The Integers as a Subset of the Real Numbers

Let $\mathbb{R}$ be the real numbers, an ordered field whose additive identity is $0$ and in which $-a$ denotes the additive inverse of $a$, let $\mathbb{N}$ be the set of natural numbers, and let $\iota:\mathbb{N}\to\mathbb{R}$ be the canonical map of $\mathbb{R}$.

The set of integers is

$$\mathbb{Z}=\{0\}\cup\{\iota(n):n\in\mathbb{N}\}\cup\{-\iota(n):n\in\mathbb{N}\},$$

a subset of $\mathbb{R}$. An element of $\mathbb{Z}$ is called an integer.