Rational Numbers

Integers allow us to subtract a larger number from a smaller one.

1 − 2 = −1

But what happens when we divide?

1 ÷ 2 = ?

The result is:

1/2

But 1/2 is not an Integer.

So Integers are not enough to represent every result of division.

KEY IDEA

A Rational Number can be written as a fraction of two integers.

p/q

where p and q are integers and q ≠ 0.

EXAMPLES

1/2 3/4 −5/2

Even a whole number such as 6 is Rational because it can be written as:

6 = 6/1

Why can't q be zero?

Division can be understood as multiplication by a multiplicative inverse.

6 ÷ 2 = 6 × 1/2

So dividing by zero would require a number that acts as the multiplicative inverse of zero.

0 × ? = 1

But every number multiplied by zero is zero.

0 × any number = 0

Therefore, no number can be multiplied by zero to give 1.

q ≠ 0

TAKEAWAY

Rational Numbers are numbers that can be written in the form p/q, where p and q are integers and q ≠ 0.

BUT IS THIS ENOUGH?

We now have numbers that can be written as fractions of integers.

But can every number be written in this form?

Can every number be written as p/q?

The answer will lead us to another kind of number.