The Factorial Notation | Christian Kramp | 1808

n! = 1 x 2 x 3 x … x (n-1) x n

The factorial notation as we know it today was introduced in 1808 by a French mathematician Christian Krump. Previously he used the word “faculty” in his work on the topic but made up his mind and in the preface to Éléments d'arithmétique universelle Ⓣ (1808) he wrote:-

I have given it the name 'faculty'. Arbogast has substituted the name 'factorial' which is clearer and more French. In adopting his idea, I congratulate myself on paying homage to the memory of my friend.

Was the notation necessary? We would all agree that yes. Moreover, it’s simple, elegant and, therefore, easy to remember.

I use the very simple notation n!n! to designate the product of numbers decreasing from nn to unity, i.e. n(n - 1)(n - 2) ... 3 . 2 . 1n(n−1)(n−2)...3.2.1. The constant use in combinatorial analysis, in most of my proofs, that I make of this idea, has made this notation necessary.

Anything new will receive its dose of criticism, and the factorial notation wasn’t an exception. In Penny Cyclopaedia (1842) in the article on 'symbols' August De Morgan wrote:-

There is no a portrait of Christian Kramp (1760-1826), so no idea how he looked but, according to the main source (MacTutor History of Mathematics Archive) for this short writing, he didn’t lack self-confidence. Good for him! :-)

Many letters written by Kramp survive and these show that he was a man with a great deal of confidence in his own abilities. If one wishes to be a little unkind this self-confidence could reasonably be described as boasting. However, he worked with tireless energy on a wide range of topics, making many worthwhile contributions.

You can find The Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge on Google Books. De Morgan’s discussion on the use of mathematical symbols is on pages 442-445.