Math 7 · Unit 3, Lesson 6: Fractions to Decimals · Tile 2 of 3
Decimals That Repeat
When the remainders start over
When the remainders start over
Try 23. 20 ÷ 3 is 6, remainder 2. Bring down a zero: 20 ÷ 3 is 6, remainder 2. Again. And again. The remainder keeps coming back as 2, so the 6s never stop. When a remainder shows up a second time, the digits after it will repeat forever.
- 20 ÷ 3 is 6, remainder 2.
- Bring down a zero: 20 ÷ 3 is 6, remainder 2. Again.
- And again: 20 ÷ 3 is 6, remainder 2.
- The remainder keeps coming back as 2, so the 6s never stop. 0.666... is written 0.6 repeating.
Words we use
| Elle (math-speak) | Finn (plain English) |
|---|---|
| Repeating decimal | A decimal where a digit or a group of digits repeats forever. A bar over the digits shows which ones repeat. |
Elle and Finn
When a remainder repeats in long division, the decimal repeats. Use a bar over the repeating digits.
Watch the remainders. Once one shows up again, you're in a loop. Put a bar over the part that loops.
Why it has to stop or repeat
When you divide by 7, the only remainders you can get are 0, 1, 2, 3, 4, 5, and 6. If you ever get 0, the decimal stops. If you never get 0, you only have 6 remainders to use, so one of them has to come back. Then the digits start over. That's why 17 = 0.142857 repeating: six digits, then it repeats.
Elle and Finn talk it over
So the threes in one third really go on forever?
Forever. They'll still be going after your homework, your graduation, and the sun burning out.
Then I'm drawing the bar.
Watch it work
In 0.83 repeating, only the 3 repeats, so only the 3 gets the bar.
NC standard: NC.7.NS.2.