Integer

The Integers are the whole numbers, both positive and negative and including zero. The usual symbol for the set of integers is a "blackboard bold" Z - a Z with two strokes on the diagonal.

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

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

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The integers can be constructed from the counting numbers (the natural numbers not including 0) as follows:

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.

The same technique of equivalence classes of pairs can be used to create the rational numbers from the integers.

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