Is #71/43# a rational number?

1 Answer
Sep 28, 2016

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)#