Complex numbers are defined as tuples (x,y) or equivalently x+iy. Therefore is is natural to visualize them as objects in the two-dimensional plane.

Definition 1.2.1: The Complex Plane
	The field of complex numbers is represented as points or vectors in the two-dimensional plane. If z = (x,y) = x+iy is a complex number, then x is represented on the horizonal, y on the vertical axis. The horizontal axis is called real axis while the vertical axis is the imaginary axis.

The complex plane

Visually, C looks like R², and complex numbers are represented as "simple" 2-dimensional vectors. Even addition is defined just as addition in R². The big difference between C and R², though, is the definition of multiplication. In R² no multiplication of vectors is defined. To be precise, there is the idea of dot-product and (if you think of R² as embedded in R³) cross-product, but the definition of complex multiplication gives C its special structure.

Example 1.2.2: Multiplication in 2-D and C
	Explore the differences between complex multiplication in C, the dot-product in R2, and the cross-product in R2 as embedded in R3.

Definition 1.2.3: Re, Im, Conj, and Modulus

If z = x + iy is a complex number, then we define the following functions: The Real Part Re: C R, Re(z) = Re(x+iy) = x is the real part of z The Imaginary Part Im: C R, Im(z) = Im(x+iy) = y is the imaginary part of z The Conjugate or z-bar conj: C C, conj(z) = conj(x+iy) = x - iy is the conjugate of z. The conjugate of z is sometimes also denoted as (z-bar) The Modulus, or Absolute Value |.|: C R, |z| = |x+iy| = is the modulus, absolute value, or length of z

Geometrically, these operations are easy to visualize:

Re(z), Im(z)

conj(z), |z|

• Re(z) is the projection of z onto the real axis
• Im(z) is the projection of z onto the imaginary axis
• conj(z) is the vector z reflected around the real axis
• |z| is the standard Euclidean length of the vector z, using good old Pythagoras

Here are a few concrete (simple) examples:

Example 1.2.4: Complex numbers in the plane
	Let u = 3 - 4i and v = i, and w = -2 Find algebraically and visually: Re(z), Im(z), |z| and conj(z) for z = u, v, w, respectively. Draw -u, -v, -w and then describe how -z looks in relation to z for arbitrary complex numbers. Draw 1/u, 1/v, 1/w and then describe how 1/z looks in relation to z for arbitrary complex numbers.

Our new operations can also be composed with one another:

Example 1.2.5: Complex operations composed
	Show that Re(Re(z)) = Re(z) Re(Im(z)) = Im(z) Im(Re(z)) = Im(Im(z)) = 0

Similar relations hold when composing the conjugate function, but since we will need these identities frequently we will state them in the form of a proposition:

Proposition 1.2.6: The conjugate operator
	For every complex number z we have: (1)       |z| = |conj(z)| = || (2)       |z|2 = z*conj(z) = z (3)       conj(conj(z)) = = z Proof

Addition and in particular multiplication are what really define the structure of the complex plane and we need to figure out how to visualize those operations. Addition (and subtraction) will be familiar from vectors in the plane:

Proposition 1.2.7: Adding complex numbers geometrically
	For any complex numbers z and w you can find the sum z+w by attaching the vector w to the tip of the vector z and completing a paralellogram. The diagonal of that paralellogram is the vector z+w. To find the difference z-w, reverse the vector w and find z+(-w). Proof

Adding vectors z and w

Subtracting vectors z and w

Multiplication is more difficult to figure out, but feel free to come up with a conjecture before reading on.

Example 1.2.8: Conjecture for multiplying geometrically
	Let z=1+i, w=1-i, and v=i. Compute and visualize z*w, v*z, v*w, z2, w2, and v2. Try to come up with a conjecture how to visualize multiplication of two general complex numbers.

