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

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 $a,b\in\mathbb Z$ ?

• yes, sorry. I corrected it.
– idm
Commented May 22, 2015 at 10:15
• Use the geometric series. Commented May 22, 2015 at 10:16
• Call the first one $x$. What is $100x$? What is $13+x$? Commented May 22, 2015 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.
– user65203
Commented May 22, 2015 at 10:22
• 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. Commented May 22, 2015 at 12:56

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.

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.

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.