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    Prove irrationality

    Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
    is not rational.
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    Quote Originally Posted by fobos3 View Post
    Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
    is not rational.
    Since we know that there are infinite natural numbers, the decimal is non-terminating.

    Since the decimal is written such that the natural numbers are in increasing order, numbers will not repeat themselves.

    Hence 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ... is a non-terminating, non-repeating decimal, hence not a rational.
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    Quote Originally Posted by fobos3 View Post
    Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
    is not rational.
    The harder question is to prove that this number is transcendental.

    By the way, the number is called Champernowne's number.
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    Quote Originally Posted by Isomorphism View Post
    Since we know that there are infinite natural numbers, the decimal is non-terminating.

    Since the decimal is written such that the natural numbers are in increasing order, numbers will not repeat themselves.

    Hence 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ... is a non-terminating, non-repeating decimal, hence not a rational.
    I'm sure the numbers won't repeat themslves but what about the digits. For example 1 2 3 and 123 the digits 1,2,3 repeat. Let me give you another example. Consider the number 0.1 23 123 1231 2312 3123 12312...=0.123 123 123 123...
    Numbers don't repeat but digits do.
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    Quote Originally Posted by fobos3 View Post
    I'm sure the numbers won't repeat themslves but what about the digits. For example 1 2 3 and 123 the digits 1,2,3 repeat. Let me give you another example. Consider the number 0.1 23 123 1231 2312 3123 12312...=0.123 123 123 123...
    Numbers don't repeat but digits do.
    0. 123 123 123 123 123 ... (Usually expressed as 0.\overline{123}) is a rational number because it can be written in terms of \frac{p}{q}. How? Using infinite geometric series:

     \text{S} = \frac{t_{1}}{1-\text{r}}

    Where t_1 is the first term in the series and r is the common ratio between each consecutive term.

    0.\overline{123} = \frac{123}{1000} + \frac{123}{1000000} + \ldots

    r = \frac{\frac{123}{1000000}}{\frac{123}{1000}} = \frac{1}{1000}

    Now substitute in the equation above:
    \text{S} = \frac{t_{1}}{1-\text{r}} =  \frac{\frac{123}{1000}}{1-\frac{1}{1000}} = \frac{\frac{123}{1000}}{\frac{999}{1000}} = \frac{123}{999}
    Last edited by Chop Suey; July 15th 2008 at 09:26 AM.
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    Quote Originally Posted by fobos3 View Post
    I'm sure the numbers won't repeat themslves but what about the digits. For example 1 2 3 and 123 the digits 1,2,3 repeat. Let me give you another example. Consider the number 0.1 23 123 1231 2312 3123 12312...=0.123 123 123 123...
    Numbers don't repeat but digits do.
    A rational number has a decimal expansion which is eventually periodic, that is from some point on there is a string of digits of some length that is just repeated continually.

    That the given number does not have that property just consider how far to the right the digit strings: 10, 100, 1000, 10000, ... first appear. What this shows is that for any given N the expansion has not become periodic before reaching N digits (since one of these strings first appears to the right of the N'th digit. Hence it does not become periodic period.

    RonL
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