Factorial calculator

Definition

Let n be a non-negative integer (n ∈ ℕ₀). The factorial n! is the product 1·2·…·n when n ≥ 1, and the value 1 when n = 0. Equivalently: 0! := 1 and n! := n·(n−1)! for n ≥ 1. This is standard school algebra and enumerative combinatorics: Graham, Knuth, Patashnik, Concrete Mathematics; Brualdi, Introductory Combinatorics; MathWorld (Factorial); NIST Digital Library of Mathematical Functions, §5.2 (factorial / gamma). The identity n! = Γ(n+1) links the integer factorial to the gamma function for n ∈ ℕ₀ — this calculator does not evaluate Γ, but it respects the integer values from the product definition.

Computation on this page: n = max(0, ⌊a⌋); result = the product of the integers from 2 to n. For n = 0 and n = 1 the loop does not multiply, so r stays 1. A negative input, after rounding down and clamping to zero, becomes 0, so the result is 0! = 1. The calculator does not return factorials of negative integers (where Γ has poles) or fractional factorials.

The result is only the integer n! (in the browser’s floating-point arithmetic). The panel does not emit a symbolic expansion, does not use big-integer arithmetic, and does not apply Stirling’s approximation.

Historical and conceptual context

Products of successive natural numbers appear when counting arrangements (permutations) from ancient Indian texts through medieval multiplication rules. Modern notation n! was popularized by Christian Kramp (1808); the link to the gamma function (Euler, Legendre) and Stirling’s approximation for large n were systematized in analysis. NIST DLMF §5.2 codifies n! = Γ(n+1) on the naturals. In classical probability, n! enters denominators of uniform models on Sₙ; in series, it appears in eˣ = Σ xᵏ/k!.

Conceptually separate: (1) the integer factorial as a product — this page; (2) the permutation count P(n) = n! — same number, different meaning; (3) k-permutations / variations V(n,k) = n!/(n−k)!; (4) combinations C(n,k) = n!/(k!(n−k)!); (5) Stirling and gamma — context, not panel output. The permutations sibling shares the product but returns 0 for negatives instead of replacing n with 0.

In computer science, n! appears in the complexity of generating all orders and in comparison-sort lower bounds (log₂(n!)). In practice, 20! already exceeds the exactly representable integers — hence the panel warning for large n.

Formal properties

Mathematical domain: n ∈ ℕ₀. Codomain: {1, 1, 2, 6, 24, 120, 720, …}. Recurrence: 0! = 1, n! = n·(n−1)! for n ≥ 1. Monotonicity: for n ≥ 2, n! > (n−1)!.

Identities: P(n) = n!; V(n,n) = n!; C(n,k)·k! = P(n,k). Stirling’s formula n! ∼ √(2πn)·(n/e)ⁿ describes growth (Concrete Mathematics; Feller) — this calculator does not apply it. Gamma: Γ(n+1) = n! for n ∈ ℕ₀ (NIST DLMF §5.2); for non-positive integers Γ has poles, so a “negative factorial” does not exist in ℝ.

In this calculator the practical domain includes non-integer inputs (floor) and negatives replaced by 0. That is a convention of this calculator, not a theorem about the Γ function.

Full derivation

Step 1 (product definition). For n ≥ 1 set n! := 1·2·…·n. For n = 0 set 0! := 1 (empty product).

Step 2 (recurrence). For n ≥ 1 factor out n: n! = n·(n−1)!. The base 0! = 1 closes the induction.

Step 3 (computation). After n = max(0, ⌊a⌋) set r = 1 and multiply by 2, 3, …, n. When n ∈ {0,1}, the loop does not run and r = 1.

Step 4 (example). For a = 5: n = 5, r = 2·3·4·5 = 120. For a = 5.7: ⌊5.7⌋ = 5 → 120. For a = −3: max(0,−3) = 0 → 1.

Step 5 (link to P(n)). The number of linear arrangements of n distinct objects is n! (multiplication rule n·(n−1)·…·1). Hence P(n) = n! for n ≥ 0 — the same number, different narrative stress on the permutations page.

Special cases

0! = 1 and 1! = 1 — same value, different reading (empty product vs single factor). 2! = 2. 3! = 6. 4! = 24. 5! = 120. 6! = 720. 7! = 5040. 10! = 3,628,800.

Input 5.7 → 120. Input −1 → 1 on this page (replaced by 0). A fractional factorial (e.g. 1.5!) needs Γ — not available here. Stirling is not shown as a result.

When n is very large, the result first loses integer bit precision (above 2⁵³), and above 170! it would exceed the floating-point range — for n ≥ 171 the calculator reports an error. That is a floating-point arithmetic limit, not a limit of the definition of n!.

Numerical notes

Computation: n = max(0, ⌊a⌋); n! = 1·2·…·n. Multiplication is exact for integer n! while the value stays within the exactly representable integers, that is through 18!. From 19! the panel shows the result as an approximation ≈ m × 10^e, and n ≥ 171 gives an error message.

The calculator accepts a comma or a period as the decimal separator. The calculator does not use big-integer arithmetic or arbitrary-precision libraries. Sibling contrast: the permutations page returns 0 for negatives; here a negative yields 1 after being replaced by 0.

In teaching practice, distinguish the mathematical formula from floating-point limits: this calculator shows the browser’s computed result, not a symbolic paper proof.

Related notions and sources

Related on CalcBoxer: permutations P(n) = n!, variations V(n,k), combinations C(n,k). Related in theory: the gamma function, Stirling’s formula, Stirling numbers, the series for eˣ.

Sources: Graham, Knuth, Patashnik — Concrete Mathematics; Brualdi — Introductory Combinatorics; NIST DLMF §5.2; MathWorld — Factorial; Wikipedia — Factorial; Kramp (1808) — n! notation. Educational tool.