To see if your conjecture about multiplication is correct, we first need to introduce a more suitable coordinate system. Similar to vectors in the 2-D plane, we define polar coordinates as follows:

Definition 1.2.9: Polar coordinates and cis
	A complex, non-zero number z=x+iy with rectangular coordinates (x,y) can also be represented in polar coordinates (r,t). Let r = |z| = and t be an angle such that x = r cos(t) und y = r sin(t) Then z = x+iy = r cos(t) + i r sin(t) = r (cos(t)+i sin(t)) = r cis(t) where we define cis(t) = cos(t) + i sin(t). r is called the radius or length, t the angle or argument of z.

	(x,y) rectangular (r,t) polar (1,0)=1 (1,0)=1 (0,1)=i (1,/2)=cis(/2) (-1,0)=-1 (1,)=cis() (0,-1)=-i (1,3/2)=cis(3/2) (1,1)=1+i (,/4)=cis(/4) (1,-1)=1-i (,-/4)=cis(-/4)
Polar coordinates of a complex number	Sample points

It is easy to convert a number z from polar coordinates (r,t) to rectangular coordinates (x,y): simply use the fact that z = r cis(t) = r(cos(t) + i sin(t)), i.e. compute

• x = r cos(t)
• y = r sin(t)

You can verify, for example, that the complex number with polar coordinates r=2 and angle t=/4 can be written in rectangular coordinates as

z = 2 cis(/4) = 2 (cos(/4) + i sin(/4)) = 2 (1/ + i 1/) = + i

Going the other way, i.e. converting from rectangular to polar coordinates, is a little more confusing since the angle in the above definition is not unique. For example, the complex number 1+i clearly has a radius of but any angle /4 + 2k will work. In other words, z=1+i=(1,1) in rectangular coordinates is equal to

(,/4) = (,9/4) = (,17/4) = ...

in polar coordinates. To avoid confusion, we will often (somewhat arbitrarily) restrict the angle of a complex number in polar coordinates to be between - and and call it principle angle.

Definition 1.2.10: The Principle Argument Arg
	For any complex number z 0 we define the principle argument or Arg(z) as the angle which the vector z makes with the positive (real) x-axis and for which - < Arg(z)

In other words, a non-zero complex number has many arguments, but only one principle argument. To find a principle argument we use the fact that

tan(t) = ^sin(t)/_cos(t) = ^{r sin(t)}/_{r cos(t)} = ^y/_x = ^Im(z)/_Re(z)

so that we can use the tan^-1 = arctan adjusted in such a way as to give us the principle angle: take any non-zero complex number z:

If Re(z) = 0 then

• if Im(z) > 0 then Arg(z) = /2
• if Im(z) < 0 then Arg(z) = -/2

Example 1.2.11: Polar coordinates examples
	Convert the following numbers into the indicated coordinates and draw them in the complex plane: z=(2,0), w=(3,), v=(2,5/6), u=(2,-3/4) from polar to rectangular z=(-2,0), w=(0,-2), v=(3,4), u=(3,-4) from rectangular to polar Prove that if z = r cis(t) then = r cis(-t)

Using polar coordinates will finally allow us to understand how multiplication of complex numbers works geometrically:

Theorem 1.2.12: Multiplying complex numbers geometrically
	If z = r1 cis(s) and w = r2 cis(t) in polar coordinates then z*w = r1 r2 cis(s + t) and zn = rn cis(ns) Proof

You might wonder why this theorem is called "multiply complex numbers geometrically" when in fact it shows only formulas. But what the formulas mean is:

• two complex numbers are multiplied by multiplying their length and adding their angles
• a complex number is raised to a power by raising the length to the power and multiplying the angle

Multiplying vectors z and w

A similar trick works for division of two complex numbers:

Corollary 1.2.13: Dividing complex numbers geometrically
	If z = r1 cis(s) and w = r2 cis(t) in polar coordinates then z/w = r1 / r2 cis(s - t) Proof

Here are some examples that should convince you that in cases where the angle of a complex number is easy to determine, multiplication and division in polar coordinates can be quite simple.

