Negative Number - Formal Construction of Negative Integers

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:

(a, b) + (c, d) = (a + c, b + d)

(a, b) × (c, d) = (a × c + b × d, a × d + b × c)

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

(a, b) ~ (c, d) if and only if a + d = b + c.

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, 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

(a, b) ≤ (c, d) if and only if a + d ≤ b + c.

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

(a, b) − (c, d) = (a + d, b + c).

This construction is a special case of the Grothendieck construction.