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| In [[computer science]], '''arbitrary-precision arithmetic''', also called '''bignum arithmetic''', '''multiple precision arithmetic''', or sometimes '''infinite-precision arithmetic''', indicates that [[calculation]]s are performed on numbers which [[numerical digit|digits]] of [[precision (arithmetic)|precision]] are limited only by the available [[memory (computers)|memory]] of the host system. This contrasts with the faster fixed-precision arithmetic found in most [[arithmetic logic unit]] (ALU) hardware, which typically offers between 8 and 64 [[bit]]s of precision.
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| Several modern [[programming language]]s have built-in support for bignums, and others have libraries available for arbitrary-precision [[integer]] and [[floating-point]] math. Rather than store values as a fixed number of binary [[bit]]s related to the size of the [[processor register]], these implementations typically use variable-length [[array data structure|arrays]] of digits.
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| Arbitrary precision is used in applications where the speed of [[arithmetic]] is not a limiting factor, or where precise results with very large numbers are required. It should not be confused with the [[symbolic computation]] provided by many [[computer algebra system]]s, which represent numbers by expressions such as ''π''·sin(2), and can thus ''represent'' any [[computable number]] with infinite precision.
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| ==Applications==
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| A common application is [[public-key cryptography]] (such as that in every modern [[web browser]]), whose algorithms commonly employ arithmetic with integers having hundreds of digits.<ref>[http://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]</ref><ref>http://www.rsa.com/rsalabs/node.asp?id=2218 recommends important RSA keys be 2048 bits (roughly 600 digits).</ref> Another is in situations where artificial limits and [[arithmetic overflow|overflows]] would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining the optimum value for coefficients needed in formulae, for example the √⅓ that appears in [[Gaussian integration]].
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| Arbitrary precision arithmetic is also used to compute fundamental [[mathematical constant]]s such as [[pi|π]] to millions or more digits and to analyze the properties of the digit strings<ref>{{cite journal |author=R. K. Pathria |authorlink=Raj Pathria |title=A Statistical Study of the Randomness Among the First 10,000 Digits of Pi |year=1962 |journal=Mathematics of Computation |volume=16 |issue=78 |pages=188–197 |url=http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0144443-7/ |accessdate=2014-01-10}} 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.</ref> or more generally to investigate the precise behaviour of functions such as the [[Riemann zeta function]] where certain questions are difficult to explore via analytical methods. Another example is in rendering [[fractal]] images with an extremely high magnification, such as those found in the [[Mandelbrot set]].
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| Arbitrary-precision arithmetic can also be used to avoid [[arithmetic overflow|overflow]], which is an inherent limitation of fixed-precision arithmetic. Similar to a 5-digit [[odometer]]'s display which changes from 99999 to 00000, a fixed-precision integer may exhibit ''wraparound'' if numbers grow too large to represent at the fixed level of precision. Some processors can instead deal with overflow by ''[[saturation arithmetic|saturation]],'' 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 [[exception handling|exception]] 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.
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| In many cases, the programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow. However, as time passes and conditions change, the bounds of the guarantee can be exceeded. For example, implementations of the [[binary search]] method that employ the form (L + R)/2 may function incorrectly when the ''sum'' of L and R exceeds the machine word size, although the individual variables themselves remain valid.
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| Some programming languages such as [[Lisp (programming language)|Lisp]], [[Python (programming language)|Python]], [[Perl]], [[Haskell (programming language)|Haskell]] and [[Ruby (programming language)|Ruby]] use, or have an option to use, arbitrary-precision numbers for ''all'' 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 [[Word (data type)|word size]]. The exclusive use of arbitrary-precision numbers in a programming language also simplifies the language, because ''a number is a number'' and there is no need for multiple types to represent different levels of precision.
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| ==Implementation issues==
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| Arbitrary-precision arithmetic is considerably slower than arithmetic using numbers that fit entirely within processor registers, since the latter are usually implemented in [[Arithmetic logic unit|hardware arithmetic]] whereas the former must be implemented in software. Even if the [[computer]] 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 and definitely not N words. There are exceptions, as certain ''variable word length'' machines of the 1950s and 1960s, notably the [[IBM 1620]], [[IBM 1401]] and the Honeywell ''Liberator'' series, could manipulate numbers bound only by available storage, with an extra bit that delimited the value.
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| Numbers can be stored in a [[fixed-point arithmetic|fixed-point]] format, or in a [[floating-point]] format as a [[significand]] 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 [[numerator]] and for the [[denominator]]. But even with the [[greatest common divisor]] divided out, arithmetic with rational numbers can become unwieldy very swiftly: 1/99 – 1/100 = 1/9900, and if 1/101 is then added the result is 10001/999900.