Example 1.2.14: Multiplying geometrically
	Explain geometrically why (1+i)(1-i), i2, and (1+i)4 are all purely real numbers, i.e. the imaginary part of each answer will be zero. Draw the vectors zn, n=1, 2, ... 8 for z=cis(/4) and z=0.85 cis(/8) Find both geometrically and algebraically and confirm that both answers agree. Compute in your head, then confirm your answer.

If a complex number z lies on the unit circle it has length one so that z=sin(t)+i cos(t) for somes t (i.e. the radius r is 1. In that case the above theorem has another well-known corollary:

Corollary 1.2.15: DeMoivre's Formula
	For any integer n and any real number t we have (cos(t) + i sin(t))n = cos(nt) + i sin(nt) Proof

DeMoivre's Formula is quite something. It says that if you take a number on the unit circle (i.e. with lenght 1) with initial argument (angle) t and multiply it by itself, it simply rotates around the unit circle by that angle t. Each time you multiply the number by itself, the vector rotates another t degrees. In other words, in this case the power operator results in a simple rotation.

Powers of a vector z with |z|=1

Two interesting questions related to this rotation, taken from the field of Complex Dynamics, are: suppose z is a complex number with |z|=1. Then:

• find conditions for Arg(z) such that zⁿ = z for some integer n. Such a point, incidentally, is called periodic of order n.
• if Arg(z)/ is irrational, what can you say about the sequence {z, z², z³, z⁴, ...}? Does it, for example, converge? Such a sequence, incidentally, is called the orbit of z.

Proposition 1.2.16: Finding Roots
	For any positive integer n and any non-zero complex number a = r cis(t) the equation zn = a has exactly n distinct roots given by: z = where k = 0, 1, 2, ... n-1. Proof

In a previous example we found the two square roots of i, which turned out to be a fair amount of work. The above proposition allows us to dispense with such a question quickly. For example, the two solutions for

z² = i = cis(/2)

z₁ = cis(/2/2) = cis(/4)
z₂ = cis((/2 + 2)/2) = cis(5/4)

which you can quickly check using DeMoivre's Formula. Here is a geometric interpretation of this proposition: let's find, for example, the three third-roots of i, i.e. we want to find all solutions to z³ = i.

1: Draw the vector i	2: Divide angle by 3 for first root	3: Draw 3 equally spaced segments, starting at the first root

Note, in particular, that the third root of i turned out to be -i, which indeed checks out:

(-i)(-i)(-i) = i*i*(-i) = (-1)*(-i) = i

This proposition is very satisfying: it says that at least every simple polynomial equation of degree n has n solutions. Later we will see that this is true in general: every n-th degree polynomial has n roots, no "if's" and "but's". This, in fact, is a sign of things to come: many theorems in complex analysis will turn out to be very "satisfying" and nicely structured, which is one reason that the study of complex analysis is a lot of fun (I think -:). But first a few more 'profane' examples.

Example 1.2.17: Finding roots geometrically
	Find the cube roots of 8 and i and draw them. Find all 4 fourth-roots of -1 and draw them geometrically Find all 5 fifth-roots of 1 and draw them geometrically Find both square-roots of 3i-2 by (a) using polar coordinates and (b) using rectangular coordinates and a formula from the previous section. Confirm that both methods result in the same answers.

Let's conclude this chapter with a result that illustrates what a 'nicely structured' theorem in complex analysis can look like.

Proposition 1.2.18: Roots of Unity
	The n n-th roots of unity are given by wnk, where k = 0, 1, 2, ... n-1 and wn = cis(2/n) They form the vertices of a regular polygon and add up to zero, i.e. they satisfy the equation: 1 + wn + wn2 + ... + wnn-1 = 0 Proof

This is neat: not only does the equation zⁿ = 1 have exactly n solutions (one of which is, of course, z=1), but the solutions have this really pretty geometric structure of forming a regular polygon, which implies algebraically that they add up to zero as vectors. Here are, for example, the eight roots of z⁸=1:

You can see their regular structure. When you add them all as vectors, you indeed get the zero vector.

We will explore some more geometric structures in the next section.