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  1. #1
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    Totally Irrational

    If someone were to advocate that the square root of 2 is actually a rational number (a number that can be written as the ratio of two whole numbers), how could you disprove this hypothesis?

    I should probably exclude HansV from this puzzle, since he's probably worked it out "on the fly" already. <img src=/S/grin.gif border=0 alt=grin width=15 height=15>

    Alan

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    WS Lounge VIP sdckapr's Avatar
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    Re: Totally Irrational

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">I would presume it were rational (some number x/y) that are whole numbers, real, and that x/y is a reduced fraction. I would then demonstrate that both x and y were even numbers negating that it could be reduced and proving it could not be true.</span hide>

    Steve

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    Re: Totally Irrational

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Assume that sqr(2) = A/B where A and B are the positive integers and B is the smallest possible. Then A > B and also B > A - B, so that we have

    (2B - A)/(A - [img]/forums/images/smilies/cool.gif[/img] = (2 - A/[img]/forums/images/smilies/cool.gif[/img]/(A/B - 1)
    = (2 - sqr(2))/(sqr(2) - 1)
    = (2 - sqr(2))(sqr(2) + 1)
    = sqr(2)


    But this contradicts the minimality of B.</span hide>
    Jerry

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    Re: Totally Irrational

    Concurs with my analysis Steve.

    Alan

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    Re: Totally Irrational

    Interesting approach Jezza. Again, the contradictions upturn the hypothesis.

    Alan

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    Re: Totally Irrational

    Alan another way is:

    X/Y=√2

    => X/Y x X/Y = √2 x √2

    =>X^2/Y^2 = 2

    => X^2 = 2 x Y^2

    OK so far? Well we can apply the next assumptions:

    1) A number that can be divided by 2 is even
    2) If you multiply an odd number by an odd number, you get another odd number
    3) If you, multiply any number (odd or even ) by an even number, you get an even number.

    As the right-hand side of the equals sign is 2 x Y^2, the result must be even. So X^2 must be even.

    If X is even then it can be divided by 2. So X^2 can be divided by 4, and we know that X^2 is the same as 2 x Y ^2. If 2 x Y ^2 can be divided by 4, then we know Y^2 can be divided by 2. So Y ^2 is even (and so Y is even too).

    Therefore I have shown that X and Y are even, so therefore X/Y is not the simplest fraction as they are both divisible by 2. A contradiction of terms.

    Therefore √2 cannot be represented by a fraction
    Jerry

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    Re: Totally Irrational

    That is correct, it is an expansion of Steve's reply.

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    Re: Totally Irrational

    I agree that it is an expansion of Steve's.

    I just wanted to share the twist when viewing the full calculation. Call it an irrational thought <img src=/S/bubbles.gif border=0 alt=bubbles width=31 height=17>
    Jerry

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    Re: Totally Irrational

    Yes the question by alan was "how could you disprove this hypothesis". He never asked for any details... <img src=/S/evilgrin.gif border=0 alt=evilgrin width=15 height=15>

    Steve

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