Have you ever noticed that some division problems seem to go on forever? 1 divided by 7 is one of them. The answer is a repeating decimal with a surprisingly neat pattern.
What Is 1 Divided by 7?
The digits 142857 repeat forever, so we can also write the answer as:
In fraction form, the answer is simply 1/7, which is called one-seventh.
How Do You Calculate 1 ÷ 7?
Using long division:
| Step | Calculation | Remainder |
|---|---|---|
| 1 | 10 ÷ 7 = 1 | 3 |
| 2 | 30 ÷ 7 = 4 | 2 |
| 3 | 20 ÷ 7 = 2 | 6 |
| 4 | 60 ÷ 7 = 8 | 4 |
| 5 | 40 ÷ 7 = 5 | 5 |
| 6 | 50 ÷ 7 = 7 | 1 |
So the digits after the decimal point are:
1 → 4 → 2 → 8 → 5 → 7
giving:
1 ÷ 7 = 0.142857142857...
The remainder eventually becomes 1 again. Once that happens, the same sequence of division steps starts over, which is why the digits repeat.
Why Does 1/7 Have a Repeating Decimal?
A decimal repeats when the long-division process returns to a remainder it has already had.
For 1 ÷ 7, the remainders cycle through:
There are six digits in the repeating block, 142857. This is a direct consequence of the long-division process: once a previous remainder returns, the following digits repeat as well.
Is 0.142857 the Exact Answer?
0.142857 is a rounded version of the answer to six decimal places. The exact decimal continues:
Therefore, the most accurate ways to write the answer are:
1/7
0.\overline{142857}
0.142857
This distinction is useful whenever a calculator displays only a limited number of decimal places.
A Fun Fact About 142857
The repeating block 142857 has an interesting pattern. When multiplied by numbers from 1 through 6, its digits cycle:
Quick Answer
1 divided by 7 equals 1/7, or 0.142857142857... as a repeating decimal. The repeating sequence is 142857.
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