factorial

factorial — noun

IPA/fækˈtɔːriəl/
IPA/fækˈtɔːriəl/
  • factorialsingular
  • factorialsplural

1. A value that you calculate by taking a given whole number and multiplying it by

1.nounB2
Definition

A value that you calculate by taking a given whole number and multiplying it by every smaller whole number down to 1. For example, the factorial of 5 (written as 5!) is 5 × 4 × 3 × 2 × 1 = 120.

Example

Tariro learned from her textbook that 5 factorial, written as 5!, equals one hundred and twenty.

5! notation: factorial written as number followed by exclamation mark

The computer science students wrote a short program to calculate the factorial of any number typed into the console.

factorial + of + number in computing context

Common collocations
  • factorial of n
  • compute / calculate the factorial
  • factorial function
  • n factorial (n!)
Synonyms
  • product

    a more general term for the result of any multiplication; not specific to the 1-to-n operation

Grammar Patterns

factorial + of + number

Usage Note

In mathematical writing, the factorial of a number n is written as n! (read as 'n factorial'). The value grows very quickly — the factorial of 20 is already over 2.4 quintillion.

Common Mistakes

The factorial of 5 is 5 + 4 + 3 + 2 + 1 = 15.
The factorial of 5 is 5 × 4 × 3 × 2 × 1 = 120.
Factorial uses multiplication, not addition.
A factorial is a number that divides another number evenly.
A factorial is a value you get by multiplying a whole number by every smaller whole number down to 1.
This confuses factor with factorial.

factorial — adjective

IPA/fækˈtɔː.ri.əl/
IPA/fækˈtɔːr.i.əl/

Word family

Synonym discussion

Sense: noun/1

These three words all relate to multiplication but differ in precision. A product is simply the result of any multiplication — it is the most general term and can apply to any numbers or expressions. A multiple is the result of multiplying a number by an integer (e.g. 15 is a multiple of 5), but it does not imply multiplication by every smaller number. A factorial is a specific kind of product: it always starts at a given whole number and multiplies by every smaller positive whole number down to 1. In conversation, saying '5 factorial' (120) is precise, whereas 'the product of 5 and all numbers below it' is a description of the same operation but not a standard term. For example, 'What is the product of 4 and 2?' has the answer 8, but 'What is the factorial of 4?' has the answer 24.

factorial ↗product ↗multiple ↗

Common collocations

Nouns

  • factorial function
  • factorial calculation

Prepositions

  • factorial of

Etymology

LatinFrenchEnglish

The word comes from the Latin 'factor' meaning 'one who acts' or 'a doer', combined with the English suffix '-ial' meaning 'relating to'. The mathematical sense was introduced by the French mathematician Christian Kramp in 1808, who chose the term because a factorial is built from the multiplication of successive factors.