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| Bounding the size of arbitrary-precision numbers is not only the total storage available, but the variables used by the software to index the digit strings. These are typically themselves limited in size.
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| Numerous [[algorithms]] have been developed to efficiently perform arithmetic operations on numbers stored with arbitrary precision. In particular, supposing that ''N'' digits are employed, algorithms have been designed to minimize the asymptotic [[Computational complexity theory|complexity]] for large ''N''.
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| The simplest algorithms are for [[addition]] and [[subtraction]], where one simply adds or subtracts the digits in sequence, carrying as necessary, which yields an ''O''(''N'') algorithm (see [[big O notation]]).
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| [[Comparison (computer programming)|Comparison]] 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 ''O''(''N''), but usually it will go much faster.
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| For [[multiplication]], the most straightforward algorithms used for multiplying numbers by hand (as taught in primary school) require O(''N''<sup>2</sup>) operations, but [[multiplication algorithm]]s that achieve O(''N'' log(''N'') log(log(''N''))) complexity have been devised, such as the [[Schönhage–Strassen algorithm]], based on [[fast Fourier transform]]s, and there are also algorithms with slightly worse complexity but with sometimes superior real-world performance for smaller ''N''. The [[Karatsuba algorithm|Karatsuba]] multiplication is such an algorithm.
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| For [[Division (mathematics)|division]], see: [[Division algorithm]].
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| For a list of algorithms along with complexity estimates, see: [[Computational complexity of mathematical operations]]
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| For examples in [[x86]]-assembly, see: [[#External links|External links]].
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| ==Pre-set precision== | |
| In some languages such as [[REXX]] the precision of all calculations must be set before doing a calculation. Other languages, such as [[Python (programming language)|Python]] and [[Ruby (programming language)|Ruby]] extend the precision automatically to prevent overflow.
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| ==Example==
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| The calculation of [[factorial]]s can easily produce very large numbers. This is not a problem for their usage in many formulae (such as [[Taylor series]]) 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, [[Stirling's approximation]] 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 [[logarithm]] of the number.
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| But if exact values for large factorials are desired, then special software is required, as in the pseudocode that follows, which implements the classic algorithm to calculate 1, 1×2, 1×2×3, 1×2×3×4, etc. the successive factorial numbers.
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| <code>
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| Constant Limit = 1000; %Sufficient digits.
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| Constant Base = 10; %The base of the simulated arithmetic.
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| Constant FactorialLimit = 365; %Target number to solve, 365!
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| Array digit[1:Limit] of integer; %The big number.
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| Integer carry,d; %Assistants during multiplication.
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| Integer last,i; %Indices to the big number's digits.
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| Array text[1:Limit] of character;%Scratchpad for the output.
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| Constant tdigit[0:9] of character = ["0","1","2","3","4","5","6","7","8","9"];
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| BEGIN
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| digit:=0; %Clear the whole array.
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| digit[1]:=1; %The big number starts with 1,
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| last:=1; %Its highest-order digit is number 1.
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| '''for''' n:=1 '''to''' FactorialLimit '''do''' %Step through producing 1!, 2!, 3!, 4!, etc.
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| carry:=0; %Start a multiply by n.
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| '''for''' i:=1 '''to''' last '''do''' %Step along every digit.
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| d:=digit[i]*n + carry; %The classic multiply.
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| digit[i]:=d '''mod''' Base; %The low-order digit of the result.
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| carry:=d '''div''' Base; %The carry to the next digit.
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| '''next''' i;
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| '''while''' carry > 0 %Store the carry in the big number.
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| '''if''' last >= Limit '''then''' croak("Overflow!"); %If possible!
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| last:=last + 1; %One more digit.
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| digit[last]:=carry '''mod''' Base; %Placed.
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| carry:=carry '''div''' Base; %The carry reduced.
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| '''Wend''' %With n > Base, maybe > 1 digit extra.
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| text:=" "; %Now prepare the output.
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| '''for''' i:=1 '''to''' last '''do''' %Translate from binary to text.
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| text[Limit - i + 1]:=tdigit[digit[i]]; %Reversing the order.
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| '''next''' i; %Arabic numerals put the low order last.
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| '''Print''' text," = ",n,"!"; %Print the result!
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| '''next''' n; %On to the next factorial up.
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| END;
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| </code>
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| 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.
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| The second most important decision is in the choice of the base of arithmetic, here ten. There are many considerations. The scratchpad variable ''d'' must be able to hold the result of a single-digit multiply ''plus the carry'' 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 ''n'' is not itself limited to a single digit. This has the consequence that the method will fail for ''n'' > 3200 or so. In a more general implementation, ''n'' 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 ''carry'' may need to be carried into multiple higher-order digits, not just one.
