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7/4/2025, 6:00:38 PM
>>17810503
It can be made somewhat intuitive if you know that multiplying two complex numbers results in their angles being added and their lengths being multiplied.
So if you want a complex number of length 1 with a certain angle θ, you can, with a bit of handwaving, get to it by multiplying (1 + iθ/∞), or 1 plus an infinitesimal imaginary fraction of your desired angle, infinitely many times. So that would look like (1 + iθ/∞)^∞. And what's the limit definition of e^x? Well, for simplicity, if you just replace the numbers approaching infinity with infinity symbols, it would be (1+ x/∞)^∞. So the previous formula is actually just e^(iθ). And if you rotate 1 by an angle of π (or 180°) through the complex plane, you end up on the on the opposite side at -1. So e^(iπ) = -1. Hope that maybe helps a little.
It can be made somewhat intuitive if you know that multiplying two complex numbers results in their angles being added and their lengths being multiplied.
So if you want a complex number of length 1 with a certain angle θ, you can, with a bit of handwaving, get to it by multiplying (1 + iθ/∞), or 1 plus an infinitesimal imaginary fraction of your desired angle, infinitely many times. So that would look like (1 + iθ/∞)^∞. And what's the limit definition of e^x? Well, for simplicity, if you just replace the numbers approaching infinity with infinity symbols, it would be (1+ x/∞)^∞. So the previous formula is actually just e^(iθ). And if you rotate 1 by an angle of π (or 180°) through the complex plane, you end up on the on the opposite side at -1. So e^(iπ) = -1. Hope that maybe helps a little.
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