**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.
Stirling's formula:
*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*:
http://sprott.physics.wisc.edu/pickover/math-book.html |