How to prove that $y = 0,273273273...,$ is a rational number?

Question

How to prove that $$y = 0.273273273...$$ is a rational number?

I don't have any experience with proofs... Can I get your help and your advice?

You could start by observing that its $273\times0.001001001001...$, and that $999\times0.001001001...=0.9999....=1$. — skyking, Oct 13 '15 at 13:31

score 6 · Answer 1 · answered Oct 13 '15 at 13:30

6

$1000y = 273.273273...$. Therefore, $1000y-y = 273$. Solving for $y$, $y = \frac{273}{999}$.

answered Oct 13 '15 at 13:30

Eric Brooks

466

score 5 · Accepted Answer · answered Oct 13 '15 at 13:33

5

Since $y$ has a repeating cycle of $3$ digits, just write: $\;1000y=273+y$, solve for $y$, and simplify.

This procedure, though intuitively clear, is not rigourously justified, but it can easily be justified with the interpretation of the positional notation for numbers via a geometric series.

answered Oct 13 '15 at 13:33

Bernard

175,478

Thank you Bernard and all! – Always learning Oct 16 '15 at 16:42

How to prove that $y = 0,273273273...,$ is a rational number?

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