Why some fractions repeat for ever as decimals.
Why 1/8 ends but 1/3 and 1/7 repeat for ever, how to tell from the bottom alone, and how long the repeating part can be.
By Suraj Kumar3 min read
Divide 1 by 8 and it stops at 0.125. Divide 1 by 7 and 142857 comes round for ever. The bottom of the fraction decides which.
Only twos and fives end.
Our decimals count in tens, and 10 is 2 × 5. A fraction's decimal ends only when its bottom, simplified, has no prime factors but 2 and 5: so 1/8, 3/20 and 7/40 end, while 1/3, 1/6 and 1/7 repeat.
Why the digits come round again.
In long division by 7, every remainder is one of 1 to 6. As soon as one comes back, the same digits follow again, so the repeat is at most six digits long. The page watches for a remainder it has already seen and marks the repeat with a line over it: 1/7 is 0.(142857).
Some repeats are long.
1/97 repeats 96 digits before coming round. The page finds them all exactly and says how many there are.
Further reading
Topics: fraction to decimal · repeating decimal · why does 1/3 repeat · terminating decimal · 1/7 as a decimal · fraction to decimal chart
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