14 CHAPTER 1. PRELIMINARIES

This is called the polar form of the complex number.The field of complex numbers is denoted as C. An important construction regarding

complex numbers is the complex conjugate denoted by a horizontal line above the number.It is defined as follows.

a+ ib ≡ a− ib.

What it does is reflect a given complex number across the x axis. Algebraically, the followingformula is easy to obtain. (

a+ ib)(a+ ib) = a2 + b2.

Definition 1.5.2 Define the absolute value of a complex number as follows.

|a+ ib| ≡√a2 + b2.

Thus, denoting by z the complex number, z = a+ ib,

|z| = (zz)1/2

.

With this definition, it is important to note the following. Be sure to verify this. It isnot too hard but you need to do it.

Remark 1.5.3 : Let z = a+ ib and w = c+ id. Then |z − w| =√(a− c)

2+ (b− d)

2. Thus

the distance between the point in the plane determined by the ordered pair, (a, b) and theordered pair (c, d) equals |z − w| where z and w are as just described.

For example, consider the distance between (2, 5) and (1, 8) . From the distance formula

this distance equals

√(2− 1)

2+ (5− 8)

2=

√10. On the other hand, letting z = 2+ i5 and

w = 1+ i8, z−w = 1− i3 and so (z − w) (z − w) = (1− i3) (1 + i3) = 10 so |z − w| =√10,

the same thing obtained with the distance formula.Complex numbers, are often written in the so called polar form which is described next.

Suppose x+ iy is a complex number. Then

x+ iy =√x2 + y2

(x√

x2 + y2+ i

y√x2 + y2

).

Now note that (x√

x2 + y2

)2

+

(y√

x2 + y2

)2

= 1

and so (x√

x2 + y2,

y√x2 + y2

)is a point on the unit circle. Therefore, there exists a unique angle, θ ∈ [0, 2π) such that

cos θ =x√

x2 + y2, sin θ =

y√x2 + y2

.

The polar form of the complex number is then

r (cos θ + i sin θ)

where θ is this angle just described and r =√x2 + y2.

A fundamental identity is the formula of De Moivre which follows.