Yeah and also that an exponential is a turning function, or that you can even take an number to an imaginary power in the first place, or that turning the right amount just happens to take the power of e (and not, say, 2)

It's a lot less surprising if you look at the functions as Taylor series, then you'd see that exponentiation is basically the same thing as cos + sin. This also explains "why e and not 2?", because the Taylor series for e, cos and sin are very nice by definition (e is defined to have trivial derivative, and radians are defined to make sin and cos have simple derivatives) while the Taylor series for exponentiation by 2 is not.

Agreed, but that is a lot of overhead to get to the insight.

> that turning the right amount just happens to take the power of e (and not, say, 2)

username90 already explained this one, but in my own rephrasing:

The premise is incorrect. It isn't that "turning just happens to take the power of e". e is defined to be the number for which this is true.

Those kind of « it’s defined like this » explanation has always confused me in math in general. there are two really different things :

- the mathematical concept was initialy created as a way to solve this particular problem, and so this should come at no surprise

And

- after centuries trying to define this concept, we found that the best way to define this concept is like that.

In your case, was « e » created to perform 2D rotations in the first place ?

We can look at history:

1. e was discovered in 1618

2. i was introduced around 1637

4. calculus was formalized by Newton in 1687

3. derivatives of sine and cosine were discovered in 1722

4. de Moivre's formula (cos x + i sin x) ^ n = cos nx + i sin nx was discovered in 1730

5. Eulers fornula e ^ ix = cos x + i sin x was discovered in 1748 by using Taylor series (requires their derivatives).

So we knew about e and i more than a century before Eulers fornula.

However we discovered it pretty soon after we started applying calculus to trigonometrical functions. Also we can note that it is a strictly more powerful than de Moivre's formula since it shows that we can easily add angles just by multiplying their complex representations, so it is not just a simplification of old knowledge.

So it seems like the discovery of Euler's fornula actually did help with working with rotations. However it was done during a time when we were still exploring the applications of Calculus so there were a lot of low hanging fruit like this to pick.

So to directly answer your question, e was not created to perform 2D rotations, instead we discovered that putting i inside of e makes 2D rotations simple. But the actual important identity you care about is that you can add angles by multiplying which is easy to prove using eulers fornula:

(cos x + i sin x) * (cos y + i sin y) = cos (x + y) + i sin (x + y)

> Eulers fornula e^ix = cos x + i sin x was discovered in 1748

Discovered by Roger Cotes in 1714, in IMO its most natural form, ix = log(cos x + i sin x)

> actual important identity you care about is that you can add angles by multiplying

Personally I think the multiplicative concept of rotations is the natural one, with rotations associated to points on a circle embedded in the plane, rather than associated to arclengths. The amazing thing is that we can compose rotations (naturally multiplicative) by first taking the logarithm of the rotations (a.k.a. angle measure) adding them, and then taking the inverse logarithm. The logarithm is a tool which turns multiplication into addition.

The protractor and the slide rule turn out to be more or less the same concept, which is why we can substitute the former for the latter in https://en.wikipedia.org/wiki/Prosthaphaeresis

If you explored the problem domain, starting with an understanding of trigonometry and calculus, it will naturally emerge as a result. It was not arbitrarily defined.

Imaginary numbers were not arbitrarily defined either, they precisely describe an actual phenomenon.

At a certain level, it is difficult to distinguish between "the reality we decided is true" for math vs. "the reality that must be true underneath". If you accept empiricism, such that you trust observations we make about the physical world, this problem becomes irrelevant to this discussion.

No, as username90 says, e was not created to perform rotations. The aspect of e that comes into play for the relation to sine and cosine is that the function e^x satisfies the differential equation df/dx = f. This makes the Taylor series convenient by not introducing factors of (ln K)^n in each term.

Thus, 2^ix = cis (x ln 2) = cos (x ln 2) + i sin (x ln 2). And since e is, by definition, the number that satisfies ln e = 1, e^ix can be stated more simply as cis x. And that identity, ln e = 1, is in fact the original motivating definition of e.

There are many valid ways to arrive at e as an interesting constant. A particular way of arriving at e as an interesting comment might provide a connection to something else (such as 2D rotation), but that doesn't mean it was created for that purpose.

Personally I think the cool part is that the complex logarithm takes the stereographic projection and turns it into the Mercator projection. That is, it conformally maps the plane onto an infinite two-ended cylinder.

The expression log(–1) = iπ more or less just tells you how to normalize the coordinate system on the cylinder. If you wanted you could pick a somewhat different coordinate system. This one is conventional and often convenient though.

Complex numbers are about rotation. That’s the only odd insight in Euler’s equation. Once you get that, it all falls into place (and in fact starts to explain all the use of complex numbers in engineering).

> Complex numbers are about rotation.

Why?

(I'm not denying that it is, just pointing out that like the person I first replied to, you are jumping over the interesting part here and stating interesting conclusions like they are definitions.)

A complex number “is”† a quotient of two Euclidean vectors in a plane. That is, Z = u\v is the object which when you multiply it by one vector u, scales and rotates that into another vector v. We can write uZ = uu⁻¹v = v.

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

https://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

† More precisely, the algebra of complex numbers is structurally the same as the algebra of quotients of planar vectors.

As I said, "that's the only odd insight." I'm not claiming it's obvious, it's not. but once explained it makes sense and the rest falls in place. Here's a better resource than I can type up:

https://betterexplained.com/articles/a-visual-intuitive-guid...

Hm does that mean you could say the complex quantum probability is just a probability over rotations?

Well you add up all the rotation like stuff and square the result to get the probability. In the book QED Feynman describes it as adding little arrows to explain it to a lay audience. In physical terms it's probably most like adding waves and then the square of the amplitude or wave energy is the probability but not quite - no one really knows what the actual reality is.

Kinda sorta, imagine that probability has a direction. Interference is then about alignment of these directions.

what does electric engineering have to do with rotation? Generators perhaps run in circles, an inductor is a coil often enough, but a capacitor?

Electromagnetism concerns movement of what appear to be moving circles. If you roll a circle with a pen marker it draws a sine or cosine curve. If you try to draw the spiral of a light wave you get perpendicular sine and cosine curves. Light is electromagnetic radiation and in turn both electricity and magnetism are fundamentally connected forces.

there is no "spiral of a light wave" lol. If anything, I'd think of transversal movement. Ironically I cannot say that nobody has ever seen a photon. But nobody has ever seen an electron. There could be a model in which electrons move transversal (at mm/s), mediated by photons, but I have not looked that closely into it. It's not part of the usual theories.

if you attach a pen to a hoop and move the hoop along a wall, you do not get a wave, you get (edit: what looks like but isn't) sharp discontinuity where the pen touches the ground (https://google.com/search?q=cos%28x-sin%28x%29%29)

Circular polarized light is a spiral, at least in terms of how we normally draw the vector annotation.

You may say that AC has a lot to do with rotation.

EM is about rotating fields. Phase is very fundamentally connected with rotation. These are where complex numbers show up.