How can you transform a decimal such as $\displaystyle 2,345345345...$ as a fraction?

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- Sep 29th 2007, 07:54 AMp.numminenDecimal into a fraction
How can you transform a decimal such as $\displaystyle 2,345345345...$ as a fraction?

- Sep 29th 2007, 08:40 AMred_dog
$\displaystyle 2,345345345\ldots=2,(345)=2+\frac{345}{999}$

and you continue... - Sep 29th 2007, 08:56 AMp.numminen
Why not $\displaystyle 2,345345345\ldots=2+\frac{345}{1000}$ ?

- Sep 29th 2007, 11:25 PMJhevon
for the obvious reason, $\displaystyle 2 + \frac {345}{1000} = 2.345$, the decimal ends there, it does not repeat. you need 999 as the denominator to get the repeating decimal

a simpler way, very common trick, i see it around here all the time...

let $\displaystyle x = 2.\overline{345}$ ..................................(1)

$\displaystyle \Rightarrow 1000x = 2345. \overline{345}$ .......................(2)

subtracting equation (1) from (2), we see that:

$\displaystyle 1000x - x = 2345. \overline{345} - 2. \overline{345}$

$\displaystyle \Rightarrow 999x = 2343$

$\displaystyle \Rightarrow x = \frac {2343}{999}$ - Sep 29th 2007, 11:26 PMFailbait
Because your number isn't finite. $\displaystyle \frac{345}{1000}$ would just give you $\displaystyle 2.345$.

- Sep 29th 2007, 11:26 PMearboth
hello,

to show you where red_dog got the denominator from I'll use the following method.

Let x = 2.345345345... that means there are 3 digits which are repeated immidiately. Therefore you have to multiply your number by 10³. Now you have

$\displaystyle \boxed{\begin{array}{lcr}1000x&=&2345.345345345... \\ x&=&2.345345345...\end{array}}$

Subtract and you have

$\displaystyle 999x = 2343~\iff~x = \frac{2343}{999}$