How Complex Could Geometry Be?

Okay I spent way too long trying to think of an interesting title. I initially wanted to make a pun based on psychological complexes--"Argand had a real zero complex" (see, I've got absolutely nothing... if you've got something, feel free to use the comments...)--but then I said the word "complex" too many times that I completely forgot what it meant, and I had to google "complex plane" to make sure that was a thing.

See, math, it's dangerous (maybe more than I thought). Oh well, this title should be sufficient. If anything, it thematically connects this post to my last post's brief mention of how complex analysis is awesome for geometry, symmetry, and pretty much everything.

Last post I gave a brief introduction to symmetry. I figured I could take the time and make another post to detail the basics of complex analysis. Again, this will be long and will simply give an introduction to the use of complex numbers, so feel free to skip through it if you know it already.. Soon, I'll make another post for group theory. 

Complex numbers are awesome. They are very similar to the real numbers, but by the way we construct them, allow for greater geometric significance. First let's begin with i.

i is the square root of -1, defined by the equation i² = -1. This allows us to define complex numbers, where we can more fully account for algebra. For example, when we are restricted to just the reals, the equation x⁴ - 1 = 0 has just two solutions--1 and -1--but in the complex numbers it has 4 (the same amount as the degree of the polynomial)--1, -1, i, and -i.

Now let's define complex numbers

Take R2

No, not you

Excuse me, the real plane, R²

It has an x-axis and a y-axis, and every point in R² can be represented by a coordinate pair, (a, b).

The complex plane is very similar, instead of an x- and a y-axis, it has a "real" and an "imaginary" axis:

Here, still every point can be represented (a, b), if we understand (a, b) --> a + ib. This is the form of a complex number--a is the real part, and ib is the imaginary part.

Every complex number, as I alluded to in my last post, has an argument and a modulus. The argument is the angle, and the modulus is the distance between (a, b) and the origin, (0, 0). Thus, the modulus of z = a + ib is |z|² = a² + b².

Complex numbers can also be represented by z = r cis(θ)= r(cos(θ)+ isin(θ)) = r e^iθ, where r is a real number and represents the modulus, and 0≤ θ≤ 2π where θ represents the argument.

The usual algebraic operations work with complex numbers, and it is easiest to view i as a variable with the special case that i² = -1.


Let w = a + ib and v = c + id be complex numbers. Addition and subtraction are clear: w+ v = (a+b) + i(b+ d). Multiplication, less so, but still intuitive: wv = (a + ib)(c + id) = ac + iad+ ibc - bd = (ac - bd) + i(ad + bc).

Now that we see multiplication, let me introduce you to our friend the complex conjugate! Each complex number z = a + ib has a complex conjugate = a - ib. Notice how this is the reflection of z across the real axis!

From wikipedia

Why this is so special is that if we ever want to get a real number from a complex one , all we have to do is multiply z by is conjugate (hence the line in the sonnet in my first post "Conjugating in C, you made me realize)! (a + ib)(a - ib) = a² + b².

Now, this gives us motivation behind division. v != 0, w/v = (a+ ib)/(c+ id), and multiplying by the conjugate of v over the conjugate of v, gives us ((ac+ db)+ i(bc - ad)) / c² + d².

In polar coordinates, these operations becomes much smoother. Multiplication of w= r₁ e^iθ₁ and v = r₂ e^iθ₂, wv= (r₁r₂) e^{i(θ₁+ θ₂)}. Notice that the multiplication of the conjugate would result in some z = (r r ) e^{i(θ- θ} = r² e⁰ = r², which is exactly what we'd think it'd be.

Inversion is easy to see. We are looking for a z^-1 such that zz^-1 = 1. If z = r e^iθ then by what we just saw before, we are looking for some z^-1= p e^iΦ such that rp = 1 and θ+ Φ= 0. Thus, p = 1/r and Φ= -θ.

This post got pretty long, so I'll give you a break and will post about mapping in the complex plane and the Riemmanian sphere in a separate post later.

Connor