Rational Numbers

Rational equivalence

Consider $\Z \times (\Z - \set{0_{\Z }})$. We say that the elements $(a, b)$ and $(c, d)$ of this set are rational equivalent if $ad = bc$. Briefly, the intuition is that $(a, b)$ represents $a$ over $b$ In the usual notation, $(a, b)$ represents “$a/b$”. So this equivalence relation says these two are the same if $a/b = c/d$ or else $ad= bc$.

Definition

The set of rational numbers is the set of equivalence classes (see Equivalence Classes) of $\Z \times (\Z - \set{0_{\Z }})$ under ratioanl equivalence. We call an element of the set of rational numbers a rational number or rational. We call the set of rational numbers the set of rationals or rationals for short.

Notation

We denote the set of rationals by $\Q $.² If we denote rational equivalence by $\sim$ then $\Q = (\Z \times (\Z - \set{0_{\Z }}))/\sim$.

1. Future editions will include an account. ↩︎
2. From what we can tell, $\Q $ is a mnemonic for “quantity”, from the latin “quantitas.” It may also be a mnemonic for quotient. ↩︎