Arbitrary-precision arithmetic (also called bignum arithmetic, multiple precision arithmetic, or infinite-precision arithmetic) indicates that calculations are performed on numbers which digits of precision are limited only by the available 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 bits of precision.
Several modern programming languages 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 bits related to the size of the processor register, these implementations typically use variable-length arrays of digits.
lisp, smalltalk, rexx and haskell, supports arbitrary precision integers (also known as infinite precision integers or bignums). Other languages which do not support this concept as a top-level construct may have libraries available to represent very large numbers using arrays of smaller variables, such as java and c# biginteger class or perl "bigint" package.
These use as much of the computer's memory as is necessary to store the numbers; however, a computer has only a finite amount of storage, so they too can only represent a finite subset of the mathematical integers. These schemes support very large numbers, for example one kilobyte of memory could be used to store numbers up to 2466 decimal digits long.
Application
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. Another is in situations where artificial limits and 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.
Big ints can also be used to compute fundamental mathematical constants such as π to millions 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.
Arbitrary-precision arithmetic can also be used to avoid overflow, which is an inherent limitation of fixed-precision arithmetic. Some processors can instead deal with overflow by saturation, which means that if a result would be unrepresentable, it is replaced with the nearest representable value.
