Decimal into a fraction

**p.numminen** · September 29th 2007, 07:54 AM

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

**red_dog** · September 29th 2007, 08:40 AM

$2,345345345\ldots=2,(345)=2+\frac{345}{999}$
and you continue...

**p.numminen** · September 29th 2007, 08:56 AM

Why not $2,345345345\ldots=2+\frac{345}{1000}$ ?

**Jhevon** · September 29th 2007, 11:25 PM

Originally Posted by p.numminen

Why not $2,345345345\ldots=2+\frac{345}{1000}$ ?

for the obvious reason, $2 + \frac {345}{1000} = 2.345$ , 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 $x = 2.\overline{345}$ ..................................(1)

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

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

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

$\Rightarrow 999x = 2343$

$\Rightarrow x = \frac {2343}{999}$

**Failbait** · September 29th 2007, 11:26 PM

Because your number isn't finite. $\frac{345}{1000}$ would just give you $2.345$ .

**earboth** · September 29th 2007, 11:26 PM

Originally Posted by p.numminen

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

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

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

Subtract and you have

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

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