The Rational Numbers

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\mathbb{Z}$, $0_{\mathbb{Z}}$ and $1_{\mathbb{Z}}$ be as in The Integers §integers and The Integers §constants, and let $Q=\mathbb{Z}\times\mathbb{N}$, $\approx$ and $[x,m]$ be as in Construction of the Rationals: Pairs of an Integer and a Natural Number up to Equal Ratios, with Sum, Product, Negation and Order §equivalence.

¶ The set of rational numbers is the quotient $\mathbb{Q}=Q/{\approx}$, which is a set whose elements are the classes $[x,m]$ with $x\in\mathbb{Z}$ and $m\in\mathbb{N}$, by Construction of the Rationals: Pairs of an Integer and a Natural Number up to Equal Ratios, with Sum, Product, Negation and Order §equal.

¶ The sum $+$, the product $\cdot$, the negation $-$ and the order $\le$ on $\mathbb{Q}$ are those of Construction of the Rationals: Pairs of an Integer and a Natural Number up to Equal Ratios, with Sum, Product, Negation and Order §operations. For $u,v\in\mathbb{Q}$, $u-v$ denotes $u+(-v)$, and $u<v$ means $u\le v$ and $u\neq v$.

¶ $0_{\mathbb{Q}}=[0_{\mathbb{Z}},1]$ and $1_{\mathbb{Q}}=[1_{\mathbb{Z}},1]$.

¶ $j:\mathbb{Z}\to\mathbb{Q}$ is the map with $j(x)=[x,1]$, given by Maps and Relations Given by Formulas §map.

Here $1\in\mathbb{N}$ by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §background and $0_{\mathbb{Z}},1_{\mathbb{Z}}\in\mathbb{Z}$ by The Integers §constants, so for every $x\in\mathbb{Z}$ the classes $[x,1]$, $[0_{\mathbb{Z}},1]$ and $[1_{\mathbb{Z}},1]$ are rational numbers by Construction of the Rationals: Pairs of an Integer and a Natural Number up to Equal Ratios, with Sum, Product, Negation and Order §equal.