Dictionary
Words, grammatical forms and meanings linked to the ontology.
factorial en · ADJ
Pronunciation
- /fækˈtɔːɹi.əl/
- (Southern-England) audio
Etymology
Etymology tree
Proto-Indo-European *dʰeh₁-
Proto-Indo-European *dʰeh₁k-
Proto-Indo-European *-yéti
Proto-Indo-European *dʰh₁kyéti
Proto-Italic *θakjō
Proto-Italic *fakjō
Latin faciō
Proto-Indo-European *-tōr
Proto-Italic *-tōr
Latin -tor
Latin factorbor.
Middle French facteurbor.
▲
Latin factorbor.
Anglo-Norman factourbor.
Middle English factour
English factor
Proto-Indo-European *h₂el-
Proto-Indo-European *-is
Proto-Indo-European *h₂élisder.?
Proto-Italic *-ālis
Latin -ālisder.
Old French -ialder.
Middle English -ial
English -ial
English factorial
From factor + -ial.
Meanings
-
Of or pertaining to a factor or factorial.
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#33; for n#92;in#92;mathbbN by the following two properties,
(i) 1#33; is defined to be 1,
(ii) for all k#92;in#92;mathbbN, we define (k#43;1)#33; to be (k#43;1)#92;timesk#33;.
1995, K. E. Hirst, Numbers, Sequences and Series, Butterworth-Heinemann, →ISBN, page 21:I shall denote, as usual, the factorial product of the numbers up to n by n#33;, but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently definen#33;#33;#61;n#92;cdot(n-2)#33;#33;.Starting with1#33;#33;#61;1,#92;quad 2#33;#33;#61;2,it follows that, for positive values of n,n#33;#33;#61;n#92;cdotn-2#92;cdotn-4#92;cdots,where the last factor is either 1 or 2, according as n is odd or even.
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:
-
Of or pertaining to a factor, a kind of business agent.
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.
2004, The Digest: Annotated British, Commonwealth, and European Cases:
-
(dated) Of or pertaining to a factory.
Forms
Deriveds
factorially (derived from adjective or noun) · bifactorial (derived from adjective or noun) · plurifactorial (derived from adjective or noun) · monofactorial (derived from adjective or noun) · factorial experiment (derived from adjective or noun) · interfactorial (derived from adjective or noun) · hyperfactorial (derived from adjective or noun) · subfactorial (derived from adjective or noun) · nonfactorial (derived from adjective or noun) · factorial table (derived from adjective or noun) · unifactorial (derived from adjective or noun) · polyfactorial (derived from adjective or noun) · factoriality (derived from adjective or noun) · trifactorial (derived from adjective or noun) · factorialization (derived from adjective or noun) · multifactorial (derived from adjective or noun) · factorialize (derived from adjective or noun) · semifactorial (derived from adjective or noun) · superfactorial (derived from adjective or noun)
Relateds
primorial
factorial en · NOUN
Pronunciation
- /fækˈtɔːɹi.əl/
- (Southern-England) audio
Etymology
Etymology tree
Proto-Indo-European *dʰeh₁-
Proto-Indo-European *dʰeh₁k-
Proto-Indo-European *-yéti
Proto-Indo-European *dʰh₁kyéti
Proto-Italic *θakjō
Proto-Italic *fakjō
Latin faciō
Proto-Indo-European *-tōr
Proto-Italic *-tōr
Latin -tor
Latin factorbor.
Middle French facteurbor.
▲
Latin factorbor.
Anglo-Norman factourbor.
Middle English factour
English factor
Proto-Indo-European *h₂el-
Proto-Indo-European *-is
Proto-Indo-European *h₂élisder.?
Proto-Italic *-ālis
Latin -ālisder.
Old French -ialder.
Middle English -ial
English -ial
English factorial
From factor + -ial.
Meanings
-
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.
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#92;prod#95;#123;k#61;1#125;ⁿk#61;1#92;cdot 2#92;cdot 3#92;cdotsn.But what does this equal?
Well, of course we can say that it is equal to n factorial and write#92;prod#95;#123;k#61;1#125;ⁿk#61;n#33; and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!
2018 September, Colin Foster, “What is the formula for factorial?”, in Mathematics in School, volume 47, number 4, page 40:The expression #123;#125;#95;nC#95;r means the number of combinations of n things taken r at a time. It is also written #92;binomnr, and is equal to#92;frac#123;n(n-1)(n-2)(n-3)#92;cdots#92;cdots(n-r#43;1)#125;#123;r#33;#125;or n#33;#47;r#33;(n-r)#33;, in which r#33; is read "factorial r" and denotes the product of all the integral numbers from 1 to r inclusive.
1916 July, M. Mott-Smith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, →DOI, page 428:
Forms
Hyponyms
rising factorial · double factorial · falling factorial
Relateds
factory · factorial prime · faction · factor · fashion · fact
Translations (41)
tl buuin (mathematical operation or its result) · mi huarea (mathematical operation or its result) · is fimm aðfaldað (mathematical operation or its result) · cs faktoriál (mathematical operation or its result) · el παραγοντικό (mathematical operation or its result) · es factorial (mathematical operation or its result) · fr factorielle (mathematical operation or its result) · bg факториа́л (mathematical operation or its result) · da fakultet (mathematical operation or its result) · pl silnia (mathematical operation or its result) · is fimm hrópmerkt (mathematical operation or its result) · he עֲצֶרֶת (mathematical operation or its result) · sv fakultet (mathematical operation or its result) · nl faculteit (mathematical operation or its result) · nn fakultet (mathematical operation or its result) · ru факториа́л (mathematical operation or its result) · gl factorial (mathematical operation or its result) · pt fatorial (mathematical operation or its result) · is aðfeldi (mathematical operation or its result) · nb fakultet (mathematical operation or its result) · et faktoriaal (mathematical operation or its result) · hy ֆակտորիալ (mathematical operation or its result) · th แฟกทอเรียล (mathematical operation or its result) · ko 계승 (mathematical operation or its result) · id faktorial (mathematical operation or its result) · de Fakultät (mathematical operation or its result) · tr faktöriyel (mathematical operation or its result) · ja 階乗 (mathematical operation or its result) · ca factorial (mathematical operation or its result) · fi kertoma (mathematical operation or its result) · ro factorial (mathematical operation or its result) · ky факториал (mathematical operation or its result) · cmn 階乘 /阶乘 (mathematical operation or its result) · is hrópmerkt tala (mathematical operation or its result) · ast factorial (mathematical operation or its result) · vi giai thừa (mathematical operation or its result) · pl faktoriał (mathematical operation or its result) · ga iolrán (mathematical operation or its result) · it fattoriale (mathematical operation or its result) · hu faktoriális (mathematical operation or its result) · eo faktorialo (mathematical operation or its result)