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 \(b \in \mathbb{N}^*\).

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