Question

Is `71/43` a rational number?

Answer 1

Yes.

Explanation:

A rational number is a number expressible in the form `p/q` for integers `p, q` with `q != 0`.

To get a decimal representation you can long divide `71/43`.

During the long division, the running remainder will take one of the values `0, 1, 2, 3,... 42`. SInce there are only finitely many possibilities, the remainder must eventually repeat. If the decimal representation is non-terminating then the running remainder will not take the value `0` and the length of the repeating decimal will probably be a factor of `42` since `43` and `10` are co-prime.

I suspected that this one would repeat after `42` decimal places, but it actually repeats after `21`.

`71/43 = 1.bar(651162790697674418604)`