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| 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 ''digit'', 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 we decided to build the representation in a space-padded text variable and then print that. The first few results (with spacing every fifth digit and annotation added here) are:
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| Reach of computer integers.
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| 1 = 1!
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| 2 = 2!
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| 6 = 3!
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| 24 = 4!
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| 120 = 5! 8-bit 255
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| 720 = 6!
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| 5040 = 7!
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| 40320 = 8! 16-bit 65535
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| 3 62880 = 9!
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| 36 28800 = 10!
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| 399 16800 = 11!
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| 4790 01600 = 12! 32-bit 42949 67295
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| 62270 20800 = 13!
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| 8 71782 91200 = 14!
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| 130 76743 68000 = 15!
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| 2092 27898 88000 = 16!
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| 35568 74280 96000 = 17!
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| 6 40237 37057 28000 = 18!
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| 121 64510 04088 32000 = 19!
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| 2432 90200 81766 40000 = 20! 64-bit 18446 74407 37095 51615
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| 51090 94217 17094 40000 = 21!
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| 11 24000 72777 76076 80000 = 22!
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| 258 52016 73888 49766 40000 = 23!
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| 6204 48401 73323 94393 60000 = 24!
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| 1 55112 10043 33098 59840 00000 = 25!
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| 40 32914 61126 60563 55840 00000 = 26!
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| 1088 88694 50418 35216 07680 00000 = 27!
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| 30488 83446 11713 86050 15040 00000 = 28!
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| 8 84176 19937 39701 95454 36160 00000 = 29!
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| 265 25285 98121 91058 63630 84800 00000 = 30!
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| 8222 83865 41779 22817 72556 28800 00000 = 31!
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| 2 63130 83693 36935 30167 21801 21600 00000 = 32!
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| 86 83317 61881 18864 95518 19440 12800 00000 = 33!
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| 2952 32799 03960 41408 47618 60964 35200 00000 = 34! 128-bit 3402 82366 92093 84634 63374 60743 17682 11455
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| 1 03331 47966 38614 49296 66651 33752 32000 00000 = 35!
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| We could try to use the available arithmetic of the computer more efficiently. 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 ''mod'' and ''div'' as in the example, and nearly all arithmetic units provide a ''[[carry flag]]'' 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 much faster than the result of the compilation of a high-level language, which does not provide access to such facilities.
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| For a single-digit multiply the working variables must be able to hold the value (base-1)² + 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'', digit ''j'') where ''i'' and ''j'' 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.
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| ==History==
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| IBM's first business computer, the [[IBM 702]] (a [[vacuum tube]] machine) of the mid-1950s, implemented integer arithmetic ''entirely in hardware'' on digit strings of any length from one to 511 digits. The earliest widespread software implementation of arbitrary precision arithmetic was probably that in [[Maclisp]]. Later, around 1980, the [[operating system]]s [[VAX/VMS]] and [[VM/CMS]] offered bignum facilities as a collection of [[literal string|string]] [[subprogram|function]]s in the one case and in the languages [[EXEC 2]] and [[REXX]] in the other.
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| An early widespread implementation was available via the [[IBM 1620]] of 1959–1970. The 1620 was a decimal-digit machine which used discrete transistors, yet it had hardware (that used [[lookup table]]s) 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 sixty thousand digits, however [[Fortran]] compilers for the 1620 settled on fixed sizes such as ten, though it could be specified on a control card if the default was not satisfactory.
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| ==Arbitrary-precision software==
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| ===Stand-alone application software===
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| Software that supports arbitrary precision computations:
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| * [[bc programming language|bc]] an arbitrary-precision math program that comes standard on most [[Unix-like]] systems.
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| ** [[dc (computer program)|dc]]: the POSIX desk calculator
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| * [[Maxima (software)|Maxima]]: a [[computer algebra system]] which ''bignum'' integers are directly inherited from its implementation language [[Common Lisp]]. In addition, it supports arbitrary-precision floating-point numbers, ''bigfloats''.
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| * [[Maple (language)|Maple]], [[Mathematica]], and several other [[computer algebra]] software include arbitrary-precision arithmetic. Mathematica employs [[GNU Multi-Precision Library|GMP]] for approximate number computation.
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| * [[PARI/GP]], an [[open source]] computer algebra system that supports arbitrary precision.
