In response to wasoxygen's question about how i^i can be a real number (link), I have compiled some notes on basics of the complex numbers. There are seven more pages to scan, but I had to get out from the computer lab before I could get them. Please remind me if I would forget to update the post.

Part 1: What are complex numbers, definition.

Part 2: Polar coordinate system,

Part 3: Complex numbers in polar coordinates, n-th powers of complex numbers.

Part 4: Euler's Theorem

Prerequisite of power series expansion.

Part 5: Consequences of the Euler's Theorem.

Regardless of its incomplete status: should I clarify what's already here? Any questions or observations?

I am trusting you on the trig identities used to get z³ in part 1.3, because the list is too long and ctrl-f doesn't work for images.

The power series expansion in 1.4 a) looks familiar, but also requires a bit of faith since I could not hope to derive it myself.

The only step that left me really perplexed is 1.5 2). Are you just rearranging the equations and substituting? I didn't follow.

From there it is clear how you get to cos(i) = 1.543..., and I suppose I should not be surprised that this falls outside the usual range of [-1, 1] for cosine.

I was pleased with how much sense I was about to make from your notes, thanks for putting it together.

- trig identities

The only ones you need here are the ones about the sum of angles:

sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y)

cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y)

**EDIT:** By the way, while Wikipedia's list is amazing most of them are there only for the sake of completeness. Let me put it like that: my typical day consists of crunching weird maths for at the very least four hours (even more during weekdays since, you know, classes). Aside from having to find if an expression like sin(3pi/17) has an exact value or needing to see before my eyes if something that I'm calculating is similar to what's in the list of identities (surprisingly not as often as you could think) I don't really have much use of it. On the other hand this list once memorised in junior high serves literally 99.9% of my trigonometric needs to this day. It's actually more than enough for the purposes of this primer.

- Power expansion

In this case it's a Maclaurin series (which is literally the Taylor series with a = 0).

- The only step that left me really perplexed is 1.5 2). Are you just rearranging the equations and substituting? I didn't follow.

Take the first equation, add the other, divide by two. You've found the cos(x). Sine can be obtained by subtracting the second equation from the first and dividing by 2i.

- From there it is clear how you get to cos(i) = 1.543..., and I suppose I should not be surprised that this falls outside the usual range of [-1, 1] for cosine.

Well, we are no longer working on *real* numbers. We made an extension of the existing definition. It will still work for real values as intended. That's pretty much maths in essence: we have used existing formulas (power series, sin/cos functions) in an unusual context, and our definitions got somewhat updated. ;)

**EDIT:** By the way, note that in normal sine and cosine we use real numbers as arguments!