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Math Help - Is 0.999... the same as 1??

  1. #1
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    Is 0.999... the same as 1??

    To convert an infinitely recurring decimal to a fraction looks reasonably easy, e.g.:
    let n = 0.222... (i)
    then 10n = 2.222... (ii)
    therefore 9n = (ii) - (i) = 2 ; so n = 2/9
    Thats why they say: if it's a single digit recurring, just put it over 9. But this means that 0.999... = 9/9 = 1!
    I can see that, for all practical purposes, this is so, but surely if we're being rigorous 0.999... is not 1; it never quite gets there.
    This make suspect that the whole system of conversion is suspect. I even think that a recurring decimal didn't ought to be classed as a "rational" number because it can't properly be expressed as a fraction. My thinking must be flawed. Please help me before I go mad.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by MadSnail View Post
    To convert an infinitely recurring decimal to a fraction looks reasonably easy, e.g.:
    let n = 0.222... (i)
    then 10n = 2.222... (ii)
    therefore 9n = (ii) - (i) = 2 ; so n = 2/9
    Thats why they say: if it's a single digit recurring, just put it over 9. But this means that 0.999... = 9/9 = 1!
    I can see that, for all practical purposes, this is so, but surely if we're being rigorous 0.999... is not 1; it never quite gets there.
    This make suspect that the whole system of conversion is suspect. I even think that a recurring decimal didn't ought to be classed as a "rational" number because it can't properly be expressed as a fraction. My thinking must be flawed. Please help me before I go mad.
    Think about this: What does 0.999.. mean?

    The answer is that it is the sum of the series:

    9/10+9/100+ ... = 9(1/10+1/100+ ...)

    which is 1. It means nothing more and nothing less.

    RonL
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