What is a factorial

A factorial is represented by the following pattern:

1!=1

2!=2*1

3!=3*2*1

More generally the algorithm is as follows:

n!=(n-1)(n-2)(n-3)... for n=1,2,3....

Notice that I purposely skipped 0! because it might not make sense yet since 0!=1. Moreover all fractions and all negative numbers were not included in the definition of n! This definition is very limiting so there is a need for a better definition of a factorial for such applications.

A more versatile definition of a factorial

This definition extends what you can do significantly and now the 0! can be shown as follows:

There is a lot more you can do with this formula for exmaple now, it is possible to move away from integers and begin to use fractions of the factorial.

1/2! is solved as follows:

The n can essentially be anything just for show lets plug into mathematica 1!, 2!, and 3!

1!

2!

3!

Honestly I really did not want to do this by hand because it was too much work hopefully you all prove this to yourselves by hand if you don't believe me.