We prove that every rational number \(\dfrac{a}{b}\) can be written as a terminating or recurring decimal. The above examples serve as illustrations of the proof.

Proof

At each step of the division of a by b there is a remainder. If the remainder is 0, then the division algorithm stops and the decimal is a terminating decimal.

If the remainder is not 0, then it must be one of the numbers 1, 2, 3, …, b − 1. (In the above example, b = 7 and the remainders can only be 1, 2, 3, 4, 5 and 6.) Hence the remainder must repeat after at most b − 1 steps.

Only terminating decimals can be entered in a calculator.

Example 3

Find 4.543 ÷ 0.6.

Solution

Multiply each number by 10 to get a divisor that is a whole number.

Hence:

4.543 ÷ 0.6 becomes 45.43 ÷ 6.

Hence \(4.543 ÷ 0.6 = 7.571\dot{6}.\)