Recurring Decimals

What a recurring decimal is

When you turn a fraction into a decimal by dividing, one of two things happens. Either the division stops (a terminating decimal like 1/4 = 0.25), or it never stops and a pattern repeats forever (a recurring decimal like 1/3 = 0.3333...).

A recurring decimal has a digit, or a block of digits, that repeats without end.

Dot notation

Writing endless digits is clumsy, so we place dots over the repeating part:

Decimal	Dot notation	Meaning
0.3333...	0.3̇	the 3 repeats
0.7777...	0.7̇	the 7 repeats
0.121212...	0.1̇2̇	the block "12" repeats
0.41666...	0.416̇	only the 6 repeats
0.142857142857...	0.1̇42857̇	the block "142857" repeats

A dot sits over the first and last digit of the repeating block; everything between repeats too.

Why some fractions recur

Divide 1 ÷ 3 by long division and you keep getting a remainder of 1, which forces the 3 to repeat forever. Whether a fraction terminates depends entirely on its denominator (in simplest form):

• If the denominator's only prime factors are 2 and 5, the decimal terminates (because 10 = 2 × 5).
• If the denominator has any other prime factor (3, 7, 11...), the decimal recurs.

Fraction	Denominator factors	Result
1/8	2 × 2 × 2	0.125 (terminates)
1/20	2 × 2 × 5	0.05 (terminates)
1/3	3	0.3̇ (recurs)
1/6	2 × 3	0.16̇ (recurs)
1/7	7	0.1̇42857̇ (recurs)

Because a recurring decimal always comes from a fraction, it is a rational number — it can be written exactly as a fraction.

Converting a recurring decimal to a fraction

The standard method uses algebra to cancel the repeating tail.

Worked example 1: write 0.7̇ as a fraction

1. Let x = 0.7777...
2. One digit repeats, so multiply by 10: 10x = 7.7777...
3. Subtract the first equation from the second:
4. 10x − x = 7.7777... − 0.7777...
5. 9x = 7
6. So x = 7/9.

The repeating tails are identical, so subtracting wipes them out — that is the whole trick.

Worked example 2: write 0.1̇2̇ as a fraction

1. Let x = 0.121212...
2. Two digits repeat, so multiply by 100: 100x = 12.121212...
3. Subtract: 100x − x = 12.121212... − 0.121212... → 99x = 12.
4. x = 12/99 = 4/33 (dividing top and bottom by 3).

Choosing the power of 10: match it to the length of the repeating block — multiply by 10 for one repeating digit, 100 for two, 1000 for three, and so on, so that one full block shifts past the point.

A harder case: non-repeating part first

Worked example 3: write 0.41666... (0.416̇) as a fraction

Here "41" does not repeat but "6" does. Shift so the repeating parts line up:

1. Let x = 0.41666...
2. Multiply so the point sits just before the repeat: 10x = 4.1666...
3. Multiply again to move one repeat block: 100x = 41.666...
4. Subtract: 100x − 10x = 41.666... − 4.1666... → 90x = 37.5.
5. x = 37.5/90 = 375/900 = 5/12.

A famous surprise: 0.9̇ = 1

Apply the method to 0.9999...:

• x = 0.9999..., 10x = 9.9999..., subtract → 9x = 9 → x = 1.

So 0.9̇ is exactly equal to 1 — not "almost". This is a true equality, and a great talking point.

Try it yourself

Convert these to fractions using the algebra method, then simplify:

1. 0.5̇ (0.555...)
2. 0.2̇7̇ (0.272727...)
3. 0.83̇ (0.8333...)

(Answers: 5/9; 3/11; 5/6.)

Why this matters

Recurring decimals connect decimals, fractions and rational numbers, and the conversion trick is a favourite exam topic. Build on it with decimals explained and the rational-versus-irrational ideas in surds and irrational numbers.