <a href="/wiki/EXEC_2" title="EXEC 2">EXEC 2</a> and <a href="/wiki/REXX" class="mw-redirect" title="REXX">REXX</a> in the other. </p><p>An early widespread implementation was available via the <a href="/wiki/IBM_1620" title="IBM 1620">IBM 1620</a> of 1959–1970. The 1620 was a decimal-digit machine which used discrete transistors, yet it had hardware (that used <a href="/wiki/Lookup_table" title="Lookup table">lookup tables</a>) to perform integer arithmetic on digit strings of a length that could be from two to whatever memory was available. For floating-point arithmetic, the mantissa was restricted to a hundred digits or fewer, and the exponent was restricted to two digits only. The largest memory supplied offered 60 000 digits, however <a href="/wiki/Fortran" title="Fortran">Fortran</a> compilers for the 1620 settled on fixed sizes such as 10, though it could be specified on a control card if the default was not satisfactory. </p> <div class="mw-heading mw-heading2"><h2 id="Software_libraries">Software libraries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=6" title="Edit section: Software libraries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">List of arbitrary-precision arithmetic software</a></div> <p>Arbitrary-precision arithmetic in most computer software is implemented by calling an external <a href="/wiki/Library_(computer_science)" class="mw-redirect" title="Library (computer science)">library</a> that provides <a href="/wiki/Data_type" title="Data type">data types</a> and <a href="/wiki/Subroutine" class="mw-redirect" title="Subroutine">subroutines</a> to store numbers with the requested precision and to perform computations. </p><p>Different libraries have different ways of representing arbitrary-precision numbers, some libraries work only with integer numbers, others store <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> numbers in a variety of bases (decimal or binary powers). Rather than representing a number as single value, some store numbers as a numerator/denominator pair (<a href="/wiki/Rational_number" title="Rational number">rationals</a>) and some can fully represent <a href="/wiki/Computable_number" title="Computable number">computable numbers</a>, though only up to some storage limit. Fundamentally, <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a> cannot represent all <a href="/wiki/Real_number" title="Real number">real numbers</a>, as the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> exceeds the cardinality of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/F%C3%BCrer%27s_algorithm" class="mw-redirect" title="Fürer's algorithm">Fürer's algorithm</a></li> <li><a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba algorithm</a></li> <li><a href="/wiki/Mixed-precision_arithmetic" title="Mixed-precision arithmetic">Mixed-precision arithmetic</a></li> <li><a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen algorithm</a></li> <li><a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook multiplication</a></li> <li><a href="/wiki/Little_Endian_Base_128" class="mw-redirect" title="Little Endian Base 128">Little Endian Base 128</a></li></ul> <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=Arbitrary-precision_arithmetic&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <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 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.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 id="CITEREFdotnet-bot" class="citation web cs1">dotnet-bot. <a rel="nofollow" class="external text" href="https://docs.microsoft.com/en-us/dotnet/api/system.numerics.biginteger">"BigInteger Struct (System.Numerics)"</a>. <i>docs.microsoft.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-02-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.microsoft.com&rft.atitle=BigInteger+Struct+%28System.Numerics%29&rft.au=dotnet-bot&rft_id=https%3A%2F%2Fdocs.microsoft.com%2Fen-us%2Fdotnet%2Fapi%2Fsystem.numerics.biginteger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" 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://peps.python.org/pep-0237/">"PEP 237 -- Unifying Long Integers and Integers"</a>. <i>Python.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Python.org&rft.atitle=PEP+237+--+Unifying+Long+Integers+and+Integers&rft_id=https%3A%2F%2Fpeps.python.org%2Fpep-0237%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" 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://docs.oracle.com/javase/7/docs/api/java/math/BigInteger.html">"BigInteger (Java Platform SE 7 )"</a>. <i>docs.oracle.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-02-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.oracle.com&rft.atitle=BigInteger+%28Java+Platform+SE+7+%29&rft_id=https%3A%2F%2Fdocs.oracle.com%2Fjavase%2F7%2Fdocs%2Fapi%2Fjava%2Fmath%2FBigInteger.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" 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://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/BigInt">"BigInt - JavaScript | MDN"</a>. <i>developer.mozilla.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-02-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=developer.mozilla.org&rft.atitle=BigInt+-+JavaScript+%7C+MDN&rft_id=https%3A%2F%2Fdeveloper.mozilla.org%2Fen-US%2Fdocs%2FWeb%2FJavaScript%2FReference%2FGlobal_Objects%2FBigInt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" 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 id="CITEREFJacqui_Cheng2007" class="citation web cs1">Jacqui Cheng (May 23, 2007). <a rel="nofollow" class="external text" href="https://arstechnica.com/news.ars/post/20070523-researchers-307-digit-key-crack-endangers-1024-bit-rsa.html">"Researchers: 307-digit key crack endangers 1024-bit RSA"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Researchers%3A+307-digit+key+crack+endangers+1024-bit+RSA&rft.date=2007-05-23&rft.au=Jacqui+Cheng&rft_id=https%3A%2F%2Farstechnica.com%2Fnews.ars%2Fpost%2F20070523-researchers-307-digit-key-crack-endangers-1024-bit-rsa.