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# Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product. [1]

127 relations: Algebra, Alternating factorial, Analytic function, Ancient Society of College Youths, Arbitrary-precision arithmetic, Average, Bhargava factorial, Big O notation, Binary logarithm, Binary tree, Binomial coefficient, Binomial theorem, Bohr–Mollerup theorem, Calculator, Calculus, Carl Friedrich Gauss, Catalan number, Change ringing, Christian Kramp, Clifford A. Pickover, Combination, Combinatorics, Common logarithm, Comparison sort, Complete graph, Complex number, Composite number, Computational complexity theory, Computer algebra system, Concrete Mathematics, Continued fraction, Convergent series, Derangement, Derivative, Determinant, Digamma function, Dimension, Divide and conquer algorithm, Double exponential function, E (mathematical constant), Empty product, Empty set, Entire function, Enumerative combinatorics, Euclid's theorem, Euler–Mascheroni constant, Exponential factorial, Exponential function, Exponential growth, Exponentiation, ... Expand index (77 more) »

## Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Alternating factorial

In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Ancient Society of College Youths

The Ancient Society of College Youths (ASCY) is a change ringing society, founded in 1637 and based in the City of London.

## Arbitrary-precision arithmetic

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system.

## Average

In colloquial language, an average is a middle or typical number of a list of numbers.

## Bhargava factorial

In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his thesis in Harvard University in 1996.

## Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

## Binary logarithm

In mathematics, the binary logarithm is the power to which the number must be raised to obtain the value.

## Binary tree

In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the and the.

## Binomial coefficient

In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.

## Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

## Bohr–Mollerup theorem

In mathematical analysis, the Bohr–Mollerup theorem is a theorem named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it.

## Calculator

An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Catalan number

In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects.

## Change ringing

Change ringing is the art of ringing a set of tuned bells in a controlled manner to produce variations in their striking sequences.

## Christian Kramp

Christian Kramp (8 July 1760 – 13 May 1826) was a French mathematician, who worked primarily with factorials.

## Clifford A. Pickover

Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity and is employed at the IBM Thomas J. Watson Research Center in Yorktown, New York.

## Combination

In mathematics, a combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter.

## Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

## Common logarithm

In mathematics, the common logarithm is the logarithm with base 10.

## Comparison sort

A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator or a three-way comparison) that determines which of two elements should occur first in the final sorted list.

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## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Composite number

A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.

## Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

## Computer algebra system

A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

## Concrete Mathematics

Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatment of the analysis of algorithms.

## Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

## Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

## Derangement

In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Divide and conquer algorithm

In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion.

## Double exponential function

A double exponential function is a constant raised to the power of an exponential function.

## E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

## Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.

## Enumerative combinatorics

Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.

## Euclid's theorem

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.

## Euler–Mascheroni constant

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

## Exponential factorial

The exponential factorial of a positive integer n, denoted by n\$, is n raised to the power of n &minus; 1, which in turn is raised to the power of n &minus; 2, and so on and so forth, that is, The exponential factorial can also be defined with the recurrence relation The first few exponential factorials are 1, 1, 2, 9, 262144, etc.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Exponential growth

Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function is proportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent.

## Exponentiation

Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

## Fabian Stedman

Fabian Stedman (1640–1713) was a British author and a leading figure in the early history of campanology, particularly in the field of method ringing.

## Factorial number system

In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations.

## Factorial prime

A factorial prime is a prime number that is one less or one more than a factorial (all factorials > 1 are even).

## Factorion

A factorion is a natural number that equals the sum of the factorials of its decimal digits.

## Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n.

## Floating-point arithmetic

In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.

## Floor and ceiling functions

In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).

## Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

## Functional equation

In mathematics, a functional equation is any equation in which the unknown represents a function.

## Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## Generating primes

In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently.

## Glaisher–Kinkelin constant

In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function.

