The Rational Numbers

by George E. Hrabovsky, President, MAST

Where We Have Been

We have been continuing our focus on the foundations of mathematics. We have discussed the basic ideas of set theory and logic, proof techniques, quantifiers, and have begun to develop the number system. Last column we discussed the properties of natural numbers and integers, and we introduced the idea of a preimage problem.

Where Will We Go in This Column

In this column we will continue our exploration of preimage problems to include the problem of division.

Division

In general if we have x and y and we want to divide x by y we get z. Another way of saying this is that

x ÷ y = z ==> x × z = y .

This is all well and good so long as z ∈ ÷½. What happens if we decide to take 4 ÷ 3? Or even 1 ÷ 3? Clearly the results of these divisions are not within the set of integers. In the first example we can choose to do the division and leave a remainder,

4 ÷ 3 = 1 with 1 remainder .

This is fine if we allow the remainder, but what about our second example? There is no way to break a number into smaller pieces. Once again, it seems as though we need to invent a new kind of number.

Rational Numbers

If we use division as a base, we can make the following definition,

x ÷ y ≡ x/y .

This is called a fraction. If we further state that both x and y must be integers then the fraction is a member of the set of rational numbers, ÷´. The set of integers is then the preimage of the set of integers.

Some Properties of Rational Numbers

Here are some operations on rational numbers. We begin with addition,

x/y + a/b = x × b/y × b + a × y/y × b .

To shorten the notation for multiplication we will adopt the convention of stating that in formulas a space between symbols representing numbers will indicate multiplication, 4 × 4 = 4 4 = 16.

x/y + a/b = (x b)/(y b) + (a y)/(y b)                     = (x b + a y)/(y b) .

Subtraction works in a similar way,

x/y - a/b = (x b)/(y b) - (a y)/(y b)                     = (x b - a y)/(y b) .

Multiplication is easy,

x/y a/b = (x a)/(y b) .

Division is easy, too. Since division is just the inverse of multiplication, we can treat it as

x ÷ y = x 1/y = x/y .

So,

x/y ÷ a/b = x/y 1/(a/b) = x/y b/a = (x b)/(y a) .

•Suggested Practice Problems

1. Are the operations of addition, subtraction, multiplication, and division closed for the set of rational numbers?
2. Are the operations of addition, subtraction, multiplication, and division associative for the set of rational numbers?
3. Are the operations of addition, subtraction, multiplication, and division commutative for the set of rational numbers?

Converted by Mathematica  (January 2, 2003)