The Rational Numbers as a Subset of the Real Numbers

Let $\mathbb{R}$ be the real numbers, a field in which $b^{-1}$ denotes the multiplicative inverse of a nonzero element $b$, and let $\mathbb{Z}$ be the set of integers.

The set of rational numbers is

$$\mathbb{Q}=\bigl\{a\,b^{-1}\;:\;a\in\mathbb{Z},\ b\in\mathbb{Z},\ b\ne0\bigr\},$$

a subset of $\mathbb{R}$. An element of $\mathbb{Q}$ is called a rational number.