faktoriell

Faktoriell is a term used in some linguistic contexts to denote the factorial function in mathematics. The factorial of a non-negative integer n, written as n!, is the product of all positive integers up to n. By convention, 0! equals 1. The concept is defined for integers n ≥ 0, and it can be extended to complex numbers through the gamma function, with the relation Γ(n+1) = n! for nonnegative integers n.

Factorials grow rapidly and have wide applicability in counting and analysis. For example, the number of ways to arrange n distinct items is n!, and factorials appear in binomial coefficients, probability distributions, and series expansions such as e^x = sum x^n / n!. Recurrence relations are also common, notably (n+1)! = (n+1)·n!. Variants and related notions include the double factorial n!! and the rising and falling factorials used in polynomial expansions and finite differences.

Historically, the factorial notation n! was introduced by Christian Kramp in the early 19th century and became standard in many languages, with the related French term factoriel influencing terminology. While the exact spelling and usage of Faktoriell vary by language, the underlying concept remains the same: it is a compact way to denote products of consecutive integers and their generalizations. See also gamma function, Stirling’s approximation, combinatorics, and binomial coefficients.