Question

Why do rational numbers repeat?

Answer 1

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` ...

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