Rational Numbers for the CSET

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Excerpt from the CSET study guide

The system of integers has a defect in that given integers m ¹ 0 and s, the equation mx = s may not have a solution. For example, 3x = 6 has the solution x = 2 but 4x = 6 has no solution. This defect is remedied by adjoining to the integers additional numbers (common fractions) to form the system Q of rational numbers.

A rational number is a number that can be expressed in the form, a/b where a and b are integers and b is not equal to 0.

We begin with the set of ordered pairs K = I x (I - {0}) = {(s, m): s Î I, m Î I - {0}} and define the binary relation on all (s, m), (t, n) Î K by (s, m) (t, n) if and only if sn = mt

Now is an equivalence relation and therefore partitions K into a set of equivalence classes.

J = {[s, m], [t, n],…}
where [s, m] = {(a, b): (a, b) Î K, (a, b) (s, m)}We shall call the equivalence classes of J rational numbers and in the following sections indicate that J is isomorphic to the system Q as we know it.ADDITION AND MULTIPLICATION

Addition and multiplication on J will be defined by
(i) [s, m] + [t, n] = [(sn + mt), mn]
(ii) [s, m] · [t, n] = [st, mn]

SUBTRACTION AND DIVISION

Subtraction and division are defined by J by (iii) x - y = x + (-y) for all x, y Î J (iv) x ¸ y = x · y^-1 for all x Î J, y ¹ 0 Î JExample

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