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Math Help - Proof repeated decimals are rational

  1. #1
    Member Greengoblin's Avatar
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    Proof repeated decimals are rational

    I've just been reviewing stuff on the number sets/lines, and I was looking at why repeated decimals are rational numbers. I knew this before, but have never seen a proof. I know there is one method you can use to prove specific examples:

    e.g:


    r = 4.27519191919...
    100r = 427.519191919...
    100r-r = 423.244 = 99r
    r = 423244/99000
    423244 and 99000 are both integers, so r is rational.

    I was wondering whats the general proof of this for any repeating decimal....since this method requires that you pick an exmple.

    Another thing I don't understand is how this is solved in the first place, since when you input it in the calculator, you only input the value to a certain number of deciml places, and it only computes the answer to a finite number of decimal places too, so you always get a terminating decimal which can be expressed as a rational number p/q where p,q are integers. For example, I got the above example from a book, but when I typed it in myself, the calculator gave a longer decimal expansion (the above answer, 423.244 is rounded).
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Greengoblin View Post
    I've just been reviewing stuff on the number sets/lines, and I was looking at why repeated decimals are rational numbers. I knew this before, but have never seen a proof. I know there is one method you can use to prove specific examples:

    e.g:


    r = 4.27519191919...
    100r = 427.519191919...
    100r-r = 423.244 = 99r
    r = 423244/99000
    423244 and 99000 are both integers, so r is rational.

    I was wondering whats the general proof of this for any repeating decimal....since this method requires that you pick an exmple.

    Another thing I don't understand is how this is solved in the first place, since when you input it in the calculator, you only input the value to a certain number of deciml places, and it only computes the answer to a finite number of decimal places too, so you always get a terminating decimal which can be expressed as a rational number p/q where p,q are integers. For example, I got the above example from a book, but when I typed it in myself, the calculator gave a longer decimal expansion (the above answer, 423.244 is rounded).
    This is by no means a proof, but I would say that all repeating decimals may be represented as an infinite series. Which entails an infinite amount of rational numbers being added together. Thus rational.
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  3. #3
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    To add more on Mathstud's post, take a look at this:
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Mathstud28 View Post
    This is by no means a proof, but I would say that all repeating decimals may be represented as an infinite series. Which entails an infinite amount of rational numbers being added together. Thus rational.
    A repeating decimal can be represented as a rational and the sum of a rational geometric series, which is known has a rational sum.

    This:

    "Which entails an infinite amount of rational numbers being added together. Thus rational"

    is nonesense, since every real can be represented as a countable sum of rationals.

    RonL
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  5. #5
    Member Greengoblin's Avatar
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    Ahhh, thanks, I didn't think about using infinite series. Thanks alot.
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