## GNU Multiple Precision Arithmetic Library

GNU Multiple Precision Arithmetic Library (GMP) is a free library for arbitrary-precision arithmetic, operating on signed integers, rational numbers, and floating point numbers.

## Googol

A googol is the large number 10100.

## Hadamard's gamma function

In mathematics, the Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the gamma function.

## Half-integer

In mathematics, a half-integer is a number of the form where n is an integer.

## Hypersphere

In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

## Irrationality sequence

In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series exists (that is, it converges) and is an irrational number.

## K-function

In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function.

## Knuth's up-arrow notation

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.

## Legendre's formula

In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Linear function

In mathematics, the term linear function refers to two distinct but related notions.

## List of factorial and binomial topics

This is a list of factorial and binomial topics in mathematics.

## List of integrals of trigonometric functions

The following is a list of integrals (antiderivative functions) of trigonometric functions.

## Logarithmically convex function

In mathematics, a function f defined on a convex subset of a real vector space and taking positive values is said to be logarithmically convex or superconvex if \circ f, the composition of the logarithmic function with f, is a convex function.

## Maple (software)

Maple is a symbolic and numeric computing environment, and is also a multi-paradigm programming language.

## Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas.

## Mathematical software

Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric data.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.

## Multiplication algorithm

A multiplication algorithm is an algorithm (or method) to multiply two numbers.

## Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x&minus;1, is a number which when multiplied by x yields the multiplicative identity, 1.

## N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

## Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Neil Sloane

Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Operator (mathematics)

In mathematics, an operator is generally a mapping that acts on the elements of a space to produce other elements of the same space.

## Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

## Personal computer

A personal computer (PC) is a multi-purpose computer whose size, capabilities, and price make it feasible for individual use.

## Peter Borwein

Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953) is a Canadian mathematician and a professor at Simon Fraser University.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Power rule

In calculus, the power rule is used to differentiate functions of the form f(x).

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Primorial

In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, only prime numbers are multiplied.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

## SageMath

SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.

## Schönhage–Strassen algorithm

The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers.

## Scientific notation

Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

## Shannon number

The Shannon number, named after Claude Shannon, is a conservative lower bound (not an estimate) of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for White followed by one for Black, and a typical game lasting about 40 such pairs of moves.

## Simon Plouffe

Simon Plouffe (born June 11, 1956, Saint-Jovite, Quebec) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the nth binary digit of &pi;, in 1995.

## Sorting algorithm

In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order.

## Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Srinivasa Ramanujan

Srinivasa Ramanujan (22 December 188726 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable.

## Stirling's approximation

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials.

## Symmetrization

In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.

## Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.

## Tetration

In mathematics, tetration (or hyper-4) is the next hyperoperation after exponentiation, and is defined as iterated exponentiation.

## Thomas Joannes Stieltjes

Thomas Joannes Stieltjes (29 December 1856 – 31 December 1894) was a Dutch mathematician.

## Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right.

## Vandermonde matrix

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m &times; n matrix 1 & \alpha_1 & \alpha_1^2 & \dots & \alpha_1^\\ 1 & \alpha_2 & \alpha_2^2 & \dots & \alpha_2^\\ 1 & \alpha_3 & \alpha_3^2 & \dots & \alpha_3^\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & \alpha_m & \alpha_m^2 & \dots & \alpha_m^ \end, or for all indices i and j. (Some authors use the transpose of the above matrix.) The determinant of a square Vandermonde matrix (where m.

## Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

## Wilson's theorem

In number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), one has that the factorial (n - 1)!.

## Wolfram Alpha

Wolfram Alpha (also styled WolframAlpha, and Wolfram|Alpha) is a computational knowledge engine or answer engine developed by Wolfram Alpha LLC, a subsidiary of Wolfram Research.

## Wolfram Mathematica

Wolfram Mathematica (usually termed Mathematica) is a modern technical computing system spanning most areas of technical computing — including neural networks, machine learning, image processing, geometry, data science, visualizations, and others.

## References

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