Why do rational numbers repeat?

1 Answer
George C.
Apr 10, 2016

See explanation...

Explanation:

Suppose p/q is a rational number, where p and q are both integers and q > 0.

To obtain the decimal expansion of p/q you can long divide p by q.

During the process of long division, you eventually run out of digits to bring down from the dividend p. From that point on, the digits of the quotient are determined purely by the sequence of values of the running remainder, which is always in the range 0 to q-1.

Since there only q different possible values for the running remainder, it will eventually repeat, and so will the digits of the quotient from that point.

For example: 186/7 ...

enter image source here

Notice the sequence of remainders: 4, color(blue)(4), 5, 1, 3, 2, 6, color(blue)(4), 5 which starts to repeat again.

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