A block that repeats is the number written out
Divide 2 by 7. The digits keep going: 0.285714285714… Why do they come back?
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- Start: divide 2 by 7. The remainder is 2.
- Digit 2. New remainder 6.
- Digit 8. New remainder 4.
- Digit 5. New remainder 5.
- Digit 7. New remainder 1.
- Digit 1. New remainder 3.
- Digit 4. The remainder is 2 again. The block repeats.
The remainder is what repeats
The digits repeat because a remainder comes back. A remainder is what is left at each step of the division. The remainder decides the next digit and nothing else. When a remainder comes back, the next digits come back too.
The digits between the two matching remainders are the repeating block. Here it is 285714.
Every step leaves a remainder below 7. A remainder of zero would end the division. So a repeating division uses only 1 to 6. There are only six choices. Within seven steps, a remainder must come back.
| Dividing by | Possible remainders | A block is never longer than |
|---|---|---|
| 7 | 1 to 6 | 6 |
| 17 | 1 to 16 | 16 |
| 19 | 1 to 18 | 18 |
The cap is the longest block possible. Dividing by 7, 17 or 19 reaches it. Some divisors do not. Dividing by 3 has a cap of 2. Yet 1/3 = 0.333… has a block of 1.
2/7 = 0.285714285714… This equality is exact. The decimal is not a rounded guess.