Terminating and Recurring Decimals
Every fraction becomes a decimal that either stops, like \(\frac{1}{4} = 0.25\), or repeats forever, like \(\frac{1}{3} = 0.\dot{3}\). This NSW Year 9 Core maths subtopic revises terminating and recurring decimals from Stage 4 (NESA outcome MA4‑FRC‑C‑01) and adds two new skills.
You will write repeating decimals in dot notation, for one repeating digit and for a repeating block such as \(0.\dot{4}\dot{5}\), read dot notation back as digits, and divide to convert fractions such as \(\frac{5}{12}\) and \(\frac{4}{7}\) to decimals. The new skills are deciding from the prime factors of the denominator whether a fraction terminates, without dividing, and turning a recurring decimal back into a fraction, so \(0.\dot{1}\dot{8} = \frac{2}{11}\). It is part of Year 9 Core Working with Numbers.
Theory
A decimal either stops or repeats for ever. One that stops is terminating; one that repeats is recurring. This Year 9 maths page covers dot notation, converting fractions to decimals, telling from the denominator whether a decimal terminates, and writing a recurring decimal as a fraction.
A terminating decimal stops, such as \(0.35\). A recurring decimal repeats a digit (as in \(0.2333\ldots\)) or a block of digits (as in \(0.417417\ldots\)) for ever.
Dot notation shows what repeats. One repeating digit gets one dot: \(0.2333\ldots = 0.2\dot{3}\). A repeating block gets a dot on its first digit and a dot on its last: \(0.417417\ldots = 0.\dot{4}1\dot{7}\). Digits before the dots happen once. (Some books draw a bar, called a vinculum, over the block instead.)
To change a fraction to a decimal, divide the numerator by the denominator. If a remainder of \(0\) appears, the decimal terminates. If a remainder comes back, the digits repeat.
You can also decide without dividing. Put the fraction in simplest form and break the denominator into prime factors. Only \(2\)s and \(5\)s: terminating. Any other prime: recurring.
Every terminating and every recurring decimal can be written as a fraction, so they are all rational numbers.
Dot notation:
Whether a fraction in simplest form terminates depends only on its denominator:
| Denominator | Prime factors | Decimal is |
|---|---|---|
| \(20\) | \(2 \times 2 \times 5\) | terminating |
| \(24\) | \(2 \times 2 \times 2 \times 3\) | recurring |
| \(25\) | \(5 \times 5\) | terminating |
| \(35\) | \(5 \times 7\) | recurring |
| \(50\) | \(2 \times 5 \times 5\) | terminating |
How to convert and classify decimals
- Simplify the fraction first.
- Decide from the denominator: only \(2\)s and \(5\)s means terminating; any other prime means recurring.
- Divide the numerator by the denominator, bringing down a zero each time, until the remainder is \(0\) or a remainder repeats.
- Write the answer, with dots on the first and last digits of the repeating block.
- Going back from a recurring decimal: name it \(x\), multiply by \(10\), \(100\), ..., subtract, and simplify.
(i) Write \(0.8333\ldots\) and \(0.417417\ldots\) in dot notation.
(ii) Write \(0.2\dot{1}\dot{5}\) out with its repeating digits.
(i) In \(0.8333\ldots\) the \(8\) happens once; only the \(3\) repeats, so one dot goes on the \(3\):
| \(0.8333\ldots\) | \(=\) | \(\mathbf{0.8\dot{3}}\) |
In \(0.417417\ldots\) the block \(417\) repeats, so dot its first and last digits:
| \(0.417417\ldots\) | \(=\) | \(\mathbf{0.\dot{4}1\dot{7}}\) |
(ii) The dots are on the \(1\) and the \(5\), so the block \(15\) repeats. The \(2\) happens once:
| \(0.2\dot{1}\dot{5}\) | \(=\) | \(\mathbf{0.2151515\ldots}\) |
(i) \(\dfrac{7}{20}\)
(ii) \(\dfrac{7}{30}\)
A fraction is a division. Divide the top by the bottom, bringing down a zero each time, and watch the remainders.
(i) \(7 \div 20\):
| \(70 \div 20\) | \(=\) | \(3 \text{ remainder } 10\) |
| \(100 \div 20\) | \(=\) | \(5 \text{ remainder } 0\) |
The remainder is \(0\), so the decimal stops: \(\dfrac{7}{20} = \mathbf{0.35}\), terminating.
(ii) \(7 \div 30\):
| \(70 \div 30\) | \(=\) | \(2 \text{ remainder } 10\) |
| \(100 \div 30\) | \(=\) | \(3 \text{ remainder } 10\) |
| \(100 \div 30\) | \(=\) | \(3 \text{ remainder } 10\) |
The remainder \(10\) keeps coming back, so the \(3\) repeats: \(\dfrac{7}{30} = 0.2333\ldots = \mathbf{0.2\dot{3}}\), recurring.
(i) \(\dfrac{7}{24}\)
(ii) \(\dfrac{7}{25}\)
(iii) \(\dfrac{6}{15}\)
Make sure each fraction is in simplest form, then break the denominator into prime factors.
| \(24\) | \(=\) | \(2 \times 2 \times 2 \times 3\) |
| \(25\) | \(=\) | \(5 \times 5\) |
(i) \(24\) has a factor of \(3\), so \(\dfrac{7}{24}\) is recurring. (ii) \(25\) has only \(5\)s, so \(\dfrac{7}{25}\) is terminating.
(iii) \(\dfrac{6}{15}\) is not in simplest form. Simplify first:
| \(\dfrac{6}{15}\) | \(=\) | \(\dfrac{2}{5}\) |
The denominator \(5\) has only a \(5\), so it is terminating (\(0.4\)). Testing \(15 = 3 \times 5\) without simplifying would give the wrong answer.
Give the number a name:
| \(x\) | \(=\) | \(0.363636\ldots\) |
Two digits repeat, so multiply by \(100\) to shift the block two places:
| \(100x\) | \(=\) | \(36.363636\ldots\) |
Subtract the first line from the second. The endless tails cancel exactly:
| \(100x - x\) | \(=\) | \(36\) |
| \(99x\) | \(=\) | \(36\) |
| \(x\) | \(=\) | \(\dfrac{36}{99}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{4}{11}}\) |
Check: \(4 \div 11 = 0.3636\ldots\). For one repeating digit, multiply by \(10\) instead.
Common pitfalls
Frequently asked questions
What is the difference between a terminating and a recurring decimal?
A terminating decimal stops, such as \(0.35\). A recurring decimal repeats a digit or block of digits for ever, such as \(0.2333\ldots\)
How do you write a recurring decimal with dots?
Put a dot over a single repeating digit. For a repeating block, put a dot over its first and last digits: \(0.417417\ldots = 0.\dot{4}1\dot{7}\).
How can you tell if a fraction will be a terminating decimal?
Write the fraction in simplest form and find the prime factors of the denominator. If they are only \(2\)s and \(5\)s, the decimal terminates. Any other prime factor makes it recur.
How do you convert a recurring decimal to a fraction?
Call the decimal \(x\). Multiply by \(10\), \(100\) or \(1000\) to move one repeating block past the point, subtract \(x\) so the repeating parts cancel, then solve for \(x\) and simplify.
Are recurring decimals rational numbers?
Yes. Every recurring decimal can be written as a fraction of two whole numbers, so it is rational. For example, \(0.\dot{7} = \dfrac{7}{9}\).