So you've got the “natural numbers”, $\mathbb{N}=\left\{0, 1, 2, \dots\right\}$. Or the “counting numbers”, or whatever you want to call them. Addition works, and multiplication works. But subtraction is broken, because there isn't any natural number equal to $2-3$. We might think about fixing subtraction, and one way of doing this is by moving to the integers, which allows negative numbers: $\mathbb{Z}=\left\{\dots, -2, -1, 0, 1, 2, \dots\right\}$. But …