A student wants to know why, if we can define 0! as 1, we cannot define 1 over 0

Answer 1

suppose 1/0 = x

That means that 1 = 0*x
But zero times anything is zero. There is no number x which can be defined as 1/0.

0! is useful because n! is a recursive definition.

That is,
n! = n * (n-1)!
3! = 3 * 2!
2! = 2 * 1!
1! = 1 * 0!

But, n! is also defined as the product of all the numbers from 1 to n. Since 1! is clearly 1, that means we must have 0! = 1.