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Feb15-12, 12:04 PM
P: 1
I came across Lakoff's book "where math comes from" in which there's a section devoted to explain the meaning of the Identy instead of using a "tautology" that is generally given by many a professors, text books, intimidators, etc. They almost succeeded in making me understand, but at the last moment they also fall victim to using such vague qualities such as"periodicity", "rate of change" mumbo/jumbo that I feel an intuitive understanding is still lacking. Most people will tell U that it is because it equals cos x + i sinx or because e^xi happens to equal taylor expansion when sin and cos terms are added with a complex number twist lol, yeh but why?

The best way to understand it comes first, from khalid's "betterexplained" plus page of songho ahn's where a unit helix in 3d is shown with an img, real, and an X axis respectively(where x is just a linear axis representing the angle truncated at 2PI.

Khalid's explained e^x as growth(as in compound int) ---> lim (1+1/n)^n and x represents rate/duration mix because e is the no. when u set n to infin of instantaneous 100% compounding.

Similar with the unit circle, e^0 first grows to 1x the base e ( the radius), then because of the part i in the exp and where the rate of e^x change is just e^x as a def. Then the radius R is like a vector of unit length driven by a const i vector perpen to it (i.R is vel or accel if u will,moving counterclockwise at unit increments). And, note ln(x) means at what rate/period is needed to reach x time growth in e base.

In Anh's page, he sets ln(cosx+isinx) = xi, then cosx+isinx = e^xi so xi is the rate/period needed to reach position cosx+isinx.

Now although with e^xi ,x as an angle is a very convenient indication of the position of the radius, it is hard to see how this exp resolves to something like .5678+ .8790i proportion as an example. Evidently, it is a result of the trig function and xi has a different ratio of the vertical vs horizontal depending on where the radius is pointed at along the circle, meaning different slope/tangent. So at any pos there is a changing rate of growth with regards both axes, but only in angles, not in length.

On Anh's page , the 3D diag would dispel a lot of confusion generated by using imaginary no.
Actually, with 2 extra dimensions, imag no. is just a math convenience to make sure U don't add the two nos as though they are both real and to deal with the sign when phase changes.

So if U notice, the projection on the x-Real plane is the Cos function tracing how the real shrinks and grow as the Img-x plane shadows a sin function showing how it grows and shrink on a complementary rate tracing a helix, but a 2D circle projection on the Img-Real plane.