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On a computer, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique that allows computer programs to perform calculations on integers and rational numbers with an arbitrary number of digits of precision, limited only by the available memory of the host system. It typically works by storing a number as a variable-length array of digits in some base, in contrast to most computer arithmetic which uses a fixed number of bits given by the size of the processor registers. Rational numbers can be stored as a pair of two integers for the numerator and denominator, in a fixed-point format with a fixed denominator, or in a floating point format as a significand multiplied by an arbitrary exponent. A computer is a machine for manipulating data according to a list of instructions known as a program. ...
A computer program or software program (usually abbreviated to a program) is a step-by-step list of instructions written for a particular computer architecture in a particular computer programming language. ...
A calculation is a deliberate process for transforming one or more inputs into one or more results. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics and computer science, a numerical digit is a symbol, e. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
In computer programming, an array, also known as a vector or list (for one-dimensional arrays) or a matrix (for two-dimensional arrays), is one of the simplest data structures. ...
In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ...
This article is about the unit of information; for other uses, see Bit (disambiguation). ...
In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to commonly used values—typically, the values being in the midst of a calculation at a given point in time. ...
In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...
In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...
In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after the decimal (binary or radix) point. ...
A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...
The significand (also coefficient or, more informally, mantissa) is the part of a floating-point number that contains its significant digits. ...
Perhaps the earliest widespread implementation of arbitrary precision arithmetic was in Maclisp. Later, the VAX/VMS operating system offered bignum facilities as a collection of string functions. Today, bignum libraries are available for most modern programming languages (see below). Almost all computer algebra systems implement arbitrary precision arithmetic. MacLisp is a dialect of the Lisp programming language. ...
VAX is a 32-bit computing architecture that supports an orthogonal instruction set (machine language) and virtual addressing (i. ...
OpenVMS (Open Virtual Memory System or just VMS) is the name of a high-end computer server operating system that runs on the VAX and Alpha family of computers developed by Digital Equipment Corporation of Maynard, Massachusetts (now owned by Hewlett-Packard), and more recently on Hewlett-Packard systems built...
An operating system is a special computer program that manages the relationship between application software, the wide variety of hardware that makes up a computer system, and the user of the system. ...
In various branches of mathematics and computer science, strings are sequences of various simple objects (symbols, tokens, characters, etc. ...
In computer science, a subroutine (function, procedure, or subprogram) is a sequence of code which performs a specific task, as part of a larger program, and is grouped as one, or more, statement blocks; such code is sometimes collected into software libraries. ...
A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...
Applications Arbitrary-precision arithmetic is usually much slower than arithmetic using numbers that fit entirely within processor registers, since the latter are usually implemented in hardware arithmetic whereas the former must be implemented in software. Consequently, arbitrary precision is only used in a limited range of applications that require extremely precise results or exact integer arithmetic with very large numbers. The arithmetic logic unit/arithmetic-logic unit (ALU) of a computers CPU is a part of the execution unit, a core component of all CPUs. ...
The most common application is encryption, whose algorithms commonly employ arithmetic with integers of hundreds or thousands of digits. In cryptography, encryption is the process of obscuring information to make it unreadable without special knowledge. ...
Arbitrary precision arithmetic is also used to compute fundamental mathematical constants such as pi to millions or more digits and to analyze their properties. A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...
The mathematical constant π ≈ 3. ...
Algorithms 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 complexity for large N. Flowcharts are often used to represent algorithms. ...
In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, or cost, of the computation required to solve a given problem. ...
The simplest algorithm is for addition, where one simply adds the digits in sequence, carrying as necessary, which yields an O(N) algorithm (see big O notation). 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ...
It has been suggested that Landau notation be merged into this article or section. ...
For multiplication, the most straightforward algorithms used for multiplying numbers by hand requires O(N2) operations, but O(Nlog(N)log(log(N))) multiplication algorithms have been devised (and also algorithms with slightly worse complexity but with sometimes superior real-world performance for moderate N). In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...
A multiplication algorithm is an algorithm (or method) to multiply two numbers. ...
Arbitrary-precision software Arbitrary-precision arithmetic in most computer software is implemented by calling an external library that provides datatypes and subroutines to store numbers with the requested precision and to perform computations. Illustration of an application which may use libvorbisfile. ...
In computer science, a datatype or data type (often simply a type) is a name or label for a set of values and some operations which one can perform on that set of values. ...
In computer science, a subroutine (function, procedure, or subprogram) is a sequence of code which performs a specific task, as part of a larger program, and is grouped as one or more statement blocks; such code is sometimes collected into software libraries. ...
Stand-alone application software that supports arbitrary precision computations. The GNU Multiple-Precision Library, also known as GMP, is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. ...
MPFR is a portable library written in C for arbitrary precision arithmetic on reliable floating-point numbers. ...
The C Programming Language, Brian Kernighan and Dennis Ritchie, the original edition that served for many years as an informal specification of the language The C programming language is a standardized imperative computer programming language developed in the early 1970s by Dennis Ritchie for use on the Unix operating system. ...
java. ...
Java is an object-oriented programming language developed by James Gosling and colleagues at Sun Microsystems in the early 1990s. ...
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, standardised by ANSI X3. ...
Application software is a loosely defined subclass of computer software that employs the capabilities of a computer directly to a task that the user wishes to perform. ...
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