Integer Numbers

Why

We want to subtract numbers.¹

Definition

Consider the set $\omega \times \omega $. This set is the set of ordered pairs of $\omega $. In other words, the ordered pairs of natural numbers.

We call two such pairs $(a, b)$ and $(c, d)$ of $\omega \times \omega $ integer equivalent if

\[ a + d = b + c \]

Briefly, the intuition is that $(a, b)$ represents $a$ less $b$, or in the usual notation “$a - b$”.² So this equivalence relation says these two are the same if $a - b = c - d$. Rearranging gives $a + d = b + c$.

Integer equivalence is an equivalence relation.³

The set of integer numbers is the set of equivalence classes (see Equivalence Relations) under integer equivalence on $\omega \times \omega $. We call an element an integer number (or integer).

Notation

We denote the set of integers by $\Z $. If we denote integer equivalence by $\sim$ then $\Z = (\omega \times \omega )/\mathord{\sim}$. Other notation for $\Z $ includes $\mathbb{Z}$.

Future editions will change this why. In particular, by referencing Inverse Elements and the lack thereof in $\omega $. ↩︎
This account will be expanded in future editions. ↩︎
The proof is straightforward. It will be included in future editions. ↩︎