factorial

English

Etymology

From factor +‎ -ial.

Pronunciation

• IPA^(key): /fækˈtɔːɹi.əl/

  • Audio (Southern England):

    (file)

• Rhymes: -ɔːɹiəl

Noun

factorial (plural factorials)

1. (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. Mott-Smith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, →DOI, page 428:

     The expression ${}_{n}C_{r}$ means the number of combinations of $n$ things taken $r$ at a time. It is also written ${\binom {n}{r}}$, and is equal to$${\frac {n(n-1)(n-2)(n-3)\cdots \cdots (n-r+1)}{r!}}$$or $n!/r!(n-r)!$, in which $r!$ is read "factorial $r$" and denotes the product of all the integral numbers from $1$ to $r$ 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 as$$\prod _{k=1}^{n}k=1\cdot 2\cdot 3\cdots n.$$But what does this equal?
     Well, of course we can say that it is equal to $n$ factorial and write$$\prod _{k=1}^{n}k=n!$$ and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!

Usage notes

"n!" is read as "factorial of n" or "n factorial."

Hyponyms

• double factorial
• falling factorial
• rising factorial

Related terms

• fact
• faction
• factor
• factorial prime
• factory
• fashion

Translations

mathematical operation or its result

• Armenian: ֆակտորիալ (hy) (faktorial)
• Asturian: factorial m
• Bulgarian: факториа́л m (faktoriál)
• Catalan: factorial m
• Chinese:

  Mandarin: 階乘 / 阶乘 (zh) (jiēchéng)

• Czech: faktoriál m
• Danish: fakultet
• Dutch: faculteit (nl) f
• Esperanto: faktorialo
• Estonian: faktoriaal
• Finnish: kertoma (fi)
• French: factorielle (fr) f
• Galician: factorial m
• German: Fakultät (de) f
• Greek: παραγοντικό (el) m (paragontikó)
• Hebrew: עֲצֶרֶת (he) f ('atzerét)
• Hungarian: faktoriális (hu)
• Icelandic: aðfeldi (is) n, (a factorial number) hrópmerkt tala f, (factorial of five) fimm hrópmerkt, (factorial of five) fimm aðfaldað
• Irish: iolrán m
• Italian: fattoriale (it) m
• Japanese: 階乗 (ja) (kaijō)
• Korean: 계승 (ko) (gyeseung)
• Kyrgyz: факториал (ky) (faktorial)
• Maori: huarea
• Norwegian:

  Bokmål: fakultet n

  Nynorsk: fakultet n

• Polish: silnia (pl) f, (dated) faktoriał m
• Portuguese: fatorial (pt) m
• Romanian: factorial (ro) n
• Russian: факториа́л (ru) (faktoriál)
• Spanish: factorial (es) m
• Swedish: fakultet (sv) c
• Tagalog: buuin
• Thai: แฟกทอเรียล
• Turkish: faktöriyel (tr)
• Vietnamese: giai thừa

Adjective

factorial (comparative more factorial, superlative most factorial)

1. (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 Co-ordintes to a Series of Spherical Harmonics”, in Philosophical Transactions of the Royal Society A, volume 200, →DOI, →JSTOR, page 182:

     I shall denote, as usual, the factorial product of the numbers up to $n$ by $n!$, but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently define$$n!!=n\cdot (n-2)!!.$$Starting with$$1!!=1,\quad 2!!=2,$$it follows that, for positive values of $n$,$$n!!=n\cdot n-2\cdot n-4\cdots ,$$where the last factor is either $1$ or $2$, according as $n$ is odd or even.

   • 1995, K. E. Hirst, Numbers, Sequences and Series, Butterworth-Heinemann, →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 $n!$ for $n\in \mathbb {N}$ by the following two properties,
     (i) $1!$ is defined to be $1$,
     (ii) for all $k\in \mathbb {N}$, we define $(k+1)!$ to be $(k+1)\times k!$.

2. 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.

3. (dated) Of or pertaining to a factory.

Derived terms

derived from adjective or noun

• bifactorial
• factorial experiment
• factoriality
• factorialization
• factorialize
• factorially
• factorial table
• hyperfactorial
• interfactorial
• monofactorial
• multifactorial
• nonfactorial
• plurifactorial
• polyfactorial
• semifactorial
• subfactorial
• superfactorial
• trifactorial
• unifactorial

See also

• primorial

References

• factorial on Wikipedia.Wikipedia

Further reading

• “factorial”, in Webster’s Revised Unabridged Dictionary, Springfield, Mass.: G. & C. Merriam, 1913, →OCLC.
• William Dwight Whitney, Benjamin E[li] Smith, editors (1911), “factorial”, in The Century Dictionary […], New York, N.Y.: The Century Co., →OCLC.

Portuguese

Noun

factorial m (plural factoriais)

1. Pre-reform spelling (used until 1943 in Brazil and 1990 in Portugal) of fatorial. Still used in countries where the agreement hasn't come into effect; may occur as a sporadic misspelling.

Adjective

factorial m or f (plural factoriais)

1. Pre-reform spelling (used until 1943 in Brazil and 1990 in Portugal) of fatorial. Still used in countries where the agreement hasn't come into effect; may occur as a sporadic misspelling.

Romanian

Etymology

Borrowed from French factorielle.

Adjective

factorial m or n (feminine singular factorială, masculine plural factoriali, feminine and neuter plural factoriale)

1. factorial

Declension

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

Etymology

From French factorielle. Equivalent to factor +‎ -al.

Pronunciation

• IPA^(key): /faɡtoˈɾjal/ [faɣ̞.t̪oˈɾjal]
• Rhymes: -al
• Syllabification: fac‧to‧rial

Adjective

factorial m or f (masculine and feminine plural factoriales)

1. factorial

Derived terms

• análisis factorial

Noun

factorial m (plural factoriales)

1. (mathematics) factorial

Further reading

• “factorial”, in Diccionario de la lengua española [Dictionary of the Spanish Language] (in Spanish), online version 23.8, Royal Spanish Academy [Spanish: Real Academia Española], 2024 December 10