Factorial

The product of the positive integers up to n, written n!.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaFactorial↗MathWorldFactorial↗WikidataQ120976↗Encyclopedia of MathematicsFactorial↗nLabfactorial↗Britannicatopic/factorial↗DLMF5.2↗FungrimFactorial✓↗Fungrim entryOEISA000142↗Wolfram LanguageFactorial!↗mathlib4Nat.factorial↗Rosetta CodeFactorial↗

Factorial(n)$n!$, the product of the integers from 1 to n.

SageMathenumeratio_factorial(n)SymPyfactorial(n)↗mpmathfactorial(n)Oscarfactorial(ZZ(n))Juliafactorial(ZZ(n))Rustfactorial(n)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	compute-engine	always
(complex | infinity) -> number	number-theoryreplaces compute-engine	always

Domain: Combinatorics

Details

• $0! = 1$ by convention, the empty product.
• Recurrence: $n! = n\,(n-1)!$.
• Extends to non-integers via the Gamma function, $n! = \Gamma(n+1)$, so half-integer factorials involve $\sqrt{\pi}$.
• Grows faster than any exponential; Stirling's approximation $n! \sim \sqrt{2\pi n}\,(n/e)^n$ describes the asymptotics.
• compute-engine returns ComplexInfinity at negative integers, the poles of Gamma, rather than leaving the expression unevaluated.

Plot

ComplexPlot

real slice

Examples

Basic #

live ↗

In$5!$Out$120$

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In$6!$Out$720$

The empty product is 1 by convention

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In$0!$Out$1$

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In$10!$Out$3\,628\,800$

Threads element-wise over a list

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In$\bigl\lbrack1, 2, 3, 4\bigr\rbrack!$Out$\bigl\lbrack1, 2, 6, 24\bigr\rbrack$

Scope #

$\frac{n!}{(n-1)!}$ simplifies to $n$ under Simplify; plain evaluation leaves the ratio as it stands

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In$\mathrm{Simplify}(\frac{n!}{(n-1)!})$Out$n$

Real arguments evaluate as $\Gamma(n+1)$

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In$3.5!$Out$11.631\,728\,396\,567\,457$

Negative non-integers are defined too: $(-2.5)! = \Gamma(-1.5)$

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In$(-2.5)!$Out$2.363\,271\,801\,207\,353$

Complex arguments: $(1+i)! = \Gamma(2+i)$

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In$(1+i)!$Out$(1+i)!$

$\left(-\frac12\right)! = \Gamma\!\left(\frac12\right) = \sqrt{\pi}$, kept exact (the float comes from N)

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In$(-\frac{1}{2})!$Out$(-\frac{1}{2})!$

$\infty! = \infty$

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In$∞!$Out$∞$

Applications #

The number of ways to arrange 8 distinct books on a shelf

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In$8!$Out$40\,320$

Properties #

Recurrence: $n! = n\,(n-1)!$

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In$5!=5×4!$Out$\mathrm{True}$

Extends to non-integers via the Gamma function: $\left(\frac12\right)! = \Gamma\!\left(\frac32\right) = \frac{\sqrt{\pi}}{2}$

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In$(\frac{1}{2})!$Out$(\frac{1}{2})!$

$n! = \Gamma(n+1)$. See Gamma

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In$5!=Γ(6)$Out$\mathrm{True}$

The reciprocals sum to $e$: $\sum_{k \ge 0} \frac{1}{k!} = e$

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In$\sum_{k=0}^{∞}\frac{1}{k!}$Out$e$

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In$\sum_{k=0}^2(\frac{((k+2)!)^2}{((2-k)!)^2})/(k!)^4$Out$73$

Known value · Wolfram MathematicalFunctionData, Factorial, Apéry's numbers

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In$\sum_{k=0}^3(\frac{((k+3)!)^2}{((3-k)!)^2})/(k!)^4$Out$1\,445$

Known value · Wolfram MathematicalFunctionData, Factorial, Apéry's numbers

Possible issues #

Negative integers are poles of the Gamma function, so this returns ComplexInfinity rather than staying unevaluated

live ↗

In$(-1)!$Out$\tilde∞$

Neat examples #

Factorials grow fast: $20!$ already exceeds $2 \times 10^{18}$

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In$20!$Out$2\,432\,902\,008\,176\,640\,000$

$30!$ has 33 digits and ends in seven zeros

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In$30!$Out$265\,252\,859\,812\,191\,058\,636\,308\,480\,000\,000$

About

Examples	21
Checked against	Wolfram Language 19 agree2 differSageMath 19 agree1 differ1 errorSymPy 20 agree1 errormpmath 16 agree1 errormathlib4 7 agree3 errorOscar 8 agree7 errorJulia 10 agree7 errorRust 10 agree5 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannicaDLMFFungrimFungrim entry ×42OEISWolfram LanguageSageMathSymPympmathmathlib4OscarJuliaRustRosetta Code

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedoscarpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

See also: Binomial, Factorial2, Subfactorial, Pochhammer