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Home > Negative and non-negative numbers

4.3 Division

If both numbers are of the same sign, the result is positive (even if they are both negative):

5 Formal construction of negative and non-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:

if and only if

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.

We can also define a total order on Z by writing

if and only if

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

6 See also

• hyperreal number
• integer
• negative and non-negative in binary
• rational number
• real number
• surreal number

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