Equations · Calculus and analysis
Stirling's approximation
A close estimate of n factorial for large n, without multiplying all the numbers out.
n! ≈ √(2πn) (n/e)ⁿ
Aethel says
n factorial is approximately the square root of 2 pi n, times the quantity n over e, to the power of n
What it says
Factorials grow too fast to compute directly for large n. Stirling's formula gives their size from a square root and a power, and the relative error shrinks as n grows: it is already within 1% at n = 10.
Taking logarithms gives the form used in statistical mechanics and information theory: the log of n factorial is about n log n minus n.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| n! | n factorial | 1 × 2 × 3 × … × n |
| ≈ | is approximately | approximately equal to |
| e | e | Euler's number, about 2.71828 |
Where you meet it
Combinatorics, statistical mechanics and the analysis of algorithms.
Questions
How do you say Stirling's approximation out loud?
n! ≈ √(2πn) (n/e)ⁿ is read: “n factorial is approximately the square root of 2 pi n, times the quantity n over e, to the power of n.” That is exactly what Aethel says when it meets the formula in a book.
What does n! mean in Stirling's approximation?
n! (said “n factorial”) is 1 × 2 × 3 × … × n.
What does ≈ mean in Stirling's approximation?
≈ (said “is approximately”) is approximately equal to.
Can I listen to the equations in my own textbook like this?
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