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Thread: Totally Irrational

20050615, 15:25 #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

20050615, 15:48 #2
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Re: Totally Irrational
<span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: 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

20050615, 15:59 #3
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Re: Totally Irrational
<span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: 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

20050616, 01:35 #4
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Re: Totally Irrational
Concurs with my analysis Steve.
Alan

20050616, 01:36 #5
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Re: Totally Irrational
Interesting approach Jezza. Again, the contradictions upturn the hypothesis.
Alan

20050619, 20:21 #6
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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 righthand 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 fractionJerry

20050619, 20:47 #7
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Re: Totally Irrational
That is correct, it is an expansion of Steve's reply.

20050619, 21:12 #8
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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

20050619, 23:19 #9
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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