the hyperbolic Fibonacci numbers

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[mrminer]

LOL, Michael made the whole thing so much harder by squaring, got confused, it didn't work out, he spent a minute cussing, blacked the screen while he furiously kicked himself for not being able to square a binomial, and then went right on.
Kids, that's how professional math is done!

[miraj]

Now that you've proven this, you can show that it's basically another way to write Binet's formula. First show, that (-phi)^-1 = phi-bar where phi is the golden ratio and phi-bar is it's radical conjugate. Then use the identity mentioned in the video 2sinh(log(z)) = z + 1/z where z is substituted with (i*phi)^n.

[Tenorsax]

Correction and addition:
At 6:33 the correct square is:
(a1)^2 = 1/5 * (Phi^2 + 2 + 1/Phi^2) = 1/5 * (Phi + 1 + 2 + 1 - 1/Phi) = 1/5 * (Phi - 1/Phi + 4)
Now Phi - 1/Phi =1, therefore (a1)^2 = 1/5 * (1 + 4) = 1 .//

[senseof_outrage]

14:53 not "a good place to stop"
Error on board —
sin instead of sinh

[ericbischoff]

The black screen is left to the viewer as an exercise.

More seriously, to compute (phi + 1 / phi) / sqr(5), it's simpler to replace phi by its value (1 + sqr(5)) / 2:

1 / phi = 2 / (1 + sqr(5)) = 2 (sqr(5) - 1) / (5 - 1) = (sqr(5) - 1) / 2
(phi + 1 / phi) = (1 + sqr(5)) /2 + (sqr(5) - 1) / 2 = sqr(5)
and the expression equals 1

[klausolekristiansen]

As if there was not enough hyperbole about the fibonacci numbers

[alexedwards]

Hmm, I wonder sequence you get if you use cosh instead of sinh.

[shruggzdastr-facedclown]

@Prof. Penn: Please refilm this video to correct for the minute or two of blacked-out footage (of which, by you, I'm assuming was unintentional) where you supposedly "proved" that this formula/equation yields the second seed result of 1 -- and also correct for the errors pointed-out by other viewers

[edwardlulofs]

Is this f_n written as f(z), the analytic continuation of the fibennoci sequence? I apologize for my poor spelling…

[mrtoffeewaffle]

im studying maths and physics at the moment after years out of school ive gone back to do my bsc in it and ive built up a steady supply of books websites and youtube channels to help me with my work from myorganicchemistrytutor to bprb and 3b1b. you my friend do mathematics that so far is beyond my scope but imma be coming for you >:)

[rainerzufall]

Hmm, did you want to hide the mistake, that 4/5 is not 1? Black screen!

[leewilliam]

Supposedly fibonacci travelled to egypt freece turkey and iran to learn maths for many years not sure its true although 😊