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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Integers, Rational and Real Numbers

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Before you start reading this page you should make sure that you understand that page on foundations, as many of the ideas on that page are used here. In particular, you should make sure that you're familiar with sets, ordered pairs and equivalence relations.

It might also be useful to look at the page on natural numbers, induction and counting. This page will aim to give a formal mathematical definition of the integers and of the rational and real numbers. Although the ideas from the page about natural numbers will not be used here, it will be helpful to see how we can formally prove things about the natural numbers to get an idea of what we're trying to achieve here.

Contents 1 Integers 1.1 A new construction 1.2 Addition and multiplication 2 Rational Numbers 2.1 Fields 2.2 Construction of the rationals 3 Real Numbers

Integers

Our first thought when asked to define the integers might be to start with the natural numbers N = {0, 1, 2, 3, ...} and go from there. For every number n ∈ N we could introduce a new number -n which satisfies n + (-n) = 0, and call this the inverse of n. We don't need to do this for 0 since it is already its own inverse, because 0 + 0 = 0. We could then define multiplication on this extended version of the natural numbers by writing m × (-n) = -(m × n), and (-m) × (-n) = m × n.

This construction of the integers has the advantage of being quite intuitive, but it tends to get a bit unwieldy if we try and use it to prove things. Can we do better?

A new construction

Another way of thinking about the integers is to think of them as ordered pairs of natural numbers, (a,b). Loosely, we want to be thinking of (a,b) as the integer a - b. With this in mind it is obvious that each integer will have many different representations as an ordered pair of natural numbers -- for example, the number 3 could be represented by (3,0), by (4,1), by (14,11) or by any other number of ordered pairs.

To get around this we define a relation "~" on ordered pairs of natural numbers, where we say that the pairs (a,b) and (c,d) are related if c + d = b + c. Thus, for example, (3,0) ~ (4,1), since 3 + 1 = 0 + 4. We can check that this relation is reflexive, symmetric and transitive quite easily, and thus it is an equivalence relation. We can then define the integers to be equivalence classes of ordered pairs of natural numbers, under the equivalence relation "~", and we label the equivalence class of (a,b) by [a,b].

We can recover our natural interpretation of the integers from this. Say we are looking at the equivalence class [a,b]. If a > b then there must be some natural number n such that a = b + n, and we label the equivalence class by n. If b > a then there must be some number n such that a + n = b, and we label the equivalence class by -n. Finally, if a = b then we label the equivalence class by 0. In this way we get a complete copy of the integers, defined as ordered pairs of natural numbers. For example, the integer "-3" is the equivalence class [0,3] = {(0,3), (1,4), (2,5), ...}.

It is important to realise that the integers defined in this way do not contain a copy of the natural numbers. Each integer is an equivalence class of ordered pairs of natural numbers - this is different from a natural number itself. Most of the early attempts to construct the integers from the natural numbers were held back because people believed that the integers had to contain the natural numbers as a subset. Defined in this way, they emphatically do not!

Addition and multiplication

We define the binary operation of addition as follows:

[a,b] + [c,d] = [a + c, b + d]

We need to check that this definition makes sense if we choose different values of a, b, c, d which give the same integers. That is, if [a,b] = [a′,b′] and [c,d] = [c′,d′] then we need to check that [a + c, b + d] = [a′ + b′, c′ + d′].

We define multiplication on the integers by

[a,b] × [c,d] = [ac + bd, ad + bc]

Again, we must check that this makes sense whatever member of the equivalence classes of [a,b] and [c,d] we use to do the calculation. With these rules, it is easy to check the basic rules for Z:

• Addition is commutative and associative, i.e. m + n = n + m, and m + (n + p) = (m + n) + p
• Multiplication is commutative and associative, i.e. mn = nm and m(np) = (mn)p
• Multiplication is distributive over addition, i.e. m(n + p) = mn + mp
• 0 is an identity for addition
• For every m ∈ Z there is an inverse under addition, which is -m.

Rational Numbers

Fields

Construction of the rationals

Real Numbers