How do you identify a repeating decimal?

Answer 1

In general it is far from easy. There are, of course, simple examples such as 1/3 which give 0.3 with the 3 repeating, or 1/7 = 0.142857 with the underlined string repeating, and so on. But take a look at this number: 0.01030927835051546391752577319587628865979381443298969072164948453608247422680412

371134020618556701030927835051546391752577319587628865979381443298969072164948453

6082474226804123711340206185567...

Actually the number is 1/97 and since the number is rational it must be a repeating decimal. It has a repeating string of 96 digits: the above represents two "laps". The first lap ends, in the second line, just above the letter "e" of the word "number" at the start of this paragraph.

This is the worst with a denominator less than 100. You can get a repeating string of any length by choosing a suitable denominator.