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I agree, the suggestion that anyone could/would be able to just throw out everything they know about arithmetic is sketchy at best, and even it's sketchier to m

I agree, the suggestion that anyone could/would be able to just throw out everything they know about arithmetic is sketchy at best, and even it's sketchier to make it seem as though a result like Euler's formula could purely be derived from geometric intuition. Sure, it "makes sense" in the video, but I don't think it's possible to truly grok and appreciate the result without having done any calculus. The real joy of math isn't in knowing, it's in proving. Anyway: as for proof methods, I like the Maclaurin series trick with this one. Too bad the internet hasn't come up with a universal/nice way of writing math notation in a readable format yet, I'd like to sketch that proof here...

unholiness 10y ago
The Maclaurin trick is what I'd call the "standard proof", and I think it's the source of a lot of the lack of intuition surrounding e^i... It's exactly what you call it, a "trick", where you prove two rabbits are identical by transforming them both into an infinite number of hats. On the one hand, it's all sound logic and algebra, but on the other hand, I certainly don't blame Randall Munroe for seeing that proof in high school and going on to say "I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave."[0] [0] https://xkcd.com/179/ https://xkcd.com/179/
- hasbroslasher 10y ago
  Love that xkcd, but I found the Maclaurin "trick" to be extremely common-sensical... If you're already finding the area of infinitely many small boxes under a curve then Taylor series don't seem that weird. Once you are convinced that Taylor series are O.K. then this proof is really cool. I also think it's weird to show high-schoolers this, as about 1% of high school seniors have the mathematical knowledge needed to grok this level of math. Anyways - people have differing levels of magic tolerance, and some people don't have a lot of background in calc, so I don't blame them for looking for other explanations.

unholiness 10y ago

The Maclaurin trick is what I'd call the "standard proof", and I think it's the source of a lot of the lack of intuition surrounding e^i... It's exactly what you call it, a "trick", where you prove two rabbits are identical by transforming them both into an infinite number of hats. On the one hand, it's all sound logic and algebra, but on the other hand, I certainly don't blame Randall Munroe for seeing that proof in high school and going on to say "I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave."[0] [0] https://xkcd.com/179/ https://xkcd.com/179/

hasbroslasher 10y ago
Love that xkcd, but I found the Maclaurin "trick" to be extremely common-sensical... If you're already finding the area of infinitely many small boxes under a curve then Taylor series don't seem that weird. Once you are convinced that Taylor series are O.K. then this proof is really cool. I also think it's weird to show high-schoolers this, as about 1% of high school seniors have the mathematical knowledge needed to grok this level of math. Anyways - people have differing levels of magic tolerance, and some people don't have a lot of background in calc, so I don't blame them for looking for other explanations.