factorial
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English
[edit]Etymology
[edit]Pronunciation
[edit] IPA^{(key)}: /fækˈtɔːɹi.əl/
Audio (Southern England): (file)
Noun
[edit]factorial (plural factorials)
 (mathematics, combinatorics) The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.
 1916 July, M. MottSmith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, , page 428:
 The expression means the number of combinations of things taken at a time. It is also written , and is equal toor , in which is read "factorial " and denotes the product of all the integral numbers from to inclusive.
 2018 September, Colin Foster, “What is the formula for factorial?”, in Mathematics in School^{[1]}, volume 47, number 4, page 40:
 This playing around with consecutive integers reminded one of the students of factorials, and he asked about products of integers, which I said could be expressed similarly asBut what does this equal?
Well, of course we can say that it is equal to factorial and write and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!
Usage notes
[edit]"n!" is read as "factorial of n" or "n factorial."
Hyponyms
[edit]Related terms
[edit]Translations
[edit]mathematical operation or its result

Adjective
[edit]factorial (comparative more factorial, superlative most factorial)
 (mathematics) Of or pertaining to a factor or factorial.
 1903, Arthur Schuster, “On some Definite Integrals, and a New Method of Reducing a Function of Spherical Coordintes to a Series of Spherical Harmonics”, in Philosophical Transactions of the Royal Society A, volume 200, , →JSTOR, page 182:
 I shall denote, as usual, the factorial product of the numbers up to by , but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently defineStarting withit follows that, for positive values of ,where the last factor is either or , according as is odd or even.
 1995, K. E. Hirst, Numbers, Sequences and Series, ButterworthHeinemann, →ISBN, page 21:
 The definition of addition is an example of definition by induction, sometimes called recursive definition. As another example of this, consider the following inductive definition of the factorial function.
We define for by the following two properties,
(i) is defined to be ,
(ii) for all , we define to be .
 Of or pertaining to a factor, a kind of business agent.
 2004, The Digest: Annotated British, Commonwealth, and European Cases:
 The latter sold the goods to a customer who was cashier to certain creditors of the agents without disclosing the factorial capacity in which they acted.
 (dated) Of or pertaining to a factory.
Derived terms
[edit]See also
[edit]References
[edit]Further reading
[edit] “factorial”, in Webster’s Revised Unabridged Dictionary, Springfield, Mass.: G. & C. Merriam, 1913, →OCLC.
 “factorial”, in The Century Dictionary […], New York, N.Y.: The Century Co., 1911, →OCLC.
Portuguese
[edit]Noun
[edit]factorial m (plural factoriais)
 Prereform spelling (until Brazil 1943/Portugal 1990) of fatorial. Still used in countries where the agreement hasn't come into effect; may occur as a sporadic misspelling.
Adjective
[edit]factorial m or f (plural factoriais)
 Prereform spelling (until Brazil 1943/Portugal 1990) of fatorial. Still used in countries where the agreement hasn't come into effect; may occur as a sporadic misspelling.
Romanian
[edit]Etymology
[edit]Borrowed from French factorielle.
Adjective
[edit]factorial m or n (feminine singular factorială, masculine plural factoriali, feminine and neuter plural factoriale)
Declension
[edit]Declension of factorial
singular  plural  

masculine  neuter  feminine  masculine  neuter  feminine  
nominative/ accusative 
indefinite  factorial  factorială  factoriali  factoriale  
definite  factorialul  factoriala  factorialii  factorialele  
genitive/ dative 
indefinite  factorial  factoriale  factoriali  factoriale  
definite  factorialului  factorialei  factorialilor  factorialelor 
Spanish
[edit]Pronunciation
[edit]Adjective
[edit]factorial m or f (masculine and feminine plural factoriales)
Derived terms
[edit]Noun
[edit]factorial m (plural factoriales)
Further reading
[edit] “factorial”, in Diccionario de la lengua española, Vigésima tercera edición, Real Academia Española, 2014
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 Rhymes:Spanish/al
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 es:Mathematics
 es:Combinatorics