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An irrational "aha!" puzzle


Best Answer bonanova, 10 October 2013 - 12:28 PM

Spoiler for Aha!

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#1 bonanova

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Posted 24 September 2013 - 04:33 AM

Brute force or computer simulations are often brought to bear on mathematical problems. Among the motivation for using these approaches is they  can obviate the need for careful thought. But careful thought often shows a simpler path to the answer. Occasionally, by argument from analogy to already solved problems, or invoking properties of the type of expression, for example, that the answer must possess, a solution can be found without seemingly solving the puzzle at all. Or, if an equation must be solved, it's one much simpler than first thought. Some people refer to these solutions as coming to them in what they call "Aha!" moments.
 
I will post a series of "Aha!" puzzles, starting with a rather easy one.
 
The Golden Ratio g has a number of definitions. The one that provides the best clue to our answer is to note that a rectangle of dimensions g x 1 has the property that if it is cut into a square and another rectangle, the second rectangle has the same proportions as the first.

 

Prove that g is irrational.

 

I know its usual symbol is Greek phi; but I don't know how to make one here. :unsure:


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#2 bhramarraj

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Posted 24 September 2013 - 12:01 PM

Spoiler for trial


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#3 bonanova

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Posted 24 September 2013 - 03:01 PM

Spoiler for trial

Perfect. That is a proof. And even tho not all square roots are irrational this one is.
Now, I'm looking for a proof that begins with the assumption that g is rational and show that it leads to an impossible result.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#4 phil1882

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Posted 24 September 2013 - 06:53 PM

Spoiler for

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#5 Anza Power

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Posted 25 September 2013 - 09:49 AM

Spoiler for another proof

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#6 ThunderCloud

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Posted 28 September 2013 - 06:34 PM

Spoiler for Another proof


Edited by ThunderCloud, 28 September 2013 - 06:38 PM.

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#7 bonanova

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Posted 10 October 2013 - 12:28 PM   Best Answer

Spoiler for Aha!


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell




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