What's your favorite illustration of the usefulness of complex numbers?
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yeah i sense circular reasoning here too
what i'm saying is *define*: e^{ix} := cos(x) + i*sin(x). you cannot stop me from making this definition. Think of it as a notational convenience. Whether or not you *want* to prove one before the other is a matter of taste, but logic doesn't have a preferred direction.
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I am not stopping to appeal to the Moivre formule
. But when you derive in both sides of the formule, you already know the derive of the Sin(x) and Cos(x), and after you use it to “prove” the derive of Sin(x) and Cos(x). Its a circular reasoning that obviously is invalid. -
?? i did not even use the derivative of sin(x) nor cos(x) in this at all.. that's the whole point ..
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i mean i can see your point, that nothing is stopping you from making that definition. however, i think it's a little trivial because there is definitely circular reasoning when you find the derivatives of those functions when Euler's identity was built on those derivatives(1/2)
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that's simply just a different definition of Euler's identity .. the fact is that it's "neat" and "can be done" is what is the spirit of the OP
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