Terminating and Recurring Decimals
Every question with a fully worked solution.
- Terminating and Recurring Decimals - Video - Terminating and Repeating Decimals Watch
Theory
A fraction is a division. Dividing the numerator by the denominator gives a decimal that either stops (a terminating decimal) or repeats forever (a recurring decimal). This Year 7 maths page shows how to tell which, how to write recurring decimals with dots or a bar, how to round them and why both kinds are rational numbers.
To write \(\dfrac{3}{8}\) as a decimal, work out \(3 \div 8\). Write \(3\) as \(3.000\ldots\) and divide one place at a time.
Terminating decimals stop: they have a definite number of decimal places, because the remainder becomes \(0\): \(\dfrac{3}{8} = 0.375\). Terminating means stopping.
Recurring decimals have one or more digits that repeat forever, because a remainder comes back: \(\dfrac{1}{3} = 0.333\ldots\) Recurring means repeating.
Dot notation puts a dot over the first and last digit of the repeating block: \(0.333\ldots = 0.\dot{3}\) and \(0.345345\ldots = 0.\dot{3}4\dot{5}\). Bar notation puts a bar over the whole block: \(0.\overline{345}\).
A rational number can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). Every terminating or recurring decimal is rational: \(0.9 = \dfrac{9}{10}\).
Writing recurring decimals. Only the repeating block gets dots or a bar:
| Written out | Dots | Bar |
|---|---|---|
| \(0.333\ldots\) | \(0.\dot{3}\) | \(0.\overline{3}\) |
| \(0.2666\ldots\) | \(0.2\dot{6}\) | \(0.2\overline{6}\) |
| \(0.3636\ldots\) | \(0.\dot{3}\dot{6}\) | \(0.\overline{36}\) |
| \(0.345345\ldots\) | \(0.\dot{3}4\dot{5}\) | \(0.\overline{345}\) |
| \(0.142857142857\ldots\) | \(0.\dot{1}4285\dot{7}\) | \(0.\overline{142857}\) |
Dividing \(1\) by \(7\), the remainders go \(1, 3, 2, 6, 4, 5\) and then \(1\) comes back, so the digits \(142857\) repeat. Once a remainder repeats, the digits must repeat.
How to change a fraction to a decimal
- Divide the numerator by the denominator, writing the numerator as \(5.000\ldots\)
- Remainder 0: the decimal terminates.
- A remainder comes back: the digits repeat. Mark the repeating block with dots or a bar.
How to write a repeating pattern in dot notation
- Find the digits that do not repeat, then the repeating block.
- Put a dot over the first and the last digit of the block only.
(i) \(\dfrac{5}{8}\)
(ii) \(\dfrac{5}{9}\)
(i) \(50 \div 8\) is \(6\) remainder \(2\), \(20 \div 8\) is \(2\) remainder \(4\), and \(40 \div 8\) is exactly \(5\).
(ii) \(50 \div 9\) is \(5\) remainder \(5\), and the remainder \(5\) comes back every time.
(i) \(0.625\), terminates
(ii) \(0.555\ldots = 0.\dot{5}\), recurs
Write \(4\) as \(4.0000\ldots\) \(40 \div 11\) is \(3\) remainder \(7\); \(70 \div 11\) is \(6\) remainder \(4\); then \(40\) comes back.
\(\dfrac{4}{11} = 0.363636\ldots = 0.\dot{3}\dot{6}\). It is a recurring decimal.
\(70 \div 15\) is \(4\) remainder \(10\); \(100 \div 15\) is \(6\) remainder \(10\), and \(10\) comes back. Only the \(6\) repeats.
\(\dfrac{7}{15} = 0.4666\ldots = 0.4\dot{6}\). The fourth digit is \(6\), so round up: \(0.467\).
(ii) Write \(0.\dot{6}0\dot{3}\) out to show the pattern.
(i) The \(3\) comes once; then the block \(18\) repeats.
(ii) The block is \(603\), with no digits before it.
(i) \(0.3\dot{1}\dot{8}\)
(ii) \(0.603603603\ldots\)
(ii) Given \(0.\dot{1} = \dfrac{1}{9}\), write \(0.\dot{7}\) as a fraction.
| \(0.35\) | \(=\) | \(\dfrac{35}{100} = \dfrac{7}{20}\) |
| \(0.\dot{7}\) | \(=\) | \(7 \times 0.\dot{1} = \dfrac{7}{9}\) |
Both are integers over integers, so both decimals are rational.
Common pitfalls
Frequently asked questions
What is a terminating decimal?
A decimal that stops, such as \(\dfrac{5}{8} = 0.625\).
What is a recurring decimal?
A decimal whose digits repeat forever, such as \(\dfrac{5}{9} = 0.555\ldots\)
How do you know when the digits will repeat?
When a remainder comes back during the division, the digits from that point repeat.
Where do the dots go?
Over the first and last digit of the repeating block: \(0.\dot{3}4\dot{5}\).
Is a recurring decimal a rational number?
Yes. It can be written as a fraction, for example \(0.\dot{7} = \dfrac{7}{9}\).