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| * [[Sage (mathematics software)|Sage]], an open-source [[computer algebra system]]
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| * [[SymPy]], a CAS
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| * Symbolic Math toolbox ([[MATLAB]])
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| ===Libraries===
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| Arbitrary-precision arithmetic in most computer software is implemented by calling an external [[library (computer science)|library]] that provides [[data type]]s and [[subroutine]]s to store numbers with the requested precision and to perform computations.
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| Different libraries have different ways of representing arbitrary-precision numbers, some libraries work only with integer numbers, others store floating point 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 ([[rational number|Rationals]]) and some can fully represent [[computable number]]s, though only up to some storage limit. Fundamentally, Turing machines cannot represent all real numbers, as the [[cardinality]] of <math>\mathbb{R}</math> exceeds the cardinality of <math>\mathbb{Z}</math>.
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| The Talk page for this article has a more extensive list.
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| <!-- TABLE GUIDELINES:
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| In the "Number type" column just enter one (or more) of the following: Integer, Floats, Rational, Naturals, Reals, Complex floats, Complex rational. If known, the type may be further qualified with the radix of the internal representation – decimal or binary. Don't confuse Rationals with Reals. -->
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| {| class="sortable wikitable"
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| |-
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| !Package-library name
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| !Number type
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| !Language
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| !License
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| |-
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| |[[Boost (C++ libraries)]] Multiprecision Library
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| |Integers, [[rational number|rationals]] and floats
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| |[[C++ (programming language)|C++]] and backends using GMP/MPFR
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| |[[Boost Software License]]
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| |-
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| |TTMath
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| |Integers, [[rational number|rationals]], floats and [[complex number|complex]]
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| |[[C++]]
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| |[[permissive|BSD license]]
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| |-
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| |[[GNU Multi-Precision Library]] (and [[MPFR]])
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| |Integers, rationals and floats
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| |C and C++ with bindings
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| |[[GNU Lesser General Public License|LGPL]]
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| |-
| |
| |[http://www.tc.umn.edu/~ringx004/mapm-main.html MAPM]
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| |Integers, decimal and complex floats
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| |[[C (programming language)|C]] (bindings for [[C++]])
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| |[[Freeware]]
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| |-
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| |[[MPIR (mathematics software)]]
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| |Integers, rationals and floats
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| |C and C++ with bindings
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| |[[GNU Lesser General Public License|LGPL]]
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| |-
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| |[[libgcrypt]]
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| |Integers
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| |C
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| |LGPL
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| |-
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| |[[OpenSSL]]
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| |Integers
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| |C
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| |[[BSD licenses|BSD-type]]
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| |-
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| |[http://code.google.com/p/infint/ InfInt]
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| |Integers
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| |[[C++]]
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| |LGPL
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| |-
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| |}
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| | |
| ===Languages===
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| Programming languages that supports arbitrary precision computations, either built-in, or in the standard library of the language:
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| * [[Agda (programming language)|Agda]]: the built-in ''BigInt'' datatype implements arbitrary-precision arithmetic.
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| * [[Common Lisp]]: The ANSI Common Lisp standard supports arbitrary precision integer, ratio and complex numbers.
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| * [[C Sharp (programming language)|C#]]: [http://msdn.microsoft.com/en-us/library/system.numerics.biginteger%28v=vs.100%29.aspx System.Numerics.BigInteger], from [[.NET Framework]] 4.0
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| * [[ColdFusion]]: the built-in ''PrecisionEvaluate()'' function evaluates one or more string expressions, dynamically, from left to right, using BigDecimal precision arithmetic to calculate the values of arbitrary precision arithmetic expressions.
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| * [[D (programming language)|D]]: standard library module ''[http://dlang.org/phobos/std_bigint.html std.bigint]''
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| * [[Dart (programming language)|Dart]]: the built-in ''int'' datatype implements arbitrary-precision arithmetic.
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| * [[Erlang (programming language)|Erlang]]: the built-in ''Integer'' datatype implements arbitrary-precision arithmetic.
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| * [[Go (programming language)|Go]]: the standard library package ''[http://golang.org/pkg/math/big/ math/big]'' implements arbitrary-precision integers (''Int'' type) and rational numbers (''Rat'' type)
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| * [[Haskell (programming language)|Haskell]]: the built-in ''Integer'' datatype implements arbitrary-precision arithmetic and the standard ''Data.Ratio'' module implements rational numbers.
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| * [[Idris (programming language)|Idris]]: the built-in ''BigInt'' datatype implements arbitrary-precision arithmetic.
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| * [[ISLISP]]: The ISO/IEC 13816:1997(E) [[ISLISP]] standard supports arbitrary precision integer numbers.
