1Complex numbers

IA Vectors and Matrices

1.1 Basic properties

Definition

(Complex number)

.

A complex number is a number

z ∈ C

of the

form

z

=

a

+

ib

with

a, b ∈ R

, where

i

2

=

−

1. We write

a

=

Re

(

z

) and

b

=

Im

(

z

).

We have

z

1

± z

2

= (a

1

+ ib

1

) ± (a

2

+ ib

2

)

= (a

1

± a

2

) + i(b

1

± b

2

)

z

1

z

2

= (a

1

+ ib

1

)(a

2

+ ib

2

)

= (a

1

a

2

− b

1

b

2

) + i(b

1

a

2

+ a

1

b

2

)

z

−1

=

1

a + ib

=

a − ib

a

2

+ b

2

Definition

(Complex conjugate)

.

The complex conjugate of

z

=

a

+

ib

is

a −ib

.

It is written as ¯z or z

∗

.

It is often helpful to visualize complex numbers in a diagram:

Definition

(Argand diagram)

.

An Argand diagram is a diagram in which a

complex number

z

=

x

+

iy

is represented by a vector

p

=

x

y

. Addition of

vectors corresponds to vector addition and ¯z is the reflection of z in the x-axis.

Re

Im

z

1

z

2

¯z

2

z

1

+ z

2

Definition

(Modulus and argument of complex number)

.

The modulus of

z

=

x

+

iy

is

r

=

|z|

=

p

x

2

+ y

2

. The argument is

θ

=

arg z

=

tan

−1

(

y/x

). The

modulus is the length of the vector in the Argand diagram, and the argument is

the angle between z and the real axis. We have

z = r(cos θ + i sin θ)

Clearly the pair (

r, θ

) uniquely describes a complex number

z

, but each complex

number

z ∈ C

can be described by many different

θ

since

sin

(2

π

+

θ

) =

sin θ

and cos(2π + θ) = cos θ. Often we take the principle value θ ∈ (−π, π].

When writing z

i

= r

i

(cos θ

i

+ i sin θ

i

), we have

z

1

z

2

= r

1

r

2

[(cos θ

1

cos θ

2

− sin θ

1

sin θ

2

) + i(sin θ

1

cos θ

2

+ sin θ

2

cos θ

1

)]

= r

1

r

2

[cos(θ

1

+ θ

2

) + i sin(θ

1

+ θ

2

)]

In other words, when multiplying complex numbers, the moduli multiply and

the arguments add.

Proposition. z¯z = a

2

+ b

2

= |z|

2

.

Proposition. z

−1

= ¯z/|z|

2

.

Theorem (Triangle inequality). For all z

1

, z

2

∈ C, we have

|z

1

+ z

2

| ≤ |z

1

| + |z

2

|.

Alternatively, we have |z

1

− z

2

| ≥ ||z

1

| − |z

2

||.