One number can have two decimal names
Write 0.9, then 0.99, then 0.999. Each is a little short of 1. What if the 9s never stop?
Step 1 of 5
- Call the number x. So x = 0.999…
- Multiply by 10. Then 10x = 9.999…
- Split 9.999… as 9 + 0.999…
- The 0.999… part is x. So 10x = 9 + x.
- Take x from both sides. Then 9x = 9, so x = 1.
The rule
Rule: The decimal 0.999… with endless 9s is exactly 1. A positive decimal that stops has a second name in endless 9s.
The gap to 1 is how far a decimal is below 1. After one 9, the gap is 0.1. After two 9s, the gap is 0.01. Each new 9 makes the gap ten times smaller. Endless 9s make the gap smaller than any positive number. Only 0 is that small.
| Nines | Decimal | Gap to 1 |
|---|---|---|
| 1 | 0.9 | 0.1 |
| 2 | 0.99 | 0.01 |
| 3 | 0.999 | 0.001 |
Any decimal that stops
Take 0.5. Lower its last non-zero digit by 1. That gives 0.4. Now add endless 9s: 0.4999…
Check it. 0.4999… is 0.4 plus 0.0999…. Now 0.0999… is one tenth of 0.999…, which is 1. So 0.0999… is 0.1, and the total is 0.5.
| Stops | Second name |
|---|---|
| 1 | 0.999… |
| 0.5 | 0.4999… |
| 3.25 | 3.24999… |
| 0.12 | 0.11999… |
Compare two decimals. Are they the same number?
Loading lab…