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" 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://web.archive.org/web/20120401144624/http://www.rsa.com/rsalabs/node.asp?id=2218">"RSA Laboratories - 3.1.5 How large a key should be used in the RSA cryptosystem?"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.rsa.com/rsalabs/node.asp?id%3D2218">the original</a> on 2012-04-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-03-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=RSA+Laboratories+-+3.1.5+How+large+a+key+should+be+used+in+the+RSA+cryptosystem%3F&rft_id=http%3A%2F%2Fwww.rsa.com%2Frsalabs%2Fnode.asp%3Fid%253D2218&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" class="Z3988"></span> recommends important RSA keys be 2048 bits (roughly 600 digits).</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaurent_Fousse2006" class="citation report cs1 cs1-prop-foreign-lang-source">Laurent Fousse (2006). <a rel="nofollow" class="external text" href="https://tel.archives-ouvertes.fr/tel-00477243/en">Intégration numérique avec erreur bornée en précision arbitraire. Modélisation et simulation</a> (Report) (in French). Université Henri Poincaré - Nancy I.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Int%C3%A9gration+num%C3%A9rique+avec+erreur+born%C3%A9e+en+pr%C3%A9cision+arbitraire.+Mod%C3%A9lisation+et+simulation&rft.pub=Universit%C3%A9+Henri+Poincar%C3%A9+-+Nancy+I&rft.date=2006&rft.au=Laurent+Fousse&rft_id=https%3A%2F%2Ftel.archives-ouvertes.fr%2Ftel-00477243%2Fen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" class="Z3988"></span> </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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._K._Pathria1962" class="citation journal cs1"><a href="/wiki/Raj_Pathria" class="mw-redirect" title="Raj Pathria">R. K. Pathria</a> (1962). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0144443-7/">"A Statistical Study of the Randomness Among the First 10,000 Digits of Pi"</a>. <i>Mathematics of Computation</i>. <b>16</b> (78): <span class="nowrap">188–</span>197. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-1962-0144443-7">10.1090/s0025-5718-1962-0144443-7</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">2014-01-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=A+Statistical+Study+of+the+Randomness+Among+the+First+10%2C000+Digits+of+Pi&rft.volume=16&rft.issue=78&rft.pages=%3Cspan+class%3D%22nowrap%22%3E188-%3C%2Fspan%3E197&rft.date=1962&rft_id=info%3Adoi%2F10.1090%2Fs0025-5718-1962-0144443-7&rft.au=R.+K.+Pathria&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1962-16-078%2FS0025-5718-1962-0144443-7%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" class="Z3988"></span> A quote example from this article: "Such an extreme pattern is dangerous even if diluted by one of its neighbouring blocks"; this was the occurrence of the sequence 77 twenty-eight times in one block of a thousand digits.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth2008" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (2008). <i>Seminumerical Algorithms</i>. <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 2 (3rd ed.). Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89684-8" title="Special:BookSources/978-0-201-89684-8"><bdi>978-0-201-89684-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Seminumerical+Algorithms&rft.series=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=2008&rft.isbn=978-0-201-89684-8&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" class="Z3988"></span>, Section 4.3.1: The Classical Algorithms</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDerick_Wood1984" class="citation book cs1">Derick Wood (1984). <i>Paradigms and Programming with Pascal</i>. Computer Science Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-914894-45-5" title="Special:BookSources/0-914894-45-5"><bdi>0-914894-45-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Paradigms+and+Programming+with+Pascal&rft.pub=Computer+Science+Press&rft.date=1984&rft.isbn=0-914894-45-5&rft.au=Derick+Wood&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Crandall,_Carl_Pomerance2005" class="citation book cs1">Richard Crandall, Carl Pomerance (2005). <i>Prime Numbers</i>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387252827" title="Special:BookSources/9780387252827"><bdi>9780387252827</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Numbers&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=9780387252827&rft.au=Richard+Crandall%2C+Carl+Pomerance&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArbitrary-precision+arithmetic" class="Z3988"></span>, Chapter 9: Fast Algorithms for Large-Integer Arithmetic</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arbitrary-precision_arithmetic&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20101019002107/http://oopweb.com/Assembly/Documents/ArtOfAssembly/Volume/Chapter_9/CH09-3.html#HEADING3-1">Chapter 9.3 of <i>The Art of Assembly</i></a> by <a href="/wiki/Randall_Hyde" title="Randall Hyde">Randall Hyde</a> discusses multiprecision arithmetic, with examples in <a href="/wiki/X86" title="X86">x86</a>-assembly.</li> <li>Rosetta Code task <a rel="nofollow" class="external text" href="http://rosettacode.org/wiki/Arbitrary-precision_integers_%28included%29">Arbitrary-precision integers</a> Case studies in the style in which over 95 programming languages compute the value of 5**4**3**2 using arbitrary precision arithmetic.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">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 href="/wiki/Rational_data_type" title="Rational data type">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 theory</a></li> <li><a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">Variable</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Arbitrary-precision_arithmetic&oldid=1270255274">https://en.wikipedia.org/w/index.php?title=Arbitrary-precision_arithmetic&oldid=1270255274</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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