James Stirling (1692-1770)
These days, factorials are everywhere in mathematics. For non-negative integers n, "n factorial" (written as n!), is the product of all positive integers less than or equal to n. For example, 4! = 1 x 2 x 3 x 4 = 24. The notation n! was introduced by French mathematician Christian Kramp in 1808. Factorials are important in cominatronics, for example, when determining the number of different ways of arranging objects in a sequence. They also occur in number theory, probability and calculus.
Because factorial values grow so large (for example, 70! is greater then 10^100, and 25206! is greater than 10^100,000), convenient methods for approximating large factorials are extremely useful.
n! ~= sqrt(2*pi) * (e^-n) * (n^(n+1/2)),provides an accurate estimate for n factorial. Here, the ~= symbol means "approximately equal to", and e and pi are mathematical constants e ~= 2.71828 and pi ~= 3.14159.
For large values of n, this expression results in an even simpler-looking approximation, ln(n!) ~= n*ln(n) - n, which can also be written as:
n! ~= n^n * e^-n.
In 1730, Scottish mathematician James Stirling presented his approximation for the value of n! in his most important work, Methods Differentialis. Stirling began his career in mathematics amidst political and religious conflicts. He was friends with Newton, but devoted most of his life after 1735 to industrial management.
Excerpt from The Maths Book: