complex number pt2

7/25/2019 Complex Number Pt2

1/12

Complex Numbers

(Part 2)


2/12

Geometrical Implications As a complex number has two components, x and y, we need a number

plane to represent the complex number geometrically !hus (",2) can represent# 3 2z i= +

Its con$ugate is# 3 2z i=

-1 1 2 3 4 5

-3

-2

-1

1

2

3

4

Z

-

1 1 2 3 4 5

-3

-2

-1

1

2

3

4

Z

Z'

-

1 1 2 3 4 5

-3

-2

-1

1

2

3

4

Z

Z'

O

!hey are on a circle with centre (%,%)

2 23 2

13

r= +

=-

1 1 2 3 4 5

-3

-2

-1

1

2

3

4

Z

Z'

O

!he radius is

&ith angle gi'es a lin withtrigonometry

Called the *odulus+Argument orm#

(short -orm)

( )cos sinz r i

rcis

= +

=

-

1 1 2 3 4 5

-3

-2

-1

1

2

3

4

O

Z

Z'

-

1 1 2 3 4 5

-3

-2

-1

1

2

3

4

O

Z

Z'


3/12

-1 1 2 3 4 5

-3

-2

-1

1

2

3

4

O

Z

Z'

-

1 1 2 3 4 5

-3

-2

-1

1

2

3

4

O

Z

Z'

Geometrical Implications .elationship between a complex number and its con$ugate

is the re-lection o- in the x/axisz z

Note that ( )

( ) ( )( )( )

cos sin

cos sin

cos sin

z r i

z r ir i

= +

= + =

because ( )

( )

cos cos

sin sin

=

=

adding, gi'es# ( ) ( )

( )

2

2 Re

z z x iy x iy

x

z

+ = + +

=

=


4/12

Geometrical Implications .elationship between a complex number and its con$ugate

-1 1 2 3 4 5 6

-4

-3

-2

-1

1

2

3

4

O

Z

Z'

0ubtracting gi'es# ( ) ( )

( )

2

2 Im

z z x iy x iy

iy

z

= +

=

=

&e note that this leads to a 1C!3.representation -or complex numbers#

plus 4z z ( )z z+

minus 4z z ( )z z

multiplying a complex number by /5 re'erses itsdirection

!his is the e6ui'alent o- a 57% degree rotation8


5/12

-1 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

O

A

B

-1

1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

O

A

C

B

Geometrical Implications Adding complex numbers as 'ectors 9 the parallelogram connection

-1

1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

O

I- is the 'ector1 3 2z i= + OA

-1

1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

O

A

and i- is the 'ector2 2z i= OB

then ( ) ( )1 2 3 2 2

5

z z i i

i

+ = + +

= +1

z

2z

1 2z z+

&e can also see that we reach thesame endpoint by putting the

'ectors :top to tail;

And it doesn


6/12

-3 -2 -1 1 2 3 4 5 6

-5

-4

-3

-2

-1

1

2

3

4

5

O

A

C

B

D

E

-3 -2 -1

1 2 3 4 5 6

-5

-4

-3

-2

-1

1

2

3

4

5

O

A

C

B

D

-3 -2 -1

1 2 3 4 5 6

-5

-4

-3

-2

-1

1

2

3

4

5

O

A

C

B

Geometrical Implications Adding complex numbers as 'ectors 9 the parallelogram connection

1z

2z

I- then2 2z i= 2 2z i = +

2z

so doing :top to tail; will gi'e the:resultant 'ector;# 2 2 1 3z z i = +

&e also note that, in sliding the'ectors around to :top to tail; them,there are two 'ersions#

!he :-ixed; (starts at the origin)2

z

and the :-ree; (starts elsewhere)2

z

!hese ha'e the same mod and arg, soare essentially the same

0o mo'ing 2 2z z

shows us that and arethe diagonals o- the parallelogram

2 2z z+

2 2z z

1 2z z


7/12

Geometrical Implications Naming -ree 'ectors

=ow do we name >

A

B

C

O

AC

&e need to connect A to C 'ia 3

I- we now the name o- 'ectorsand , we trace -rom A to 3 to

C, naming the 'ectors as we go

OA

OC

?et and1

OA z= 2OC z=

1z

2z

A to 3 is (re'erse direction)1

z

3 to C is (same direction)2

z+

A to 3 to C is1 2

2 1

z z

z z

+=

Note# C to 3 to A is( )

2 1

2 1

CA z z

z z

AC

= +

=

= uuurie 57%@ rotation

3z

Name AB


8/12

Geometrical Implications *odulus+Argument -orm

As indicated earlier in the circle, usingtrigonometry#

( )cos sinz r i

rcis

= +

=

-1 1 2

-1

1

O

Z1

Z2

I- is on the unit circle with

then (r45)#1

z3

=

( )

1 cos sin3 3

3

1 3

2 2

11 3

2

z i

cis

i

i

= +

=

= +

= +

*od+Arg orm

xiy -orm

2 1

24

z i

cis

=

=

xiy -orm

*od+Arg orm, where

( )

2

221 1

2

r z=

= +

=

1 1tan

1

4

=

=


9/12

Geometrical Implications *ultiplication using *odulus+Argument -orm

&hile it is easier to add or subtract using the -orm , it is easier to multiply or di'ideusing the -orm z x iy= +( )cos sinz r i = +or example, compare the pre'ious example#

( )

( ) ( )

1 31

2 21 3 1 3

2 2 2 2

1 3 1 3

2 2 2 2

1 3 1 3

2 2

i i

i i i i

i

i

+

= + + +

= + + +

+ +

= +

which is $ust a bit too messy8

with the *od+Arg 'ersion


10/12

Geometrical Implications *ultiplication using *odulus+Argument -orm

13

z cis =Bsing the *od+Arg orm 2 2 4z cis =

( )( )( )

( ) ( )( )

( )

1 2

1 2

1 2

1 2

1 2

.

cos cos sin cos sin cos sin sin

cos cos sin sin sin cos sin cos

cos sin

r cis r cis

r r i i i i

r r i

r r i

r r cis

= + + += + +

= + + +

= +

?et


11/12

-3 -2 -1 1 2 3

-3

-2

-1

1

2

O

Z

Z'

Geometrical Implications *odulus+Argument -orm geometrically>

In *od+Arg -orm, our 57%@ rotation ismultiplication by /5#

1 1 0

cos sin

i

i

= += +

!his leads to , ie =

1 cos sini

cis

= +

=xample#

(xiy) -ormD

or

mod+arg -ormD

( )1 1

1

i

i

+

=

24

24

5

2 4

cis cis

cis

cis

= +

-3 -2 -1 1 2 3

-3

-2

-1

1

2

O

Z


12/12

-3 -2 -1 1 2 3

-3

-2

-1

1

2

O

ZA

-3 -2 -1 1 2 3

-3

-2

-1

1

2

O

ZZ'

Geometrical Implications *odulus+Argument -orm geometrically>

!his then leads to the 6uestion, what doesmultiplying by do>i

0 1

cos sin

i i

i

= += +

!his leads to , ie

2

=

cos sin2 2

2

i i

cis

= +

=

Now# which is#( )1

24 2

32

4

i i

cis cis

cis

+ =

=

3 3

2 cos sin4 4

1 12

2 2

1

i

i

i

= +

= +

= +

-3 -2 -1 1 2 3

-3

-2

-1

1

2

O

Z

ie a E%@ rotation8

complex number pt2

Documents