Complex numbers

Extending the number system

2 1

The consist of the numbers of the form ,

where , and .

a bi

a b i

R R

complex numbers

0

0

Complex numbers of the form are called ;

complex numbers of the form are called .

a i

bi

real numbers

imaginary numbers

( )

( )

In a general complex number , is called the

and the . This is written Re and

Im

a bi a

b a bi a

a bi b

real part

imaginary part

The relations and cannot be used to compare pairs

of complex numbers.

Operations with complex numbers

( ) ( ) ( ) ( )a bi c di a c b d i ( ) ( ) ( ) ( )a bi c di a c b d i

and a bi c di a c b d

( ) ( ) ( ) ( )a bi c di ac bd ad bc i

2 2 2 2

a bi ac bd bc adi

c di c d c d

Do Q1-Q5, pp.226

Solving equations2 4 13 0Solve the quadratic equation .z z

2

22 2

22

4 02

40 4 0

2

44 0

2

if

if

if

bb ac

a

b b acaz bz c z b ac

a

b i ac bb ac

a

Conjugate complex numbersThe complex numbers and are

called complex numbers.

z x yi z x yi conjugate

2 22

2

The sum and the product are both

real numbers, and the difference is an imaginary

number.

z z x zz x y

z z yi

If a quadratic equation with real coefficients has two complex roots, these roots are conjugate.

( )s t s t

( )st s t

s s

t t

Conjugate roots for polynomials( ) ( )If is a complex number, and is a positive integer then .n nz n z z

( ) ( )If is a real number, then .n na az a z 1 2

1 2 1 0( )Let .n nn np z a z a z a z a z a

1 21 2 1 0

1 21 2 1 0

1 21 2 1 0

1 21 2 0

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

Then,

.

n nn n

n nn n

n nn n

n nn n n

p z a z a z a z a z a

a z a z a z a z a

a z a z a z a z a

a z a z a z a z a

p z

*( ) 0 ( ) ( ) 0 0.So, if , then p z p z p z

Conjugate roots for polynomials

( ) ( ) ( )

( ) 0

( ) 0

If is a polynomial with real coefficients, then .

If is a non-real root of the equation , then is also a root;

that is, the non-real roots of the equation occur as conj

p z p z p z

s p z s

p z

ugate

pairs.

4 4(1 ) 4 4 0.Example 1: Show that . Hence find all roots of i z

5 3 26 2 17 10 0Example 2: Solve the equation .z z z z

Do Q3-Q10, p.230

Equations with complex coefficients

3 4Example 1: Find the square roots of .i( )Note that the square roots of a complex number have form .a bi

2(2 ) (4 3 ) ( 1 3 ) 0

Example 2: Solve the quadratic equation

.i z i z i

Note that roots appear in conjugate pairs only for equationswith real coefficients.

11 1 0 1 2 1( )( ) ( )( )

For all non-constant polynomials with ,

,

w

i

n nn n n n na

a

z a z a z a a z z z z

Fundamental Theorem Of Algebra :

here i

i.e. every polynomial equation of degree has roots.n n

Geometrical representationsComplex numbers in the plane can be represented in two ways.

can be shown as a translation of the plane,

units in the -direction and in the -direction.

s a bi

a x b y

( , )

can be represented as a point

with coordinates .

s a bi S

a b

This is called an ; the -axis is

called the and the -axis is called

the .

x

y

Argand diagram

real axis

imaginary axis

Argand diagram- examples

4 4 0The roots of .z

5 3 26 2 17 10 0The roots of .z z z z

Do Q1-Q5, p.239

Vector versus Argand

Addition

Subtraction

The modulus

2 2

The distance covered by the

translation is .

In an Argand diagram this is

the distance of the point from .

s a bi a b

S O

This distance is called the of , denoted .s smodulus

2 2

( )( )

Note that

.

s a b

a bi a bi

ss

2

If is any complex number,

.

s

s ss

Note that if and , thens a bi t c di

2 2

( )( )

( ) ( )

distance

.

ST t s

t s t s

c a d b

Properties of the modulus

st s t

ss

t t

s t s t

s t s t Do Q1-Q8, pp.235-236

Do Q1, Q2, Q6, Q7, Q8, pp.239-240