How write a periodic number as a fraction? [duplicate]

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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 $a,b\in\mathbb Z$ ?

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3 Answers

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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.

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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.

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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.

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