Zero is an even number as it is divisible by 2 with no remainder. It is either negative nor positive. The number 0 may or may not be considered a natural number, but it is an integer.

0 is a palindromic number.

0 is a pronic number:

0 is the identity element with respect to addition:

0 is the absorbent element with respect to multiplication:

Also,

We cannot divide by 0 as it has no multiplicative inverse.

The expression $\frac{0}{0}$ appears sometimes when we attempt to determine the limit of an expression of the form $\frac{f\left(x\right)}{g\left(x\right)}$.

This is one of the seven indeterminate forms. 

The other indeterminate forms that includes 0 are $0\times \infty$, $0^0$ and $\infty^0$.

By definition, the factorial of 0 is 1: $0!=1$.