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Math Help - Show that rational numbers correspond to decimals

  1. #1
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    Show that rational numbers correspond to decimals

    Show that rational numbers correspond to decimals which are either repeating or terminating.

    Hint: If q = m|n, then when dividing m by n to put q into decimal form there are at most n different remainders. Conversley, if d is a repeating decimal, then find s,t such that 10^sd - 10^td is an integer.

    I took an example and understand, but don't know how to write it in a general form.

    My example:

    Part I
    2/7
    0.2857142...
    72.000...
    -14
    60
    -56
    40
    -35
    50
    -49
    10
    - 7
    30
    -28
    20
    -14
    6....
    So the my bold numbers can only be 1-6

    Part II
    R = 12.34545...
    100R = 1234.54545...
    100R - R = 1222.2
    99R = (12222 / 10)
    R = (12222 / 990)

    Any help on writing this in a general form would be greatly appreciated.
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  2. #2
    MHF Contributor

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    I would recommend that you interpret "terminating decimal" as a "repeating decimal" where the repeating part happens to be 0. That way, you can simply show that any rational number is a "repeating" decimal.

    let m and n be two integers. Then \frac{m}{n} means "m divided by n". If m> n, then there exist q and r ("quotient" and "remainder") such that m= qn+ r with r< n. That is, \frac{m}{n}= q+ \frac{r}{n} so it is sufficient to prove this for \frac{r}{n} where r is less than n. When we "long divide" n into r, we will have, at each decimal place, a "remainder" that is less than n and larger than or equal to 0, then bring down a "0" and continue. The critical point is that there can only be n different remainders, 0 to n-1 as I said. After at most n steps, a remainder must repeat, we bring down a "0" again so we are dividing n into exactly the same number as before and everything repeats.
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  3. #3
    Grand Panjandrum
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    What Halls has shown is that for any rational there is some repeating decimal which is equal to it. We can also show that any repeating decimal is rational:

    Let $$x be a repeating decimal, we can split it into two parts:

    x=a+10^{-l} p

    where $$a is the terminating non periodic part of length $$ l digits and p\in[0,1] is the purely periodic.

    Let the period be $$d digits long and the terminating decimal with $$d digits which are equal to $$p terminated after the $$d -th be $$u digit then:

    \displaystyle p=\sum_{n=0}^{\infty} u \times 10^{-n\times d}=u\sum_{n=0}^{\infty}  10^{-n\times d}

    Now the sum on the right is a convergent geometric series and so we can write down its sum, and it is rational, and as $$u is rational $$p is rational. Hence as $$a is rational $$x is rational.


    CB
    Last edited by CaptainBlack; February 10th 2011 at 10:17 PM.
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  4. #4
    MHF Contributor

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    On a different forum, many years ago, a person asked how to show that any rational number could be written as a fraction with integer numerator and denominator. It took me a moment to realize that he must have been taught that the definition of "rational number" is a number that can be written as a terminating or repeating decimal. Of course, it can be done either way.
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