## Decimals

The decimal system is a means of expressing numbers in the base of ten.

### Types of Decimals

• terminating decimal is one that has a finite number of digits.
• e.g. 0.5 and 0.875

• recurring or repeating decimal is one that has a repeating sequence of digits. Recurring decimals are shown by a dot above the recurring digits or at the beginning and end of the repeating sequence.

e.g.

 0.3333... 0.1666... 0.207207...

All rational numbers can be represented by terminating or recurring decimals.

• non-repeating decimal is a decimal that contains a non-repeating sequence of decimal digits.
• e.g. 0.810675469... does not repeat.

Irrational numbers (e.g.  and π) can be represented by non-repeating decimals.

### Operations

Calculators can be used to carry out the following operations involving decimals.

However, it is useful to be able to do these basic operations without a calculator.

Make sure that the decimal points are in line.

 Examples Answers Calculate: (a) 4.5 + 3.62 (b) 2.34 − 0.73

### Multiplication

• To multiply by multiples of 10, move the decimal point to the right.
• When multiplying two decimal numbers, carry out the calculation ignoring the decimal points. Place the decimal point in the answer so that the answer has the same number of decimal places as the total number of places in the two numbers being multiplied.

 Examples Answers Calculate: (a) 0.27 × 100 (a) (b) 3.4 × 1000 (b) (c) 0.4 × 0.5 (c) 0.4 × 0.5 = 0.20 (d) 3.6 × 6 (d) 3.6 × 6 = 21.6 (e) 1.3 × 1.2 (e) 1.3 × 1.2 = 1.56

### Division

• When dividing by multiples of 10, move the decimal point to the left.
• When dividing two decimal numbers, write the calculation as a fraction.

Move the decimal point in both the numerator and the denominator the same number of decimal places needed to make the bottom line into a whole number. Then carry out normal division.

 Example Answers Calculate: (a) 16.5 ÷ 10 (a) (b) 152 ÷ 100 (b) (c) 31.5 ÷ 5 (c) (d) 34.56 ÷ 0.4 (d)