Is the number $-1$ prime? From my understanding it's not prime because it's not greater than $0$. So my followup question is why did mathematicians exclude $-1$?
The definition of prime is having only two factors. 
$-1 \cdot 1 = -1$

Prime numbers are (by definition) elements of the natural numbers. There is, however, a natural extension of prime numbers to all the integers, known as the prime elements in the set of integers. The definition of a prime element $p$ (in the integers) is an integer that is neither zero nor $\pm1$ and whenever $p$ divides $xy$ it must also divide $x$ and $y$ (where $x$ and $y$ are both integers). 
Of course, this definition excludes $-1$ ipse dixit, so this definition may not be very satisfying. The reason for excluding $-1$ is the same as excluding $1$ from the prime numbers: both are done to ensure unique factorization at least cost.
