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ahbwramc comments on Is Scott Alexander bad at math? - Less Wrong
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But... it's just rotation! I think the thing that's weird about Euler's identity is that the symbology looks odd (especially if you're more used to degrees than radians), not that the underlying reality is odd. (Maybe I've just dealt with exponentials of complex numbers for so long that I can't be surprised by them anymore, but I don't remember being surprised by it before.)
Sure, I understand the identity now of course (or at least I have more of an understanding of it). All I meant was that if you're introduced to Euler's identity at a time when exponentiation just means "multiply this number by itself some number of times", then it's probably going to seem really odd to you. How exactly does one multiply 2.718 by itself sqrt(-1)*3.14 times?
You simply measure out a length such that, if you drew a square that many meters on a side, and also drew a square 3.1415 meters on a side, they would enclose no area between the two of them. Then evenly divide this length into meters, and for each meter write down 2.7183. Now multiply those numbers together, and you'll find they make -1. Easy!