Stirling's Approximation Calculator

All math tools

Estimate large factorials (n!) using Stirling's Formula.

Input Value

Number (n)

Approximation Results

3598695.6

Stirling's Estimate

Calculated using the asymptotic formula for n!.

Relative Error (%)0.8296%

Exact (if n ≤ 170)3.63e+6

Growth Visualization (log scale)

Neighborhood around n=10

About this calculator

What is Stirling's Approximation?

Stirling's approximation is a formula used to estimate the value of large factorials. It provides an asymptotic approximation for n!, which is particularly useful in statistical mechanics and probability theory where factorials of very large numbers frequently appear.

💡

Pro Tips

The approximation becomes more accurate as n increases.
For small n, the error is noticeable, but it decreases rapidly.
It is often used in logarithmic form: ln(n!) ≈ n ln(n) - n.

Fun Facts

"James Stirling was a Scottish mathematician who published this formula in 1730."
"The formula relates the three most famous mathematical constants: π, e, and the factorial."
"It's essential for deriving many results in thermodynamics, such as entropy in the microcanonical ensemble."

Formula: n! ≈ √(2πn) · (n/e)ⁿ