Complex Numbers 2

The Argand Diagram

Representing Complex NumbersReal numbers are usually represented as positions on ahorizontal number line.

-3 -2 -1 0 1 2 3 4 5

Real

Addition, subtraction, multiplication and division with real numbers takes place on this number line.

The Argand DiagramComplex numbers also have an imaginary part so another dimension needs to be added to the number line

1 2 3 4 5 6 7 8

1234567

-2-3-4-5-6-7-8-2-3-4-5-6-7

Real

Imaginary

-1

Complex numbers can be represented on the Argand diagram by straight lines. Putting complex numbers on an Argand diagram often helps give a feel for a problem.

Some examplesju 26 jv 72 jw 45 jz 67

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

-2-3-4-5-6-7-8

-2

-3

-4

-5

-6

-7

Real

Imaginary

-1

u

v

w

z

Complex numbers and their conjugates

jw 45 jw 45*

jz 36 jz 36*

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

-2-3-4-5-6-7-8

-2

-3

-4

-5

-6

-7

Real

Imaginary

-1

w

zw*

z*

Additionjw 45

jz 36

zw

w

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

-2-3-4-5-6-7-8

-2

-3

-4

-5

-6

-7

Real

Imaginary

-1z

Subtractionju 32 jv 36

vu

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

-2-3-4-5-6-7-8

-2

-3

-4

-5

-6

-7

Real

Imaginary

-1

The modulus of a complex number

RealO

Imaginary

y

x

x + yj22 yxyjx

The argument of a complex number

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

-2-3-4-5-6-7-8

-2

-3

-4

-5

-6

-7

Real

Imaginary

-1

θ

z=2 + 3j

w=-3 - 5j

α

jz 32 jw 53

between -180o and 180o

)arg(z )arctan( 23

56)arctan( 3

5 59

)arg(w 121

Radians

180c3

18060

c

3

61805150

c

65

Loci using complex numbersjz 46 jw 21 wz

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

The distance to a point)23( jz jz 74 )74( jz

4)23( jz

1 2 3 4 5 6 7 8

1234567

-2-3-4-5-6-7-8-2-3-4-5-6-7

Real

Imaginary

-1

4

4)23( jz

4)23( jz

4)23( jz

Loci using arguments

4)arg( z

Re

Im

4)arg( jz

Re

Im

4)arg(0 jz

Re

Im