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Formal construction of negative integers

In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers (a, b). We can extend addition and multiplication to these pairs with the following rules:

We define an equivalence relation ~ upon these pairs with the following rule:

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N ²/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. Note that Z, equipped with these operations of addition and multiplication, is a ring, and is in fact, the prototypical example of a ring.

We can also define a total order on Z by writing

This will lead to an additive zero of the form (a, a), an additive inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a + 1, a), and a definition of subtraction

This construction is a special case of the Grothendieck construction.