A terminating decimal ends at a last digit, and a repeating decimal repeats a block without end. That is why 0.999 equals 999/1000, which is less than one, while 0.9̅, with nines that never stop, equals exactly 1. There is no “almost”: no number fits between 0.9̅ and 1.
This guide answers that question and sets how to mark a repeating block. To turn any repeating decimal into a fraction, use the repeating decimal to fraction tool; to know in advance whether a fraction terminates or repeats, see the 2-and-5 test.
Three notations that get mixed up
| Notation | What it says | Value |
|---|---|---|
| 0.333 | Three decimal places and it stops | 333/1000, less than one third |
| 0.3̅ or 0.333… | The 3 repeats forever | Exactly one third |
| ≈ 0.33 | Rounded to two decimal places | An approximation, not an equality: the error is 1/300 |
The table separates three different things: a number written in full, a repeating block, and a rounding. A bare ellipsis can leave doubt about which block continues, so mark the repeating block with a bar over the digits, as in 0.2̅7̅; where you can’t draw one, use parentheses, 0.(27), which is also the form the calculator accepts. In the US decimals are written with a point; many other countries use a comma, and the calculator accepts both.
Worked example: prove that 0.9̅ = 1
Let x = 0.999… with the nines repeating forever. What is x?
- Multiply by 10. Then 10x = 9.999…, and the tail of nines is identical to the tail of x.
- Subtract. 10x − x = 9.999… − 0.999… = 9, because the tails cancel.
- Solve. 9x = 9, so x = 1.
So x = 1.
Check: 1/3 = 0.3̅. Tripling both sides, 3 × 0.3̅ gives 0.9̅ and 3 × 1/3 gives 1, the same value.
Step 2 works because a repeating block has no last digit: when the decimal point shifts one place, no nine is left over and none is missing.
Why finitely many nines never reach 1
With n nines the value is 1 − 1/10ⁿ, and each new nine cuts the distance to one tenth of what it was without ever erasing it.
| Notation | Value | Short of 1 by |
|---|---|---|
| 0.9 | 9/10 | 1/10 |
| 0.99 | 99/100 | 1/100 |
| 0.999 | 999/1000 | 1/1000 |
| 0.9̅ | 1 | 0 |
Every finite row stays below one: any finite number of nines always leaves a remainder, which is why the subtraction in the example doesn’t work for them. For the infinite repeating block, the difference 1 − 0.9̅ would be less than 1/10ⁿ for every n, and the only nonnegative number with that property is 0.
What a screen shows
A calculator set to few digits may show 1 for 0.999 and 0.67 for 2/3. That is display rounding: 0.67 = 67/100 is slightly more than 2/3. For exact equalities, keep the fraction and round only the final result.
Where to go next
- Fraction to decimal shows whether a fraction gives a terminating decimal or a repeating one.
- To turn a terminating decimal into a fraction, like 0.5, use decimal to fraction; if the decimal repeats, use repeating decimal to fraction.
- The article on 0.1 + 0.2 explains why computers don’t store 0.1 exactly.
Practice
- What is 0.3̅ + 0.6̅?
- Find the distance between 0.999 and 1.
- Write 0.125 with a tail of nines.
Show the answers
- 1, because 1/3 + 2/3 = 1
- 1/1000
- 0.1249̅
Sources
OpenStax, Prealgebra 2e, section 5.3, Decimals and Fractions, accessed October 16, 2026: explains that repeating decimals are abbreviated by marking the digits that repeat. This guide marks them with a bar.