1Complex numbers

IA Vectors and Matrices

1 Complex numbers

In

R

, not every polynomial equation has a solution. For example, there does

not exist any

x

such that

x

2

+ 1 = 0, since for any

x

,

x

2

is non-negative, and

x

2

+ 1 can never be 0. To solve this problem, we introduce the “number”

i

that

satisfies

i

2

=

−

1. Then

i

is a solution to the equation

x

2

+ 1 = 0. Similarly,

−i

is also a solution to the equation.

We can add and multiply numbers with

i

. For example, we can obtain

numbers 3 +

i

or 1 + 3

i

. These numbers are known as complex numbers. It turns

out that by adding this single number

i

, every polynomial equation will have a

root. In fact, for an

n

th order polynomial equation, we will later see that there

will always be

n

roots, if we account for multiplicity. We will go into details in

Chapter 5.

Apart from solving equations, complex numbers have a lot of rather important

applications. For example, they are used in electronics to represent alternating

currents, and form an integral part in the formulation of quantum mechanics.