mat 3730 complex variables section 1.1 the algebra of complex numbers

MAT 3730Complex Variables

Section 1.1

The Algebra of Complex Numbers

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Definitions, Notations Some materials from section 1.2

Complex Number System

In order to solve the equation

we need to expand the real number system (R) to include

12 x

1i

Definition

A Complex Number is an expression of the form

Two numbers are the same iff

Rbabiaz ,

dicbia and

dbca and

Notations

C = collection of all complex numbers

CR

biaz

Definition

bzb

aza

)Im( part,imaginary

)Re( part, real

biaz

Definition

Complex Conjugate of z

biaz

biaz

Modulus/ Absolute Value of z

)(

22

zz

baz

Operations

dicz

biaz

2

1

iadbcbdacdicbiazz

idbcazz

idbcazz

)()())((

)()(

)()(

21

21

21

Example 1

zzz

thatShow

Operations

dicz

biaz

2

1

12 2 2 2

2

z ac bd bc adi

z c d c d

Example 2

i

i

23

2

Compute

Things we take for granted…

0, ?z

zz

Properties

Geometry of Complex Numbers

We can identify z as the ordered pair (a,b).

Thus, we can represent z as the point (a,b) in the xy-plane.

biaz

y

),( ba

xa

b

Geometry of Complex Numbers

We can identify z as the ordered pair (a,b).

Thus, we can represent z as the point (a,b) in the xy-plane (Sect. 1.2)

biaz

),( ba

x

y

a

b

z

Geometry of Complex Numbers

We can also identify z as the position vector (Sect. 1.3)

biaz

ba,

x

y

a

b

ba,

Geometry of Complex Numbers

We can also identify z as the position vector (Sect. 1.3)

biaz

x

y

a

b

ba,

baz , oflength

Geometry of Complex Numbers

In these contexts, the xy-plane is referred as the Complex Plane

biaz

),( ba

xa

b

y

x

y

a

b

ba,

Example 3

0z

1e

that Prove

.0)Re(such that Let

R

zCz

Next Class

Read section 1.2