Repeating Decimals
“A repeating decimal is a fraction in different clothes.”
The formula
0.ddd… = d/9, 0.abab… = ab/99How to read it: one repeating digit becomes the digit over 9; two repeating digits become them over 99
- 0.ddd…
- — a repeating decimal where one digit d repeats forever
- d/9
- — the repeating digit divided by 9
- ab/99
- — if a two-digit block ab repeats, divide by 99
The hook
A decimal that runs on forever like 1/3 = 0.333… is really just a tidy fraction wearing a different face.
In plain words
A decimal where the same block of digits repeats forever after the point. It can always be turned back into a fraction.
The intuition
Divide 1 by 3 and the remainder keeps cycling in the same pattern, giving 0.333…. There are only finitely many possible remainders, so a repeat is inevitable — which is why every fraction becomes a decimal that either stops or repeats.
How it's built
The trick to convert back: let x = 0.ddd…, multiply by 10, and subtract. For x=0.333…, 10x=3.333…, 10x−x=3, 9x=3, x=3/9=1/3. That's where the 'digit over as many 9s as repeating places' rule comes from.
Example
Turn 0.777… into a fraction. x=0.777…, 10x=7.777…, subtract to get 9x=7, so x=7/9. And indeed 7÷9=0.777….
Common mistake
0.999… is not merely 'close to' 1 — it is exactly 1. If x=0.999…, then 10x−x=9, so 9x=9 and x=1. It's just a second way to write the same number.
Where it's used
Converting between fractions and decimals, figuring out the identity of a decimal on your calculator, and understanding why every rational is a 'stopping or repeating' decimal.
Where it came from
That dividing a fraction produces a cycling remainder has been known for ages, and notation settled on marking the repeating block with a dot or bar overhead.
Prerequisites
Quick check
Written as a fraction, what is 0.555…?
- 5/9✓
- 5/10
- 1/5
- 5/99
Practice
Written as a fraction, what is 0.888…?
Answer: 0
- One digit 8 repeats → put it over 9
- 0.888… = 8/9
Takeaway: A one-digit repeating decimal is that digit over 9.
Written as a fraction, what is 0.121212…?
Answer: 0
- The two-digit block 12 repeats → put it over 99
- 0.121212… = 12/99
Takeaway: Two repeating digits go over 99 (one 9 per repeating place).
Turn 0.444… into a fraction.
Answer: undefined
- Let x = 0.444…, then 10x = 4.444…
- 10x − x = 4, so 9x = 4
- x = 4/9
Takeaway: Multiplying by 10 and subtracting cancels the repeating tail.
Show by calculation that 0.999… = 1.
Answer: undefined
- Let x = 0.999…, then 10x = 9.999…
- 10x − x = 9.999… − 0.999… = 9
- 9x = 9 → x = 1
Takeaway: 0.999… and 1 are two notations for the same number.