EUCLIDEAN GEOMETRY

6C H A P T E R

Complex Numbers

As long as algebra and geometry have been separated, their progress have been slow
and their uses limited; but when these two sciences have been united, they have lent
each mutual forces, and have marched together towards perfection.

Joseph Louis Lagrange

In this chapter, we demonstrate the use of complex numbers to solve problems in geom-
etry. We develop some background in the first three sections. The real geometry starts in
Section 6.4, when the unit circle appears.

6.1 What is a Complex Number?

Recall some facts from high school algebra. A complex number is a number of the form

z = a + bi

where a and b are real numbers and i2 = −1. The real number a is called the real part,
denoted Re(z). The set of all complex numbers is denoted C.

We also know that every complex number can be expressed in polar form as

z = r (cos θ + i sin θ ) = reiθ

where r is a nonnegative real number and θ is a real number. (The formula eiθ = cos θ +
i sin θ is a famous result known as Euler’s formula.) A diagram may make this clearer;
much like in the xy-plane, every complex number can be plotted in the complex plane at
a point (a, b). See Figure 6.1A.

The magnitude of z = a + bi = reiθ , denoted |z|, is equal to r, or equivalently,

|z| = a2 + b2.

The number θ is called the argument of z, denoted arg z. It is the angle measured counter-
clockwise from the real axis, as shown in Figure 6.1A. Except in the special case z = 0, the
fact that r is a positive real implies θ is unique up to shifting by 360◦. (As a specific example,
cos 50◦ + i sin 50◦ = cos 410◦ + i sin 410◦.) Therefore, for the rest of this chapter we take
these arguments modulo 360◦.

95

96 6. Complex Numbers

Im z = 3 + 4i
|z| = 5

0θ Re
−1 − 2i

z = 3 − 4i

Figure 6.1A. The numbers z = 3 + 4i and −1 − 2i are plotted in the complex plane; z = 3 − 4i is
the conjugate of z.

Finally, the complex conjugate of z (or just conjugate) is the number
z = a − bi = re−iθ .

Pictorially, it represents the reflection of z over the real axis.
The conjugate has many nice properties: it behaves well with respect to basically every

operation. For example, whenever w and z are complex numbers, we have

w + z = w + z, w − z = w − z, w · z = w · z, w/z = w/z,

and so on. (Verify these.) This lets us write, for instance,

z−a = z−a
b−a b−a

and similarly reduce other arbitrarily complicated expressions. Another important relation
is that for any complex number z,

|z|2 = zz.

This is easy to prove and, as we see later, extremely useful.
Throughout this chapter, we let A denote the point in the complex plane that corresponds

to a complex number a, and adopt similar conventions for the other letters, with lowercase
letters denoting complex numbers, and uppercase letters denoting points.

6.2 Adding and Multiplying Complex Numbers

Complex numbers can be viewed a lot like vectors (u, v). We simply think about them in
the component form u + vi and note that adding them corresponds to vector addition.