# Rationals except Zero

Here’s a quick one: All rational numbers except 0 can be expressed as $(-1)^s \Pi p_i^{n_i}$ where $$s \in \{0, 1\}$$, $$p_i$$ is a prime number, and $$n_i$$ is an integer.

This reminds me of the restriction on the definition of rationals, i.e., that $$\frac{a}{b}$$ is a rational number for all integers $$a$$ and $$b$$ except $$b = 0$$. However, this can be written to make zero less exceptional: All rational numbers can be written in the form $$\frac{a}{b}$$ where $$a \in \mathbb{Z}$$ and $$b \in \mathbb{N}^*$$.