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Math Help - Decimal into a fraction

  1. #1
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    Decimal into a fraction

    How can you transform a decimal such as 2,345345345... as a fraction?
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  2. #2
    MHF Contributor red_dog's Avatar
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    2,345345345\ldots=2,(345)=2+\frac{345}{999}
    and you continue...
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  3. #3
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    Why not 2,345345345\ldots=2+\frac{345}{1000} ?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by p.numminen View Post
    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}
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  5. #5
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    Because your number isn't finite. \frac{345}{1000} would just give you 2.345.
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  6. #6
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    Quote Originally Posted by p.numminen View Post
    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 10. 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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