The general concept of factorials has a rich history dating back to the 1200s. Many different mathematicians have given their view on the subject and contributed different notations and methods to using factorials.

1200s

The first uses of factorials were used in the 1200s, and at that point they were only used to count permutations. It wasn't until hundreds of years later that the notations and term factorials were actually applied in the mathematical world(Agarwal, 2022).

1700s

In 1721, Daniel Bernoulli, a Swiss mathematician and physicist known for his pioneering work in probability and statistics, met Christian Goldbach. Christian Goldbach was a German mathematician who was referred to Daniel Bernoulli by Daniel's older brother, Nicolaus Bernoulli, while on tour in Venice. For 7 years, Bernoulli and Goldbach worked together on various mathematical subjects; factorials being one of them(O'Connor, 2006). Bernoulli and Goldbach used the gamma notation. Their notation was shown as n!=Γ(n + 1) and is read as gamma n+1. They realized that this definition of factorial was restrictive in the fact that it is defined for only non-negative integers(Agarwal, 2022). Regardless, this is still the definition typically used today.

It wasn't until the late 1720s when Leonhard Euler, another Swiss mathematician, was interested in extending the factorial to non-integer values.

In 1730, Euler was able to solve the problem of extending the factorial to all real numbers.

In 1738, Euler generalized what we know as a factorial to be a function defined by a certain integral, known as the Euler gamma function. This is shown by Γ(n + 1) = n! For all non-negative integers(Agarwal, 2022).

1800s

There are still two other notations that have been used throughout history for the factorial function. In the early 1800s, Christian Kramp, a French mathematician, introduced what is known to be the universal notation of a factorial as an exclamation mark. The first use of this notation was used in 1808 in his book "elements d'arithmetique Universelle"(O'Connor, 2012). The book describes the notation as the product of integers decreasing from n to 1. This definition was very useful to Kramp's work as he used mostly combinatorial analysis for his demonstrations. Shortly after the use of an exclamation mark, Thomas Jarrett introduced another notation, denoted by the left and bottom sides of a rectangle, which was defined the same way. Jarrett's notation was popular in Britain and America for a while, but was short-lived due to the inconvenient nature in typeset as well as writing(Agarwal, 2022).