The Rational Numbers

TODO

we have

\[(a, b) \in \mathbb{Z} \times \mathbb{Z}\]

and a binary operation $/$. Then we define equality under this operation. $a / b = c / d$ if an only if $a * d = c * b$. Now, we can define a set of equivalence classes

\[\mathbb{Q} = (\mathbb{Z}^2 / \equiv) = \{ [a/b] : a/b = c/d \text{ for } c, d \in \mathbb{Z} \}\]

The upshot is that, give any rational number $q \in \mathbb{Q}$ there exists $a, b \in \mathbb{Z}$ such that $q = a/b$.

Somewhere in here, we need to define that $a/b = a * b^{-1}$ where $b^{-1}$ refers to the multiplicative inverse of the integer $b$. So somewhere I have to define the concept of a multiplicative inverse without being recursive.

Constructing the Real Numbers Series