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Wau - Whats your thought on it?

Main post link -> http://www.youtube.com/watch?v=GFLkou8NvJo

Have a nice look at the video and comment on what you think of it? A video by Vi Hart (a Mathemusician) Does that number really exist? Or is it hypothetical? What are your views on it? From me, Its just Wow.

Note by Nishanth Hegde
4 years, 6 months ago

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It turns out that wau equals one. That's the reason why many of these sequences work. For example, take the first infinite fraction: \[\dfrac{2}{\dfrac3{\dfrac3{\dfrac3{\cdots}+\dfrac1{\cdots}}+\dfrac1{\dfrac3{\cdots}+\dfrac1{\cdots}}}+\dfrac1{\dfrac3{\dfrac3{\cdots}+\dfrac1{\cdots}}+\dfrac1{\dfrac3{\cdots}+\dfrac1{\cdots}}}}\] For simplicity, let \[x=\dfrac{1}{\dfrac3{\dfrac3{\dfrac3{\cdots}+\dfrac1{\cdots}}+\dfrac1{\dfrac3{\cdots}+\dfrac1{\cdots}}}+\dfrac1{\dfrac3{\dfrac3{\cdots}+\dfrac1{\cdots}}+\dfrac1{\dfrac3{\cdots}+\dfrac1{\cdots}}}}.\] Then we get \[x=\dfrac{1}{3x+1x}=\dfrac1{4x}\implies x^2=\dfrac14\implies x=\dfrac12,\] so \(F=2x=1\) as desired. In addition, a lot of the exponentiation infinite series crumble under the definition that \(F=1\), since \(1^{\text{anything}}=1\).

Furthermore, how legitimate is a number whose name sounds similar to the word "wow"? ;)

David Altizio - 4 years, 6 months ago

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Yeah. The most straightforward clue in the video that wau is 1 is that she says \(e^{2i \pi} = F\) which obviously is one.

Nishanth Hegde - 4 years, 6 months ago

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There were other early clues. the one with 5/6 was easy to evaluate. We get F=5/6+1/6*(F) ->F=1. That was really the moment when I went "wait a minute..."

Matthew Lipman - 4 years, 6 months ago

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@Matthew Lipman Yeah. But how to justify \(F=\frac{x+x^{y}+x^{x^{y}} \ldots }{y+y^{x}+y^{y^{x}} \ldots }\) ?

Nishanth Hegde - 4 years, 6 months ago

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@Nishanth Hegde Remember, \(x\) and \(y\) were side lengths of a rectangle that had its side lengths in the ratio \(F\). In other words, \(x=y\). Then that fraction collapses.

David Altizio - 4 years, 6 months ago

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David A. .... VERY BRILLIANT THINKING....GREAT ........

Vamsi Krishna Appili - 4 years, 6 months ago

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I have an issue with her initial definition of "wau". It appears that the continued fraction she starts with, is in fact, not convergent. As she pointed out, the partial values of the partial fraction oscillate between 1/2 and 2; so it does not converge to 1.

It would be like saying 1-1+1-1+1... = x, so 1-x = x and x = 1/2; thus making it seem like the series converges to 1/2. But this is not true under the most "normal" idea of convergence.

Marcus Neal - 4 years, 6 months ago

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The number is actually 1

Harrison Lian - 4 years, 6 months ago

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ya

Vamsi Krishna Appili - 4 years, 6 months ago

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'wau'- wow...

Bodhisatwa Nandi - 4 years, 6 months ago

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David A. got the answer

Vamsi Krishna Appili - 4 years, 6 months ago

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Yup mate! IT"S REALLY WOW!

Piyal De - 4 years, 6 months ago

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