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What I call as a periodic number is for exemple

$$0.\underbrace{13}_{period}131313...$$ or $$42.\underbrace{465768}_{period}465768465768.$$

So how can we put theses numbers like a integer fractional, i.e. of the form $\frac{a}{b}$ with $a,b\in\mathbb Z$ ?

idm
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  • yes, sorry. I corrected it. – idm May 22 '15 at 10:15
  • Use the geometric series. – AlexR May 22 '15 at 10:16
  • Call the first one $x$. What is $100x$? What is $13+x$? – Akiva Weinberger May 22 '15 at 10:16
  • Shift the decimal point by as many digits as the period length is (multiplying by a power of 10) and subtract both numbers. –  May 22 '15 at 10:22
  • This question is not a duplicate of the suggested one, I have voted to reopen. – barak manos May 22 '15 at 10:56
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    This question asks for a method for converting a repeating decimal to a ratio of integers. The linked question asks for a proof that such a ratio exists. While on the face of it that does not actually request the method for finding such a ratio, in fact the obvious answer to the linked question is to demonstrate the method, and that's what the answers did; hence answers to this question are already posted on the linked question. – David K May 22 '15 at 12:56

3 Answers3

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To illustrate the method lets take 0.13131313131313131313... as an example

Let $x = 0.13131313131313131313...$

We now multiply by a suitable power of 10 such that the fractional part is the same. In this case 100

$100x = 13.13131313131313131313... = 13 + x$

Thus $99x = 13 \Rightarrow x = \dfrac{13}{99}$

For your second example you need to multiply by a bigger power of 10 but the method is identical.

Warren Hill
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Adding to the existing answers, note that if your number isn't quite in the right form, you can get it that way easily; so if you want to know about $N=1.02371717171\cdots$, you can write $$1000N=1023+ 0.717171\cdots$$

Apply the method mentioned in the other answers to write the repeating part as a fraction $$1000N=1023 +\frac ab$$ then isolate $N$ again: $$N=\frac{1023}{1000}+\frac a{1000b}$$

Now you will need to combine as indicated to get a single fraction, but that's easy.

MPW
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Multiplying your number by a suitable power of $10$ we can make some parts the nummber jump to the left of the decimal point leaving identical fractional part. That is $10^mx$ and $x$ have the same fractional part. SO their difference is an integer $a$: That is $a= (10^k-1)x$, this shows $x$ is a rational number.

P Vanchinathan
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