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Math Help - Representing 0.999999999... as an integer fraction

  1. #1
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    Representing 0.999999999... as an integer fraction

    Well I just decided to refresh my math skills so started to reread an old Calculus book I have had for a while.

    I came across an end of chapter problem that inquired whether
    0.999999... < 1.0000000

    It asked to represent 0.9999999... as a fraction of integers to arrive at an answer

    So I did this

    x = 0.9999999999

    100 x = 99.9999999999
    - x = - 0.9999999999
    99x = 99
    x = 1

    The second part of the question referred to finding a real number between 0.9999999999 and 1.000000000

    So I did this 1/2(0.9999999999+1)
    =1/2(1.9999999999)
    =0.9999999999

    Why would I get 1 when I tried to represent 0.9999999999... ( is it a rational number with repeating digits) as a fraction of integers
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  2. #2
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    Do you understand that there is a difference between 0.99999 and 0.99999\cdots=0.9999\overline{9}?
    Now 0.999999 is a number between those two.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Do you understand that there is a difference between 0.99999 and 0.99999\cdots=0.9999\overline{9}?
    Now 0.999999 is a number between those two.
    Actually I didn't understand the difference thus the confusion

    So then, how would I represent 0.99999 as a fraction of two integers.

    Your help will be appreciated
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  4. #4
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    Quote Originally Posted by dexteronline View Post
    Actually I didn't understand the difference thus the confusion

    So then, how would I represent 0.99999 as a fraction of two integers.

    Your help will be appreciated
    Do you know how to represent 0.9 as a fraction? 0.99? 0.999? etc. ....
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    Do you know how to represent 0.9 as a fraction? 0.99? 0.999? etc. ....
    LOL

    I think I meant to ask how to represent 0.99999... ( an infinite seires of 9s ) as a fraction of two integers
    I am too slow to understand things, please don't be angered if I sound dumb
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  6. #6
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    Quote Originally Posted by dexteronline View Post
    LOL

    I think I meant to ask how to represent 0.99999... ( an infinite seires of 9s ) as a fraction of two integers
    I am too slow to understand things, please don't be angered if I sound dumb
    You found that 1 = 0.9999999......... did you not? So what's wrong with giving 1/1 as the answer?
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