All repeating numbers are rational. There is a relatively simple proof of
this.

Given a number N which ends in a repeating group of 'p' digits:

Then the number (10^p) * N ends with the same repeating block.

Now if we subtract:

     (10^p)*N - N

the 'repeating blocks' cancel out and we have a terminating decimal (the
part of N before the repeating block). Note, of course that this terminating
decimal is rational.

Thus, N(10^p-1) == N(10^p-1) is rational. If we now divide this by (10^p-1)
we have:

            N(10^p-1)
N == -------------------
              10^p-1

Note that both the numerator and denominator are rational, therefore the
entire value is rational, therefore N is rational.

By the way: this proof is from my 11th grade trigonometry book.

Bob Ammerman
RAm Systems
(contract development of high performance, high function, low-level
software)

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