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This is a very confusing way to explain a simple thing. In fact "Eulers formula simply shows how one can parametrize a helix using the exponential function",

by amai 8y ago

This is a very confusing way to explain a simple thing. In fact "Eulers formula simply shows how one can parametrize a helix using the exponential function", see https://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-ei-varphi-cos-varphi-i-sin-varphi/625151#625151 https://math.stackexchange.com/questions/3510/how-to-prove-e... And that's it.

romwell 8y ago
There are many answers in that thread; I think different things work for different people. My favorite angle on this is the following graphic/animation, also present in the thread: https://upload.wikimedia.org/wikipedia/commons/0/0e/ExpIPi.gif https://upload.wikimedia.org/wikipedia/commons/0/0e/ExpIPi.g... This shows how (1+i * Pi/N)^k, k=1..N traces out a semi-circle for large values of N. Geometrically, all it says is: * Draw a right triangle ABC with AB=1, BC=Pi/N, and ABC the right angle * Make a copy of ABC, call it A'B'C', and scale it so that A'B'(the long leg) = AC (the hypotenuse) * Put A'B'C' over ABC so that A'B' and AC coincide * Let ABC=A'B'C' * Repeat the process N times * Look where you end up when N is large enough The answer is: when N is large, Pi/N is small, and the right triangle ABC is almost isosceles, AB ~= AC. So you end up with N slices of a pie that make up a fraction of a circle. Which fraction? Well, the perimeter is N/Pi * N = Pi - so half a circle. So if A=(0,0) and B=(1,0), you end up at (-1,0). Now (1+x/n)^n approaches e^x, so it makes sense to define e^(i * Pi) to be the same limit - which we found out to be -1 + i * 0.