How can you transform a decimal such as $\displaystyle 2,345345345...$ as a fraction?
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:
$\displaystyle 1000x - x = 2345. \overline{345} - 2. \overline{345}$
$\displaystyle \Rightarrow 999x = 2343$
$\displaystyle \Rightarrow x = \frac {2343}{999}$
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
$\displaystyle 999x = 2343~\iff~x = \frac{2343}{999}$