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| * [[J (programming language)|J]]: built-in ''extended precision''
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| * [[Java (programming language)|Java]]: [http://java.sun.com/javase/6/docs/api/java/math/BigInteger.html class java.math.BigInteger] (integer), [http://java.sun.com/javase/6/docs/api/java/math/BigDecimal.html Class java.math.BigDecimal] (decimal)
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| * [[JavaScript]]: the [http://code.google.com/p/gwt-math/ gwt-math] library provides an interface to java.math.BigDecimal, and the [http://leemon.com/crypto/BigInt.html BigInt] library supports arbitrary-precision integers.
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| * [[Julia (programming language)|Julia]]: the built-in "[http://docs.julialang.org/en/release-0.2/stdlib/base/#bigfloats BigFloat]" and "BigInt" types provide arbitrary-precision floating point and integer arithmetic respectively.
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| * [[OCaml]]: The [http://caml.inria.fr/pub/docs/manual-ocaml/manual036.html Num] library supports arbitrary-precision integers and rationals.
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| * [[Perl]]: The [http://perldoc.perl.org/bignum.html bignum] and [http://perldoc.perl.org/bigrat.html bigrat] pragmas provide BigNum and BigRational support for Perl.
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| * [[PHP]]: The [http://www.php.net/manual/en/book.bc.php BC Math] module provides arbitrary precision mathemathics.
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| * [[Pike (programming language)|Pike]]: the built-in ''int'' type will silently change from machine-native integer to arbitrary precision as soon as the value exceeds the former's capacity.
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| * [[Python (programming language)|Python]]: the built-in ''int'' (3.x) / ''long'' (2.x) integer type is of arbitrary precision. The ''Decimal'' class in the standard library module ''decimal'' has user definable precision and limited mathematical operations (exponentiation, square root, etc. but no trigonometric functions). The ''Fraction'' class in the module ''fractions'' implements rational numbers. More extensive arbitrary precision floating point arithmetic is available with the third-party "mpmath" and "bigfloat" packages.
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| * [[Racket (programming language)|Racket]]: the built-in ''exact'' numbers are of arbitrary precision. Example: (expt 10 100) produces the expected (large) result. Exact numbers also include rationals, so (/ 3 4) produces 3/4.
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| * [[REXX]]: variants including Open Object Rexx and NetRexx
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| * [[Ruby (programming language)|Ruby]]: the built-in ''Bignum'' integer type is of arbitrary precision. The ''BigDecimal'' class in the standard library module ''bigdecimal'' has user definable precision.
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| * [[Scheme (programming language)|Scheme]]: R<sup>5</sup>RS encourages, and R<sup>6</sup>RS requires, that exact integers and exact rationals be of arbitrary precision.
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| * [[Scala (programming language)|Scala]]: [http://www.scala-lang.org/api/current/index.html#scala.math.BigInt Class BigInt] and [http://www.scala-lang.org/api/current/index.html#scala.math.BigDecimal Class BigDecimal].
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| * [[Seed7]]: [http://seed7.sourceforge.net/manual/types.htm#bigInteger bigInteger] and [http://seed7.sourceforge.net/manual/types.htm#bigRational bigRational].
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| * [[Smalltalk]]: variants including [[Squeak]], [[Smalltalk/X]], [[GNU Smalltalk]], [[Dolphin Smalltalk]], etc.
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| * [[Standard ML]]: The optional built-in [http://www.standardml.org/Basis/int-inf.html IntInf] structure implements the ''INTEGER'' signature and supports arbitrary-precision integers.
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| == See also ==
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| * [[Karatsuba algorithm]]
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| * [[Toom–Cook multiplication]]
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| * [[Schönhage–Strassen algorithm]]
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| * [[Fürer's algorithm]]
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| == References ==
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| {{reflist}}
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| * {{Citation |last=Knuth |first=Donald |authorlink=Donald Knuth |title=Seminumerical Algorithms |series=[[The Art of Computer Programming]] |volume=2 |year=2008 |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89684-2}}, Section 4.3.1: The Classical Algorithms
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| == External links ==
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| * [http://oopweb.com/Assembly/Documents/ArtOfAssembly/Volume/Chapter_9/CH09-3.html#HEADING3-1 Chapter 9.3 of ''The Art of Assembly''] by [[Randall Hyde]] discusses multiprecision arithmetic, with examples in [[x86]]-assembly.
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| * Rosetta Code task [http://rosettacode.org/wiki/Arbitrary-precision_integers_%28included%29 Arbitrary-precision integers] Case studies in the style in which over 47 programming languages compute the value of 5**4**3**2 using arbitrary precision arithmetic.
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| {{data types}}
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| [[Category:Computer arithmetic]]
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| [[Category:Computer arithmetic algorithms]]
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