Rational numbers: fractions, decimals and recurring decimals

2026-10-03Numbers & Estimation Ms Lam

Convert between fractions and decimals, recognise recurring decimals, and keep rational arithmetic organised.

Key formulas

a/b = \text{terminating or recurring decimal}

0.\dot{3} = \frac{1}{3}

0.\dot{1}\dot{6} = \frac{1}{6}

What makes a number rational

A rational number can be written as a fraction a/b where a and b are integers and b ≠ 0. Terminating decimals and recurring decimals are both rational; non-recurring infinite decimals such as √2 are not.

Fraction–decimal conversion

Divide numerator by denominator to get a decimal. If the remainder eventually repeats, use a dot notation over the repeating block. To convert a recurring decimal back to a fraction, use the place-value equation method taught in Secondary 2–3.

Operations with fractions

Add and subtract only after a common denominator. Multiply numerators and denominators; divide by multiplying by the reciprocal. Cancel common factors early to keep numbers small.

Exam habit

Leave answers as simplified improper fractions or mixed numbers exactly as the question asks — do not mix forms without reason.

Worked example

Recurring decimal to fraction

Problem. Express 0.\dot{4} as a fraction in simplest form.

Solution. Let x = 0.444…. Then 10x = 4.444…. Subtract: 9x = 4, so x = 4/9.

Review quiz

1 / 4

7/8 as a decimal is

Practice questions

Express 7/8 as a decimal.
Write 0.\dot{6} as a fraction in simplest form.
Calculate 2/5 + 3/4 and 5/6 ÷ 2/3.

Study notes

Is every decimal rational?

No. Only terminating or recurring decimals are rational.

Should I use a calculator for recurring decimals?

Use exact fraction methods unless the paper allows calculator rounding.