Rational Order

Why

We want to order the rationals.

Definition

Consider $\eqc{(a, b)}, \eqc{(b, c)} \in \Q $ with $0_{\Z } < b, d$ If $ad < bc$, then we say that $\eqc{(a, b)}$ is less than $\eqc{(b, c)}$.¹ If $\eqc{(a, b)}$ is less than $\eqc{(b, c)}$ or equal, then we say that $\eqc{(a, b)}$ is less than or equal to $\eqc{(b, c)}$.

Notation

If $x, y \in \Q $ and $x$ is less than $y$, then we write $x < y$. If $x$ is less than or equal to $y$, we write $x \leq y$.

1. One needs to show that this is well-defined. The account will appear in future editions. ↩︎