Integer

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${\bb~Z}=\{\ldots,{\quad}-3,{\quad}-2,{\quad}-1,{\quad}0,{\quad}1,{\quad}2,{\quad}3,{\quad}\ldots}$

The integers have a natural embedding into the rational numbers, and contain the natural numbers.

More technical stuff ...

• Let P be the collection of all pairs of whole numbers:
• $P=\{(a,b):a,b\in{N}}$
• Let (a,b) be equivalent to (c,d) if a+d=b+c.
• For any pair (a,b) we can consider the collection of all pairs equivalent to it.
• This collection is called the equivance class of (a,b), and we write it as E(a,b)
• We can now define arithmetic operations on the equivalence classes:
• The sum is obtained as E(a,b)+E(c,d) = E(a+c,b+d)
• The difference is obtained as E(a,b)-E(c,d) = E(a+d,b+c)
• The product: E(a,b)*E(c,d) = E(ac+bd,ad+bc)
• The equivalence classes can be thought of as the integers.
• We think of E(a,b) as being "the same as" a-b.
• Lots and lots of checking required to see that the arithmetic on the equivalence classes is "the same as" the arithmetic on the integers.