engineering mathematics complex numbers 2

8/18/2019 Engineering Mathematics Complex Numbers 2

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Complex Numbers

Who uses them

in real life?

The navigation system in the space

shuttle depends on complex numbers!


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A complex number z is a number of the form

where

x is the real part and y the imaginary part,written as

x Re z , y Im z.

i is called the imaginary unit

f x ", then z iy is a pure imaginary number#

iy x + $−=i


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%ome observations

n the beginning there were counting

numbers

$

&


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%ome observations


numbers

And then we needed integers$

&


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%ome observations


numbers

And then we needed integers$

&'$ '(


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%ome observations


numbers

And then we needed integers And rationals $

&'$ '(

"#)$


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%ome observations


numbers

And then we needed integers And rationals

And irrationals$

&'$ '(

"#)$

&


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%ome observations


numbers

And then integers And rationals

And irrationals

And reals

$

&

'$ '(

"#)$

"

&

π


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%o where do unreals fit in ?

We have always used them# * is not +ust * it is

* "i. Complex numbers incorporate all

numbers.

$

&

'$ '(

"#)$

( )i&i

"

&

π


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Who goes first?

' &


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Complex numbers do not have order

' &


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Wor.ed /xamples

$# %implify )−


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Wor.ed /xamples

$# %implify )−

&

) ) $

)

&

i

i

− = × −

= ×=


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Wor.ed /xamples

$# %implify

/valuate

)−

&

) ) $

)

&

i

i

− = × −

= ×=

&( ) $&

$& $

$&

i i i× == × −= −


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Wor.ed /xamples

(# %implify ( )i i+


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Wor.ed /xamples

(# %implify ( )i i+

( ) 0i i i+ =


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Wor.ed /xamples

(# %implify

)# %implify

( )i i+

( ) 0i i i+ =

( 0 ) *i i+ − +


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Wor.ed /xamples

(# %implify

)# %implify

( )i i+

( ) 0i i i+ =

( 0 ) *i i+ − +

( 0 ) * $(i i i+ − + = − +


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Wor.ed /xamples

(# %implify

)# %implify

1# %implify

( )i i+

( ) 0i i i+ =

( 0 ) *i i+ − +

( 0 ) * $(i i i+ − + = − +

2( 032( 03i i+ −


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Addition %ubtraction

4ultiplication

(# %implify

)# %implify

1# %implify

( )i i+

( ) 0i i i+ =

( 0 ) *i i+ − +

( 0 ) * $(i i i+ − + = − +

2( 032( 03i i+ −

( ) & &2( 032( 03 ( 0 5 )5 16i i i+ − = − = − − = −


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7ivision

*# %implify

The tric. is to ma.e the denominator real8

&( 0i +

& ( 0 &2( 03

( 0 ( 0 16

2( 03&5

0 (

&5

i i

i i

i

i

− −× =

+ − −

−=−−

=


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Argand 7iagrams

x

y

$ & (

$

&

(

& (i


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Argand 7iagrams

x

y

$ & (

$

&

(

& (i

We can represent complex

numbers as a point#


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Argand 7iagrams

x

y

$ & (

$

&

(


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Argand 7iagrams

x

y

$ & (

$

&

(

We can represent complex

numbers as a vector#

9

A $

& z i OA= + =


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

$& z i OA= + =

&& ( z i OB= + =


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =

$ & z z OA AC

OC

+ = +

= uuur


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =

? BA =


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =

? BA =


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =

OB BA OA+ =


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =

OB BA OA

BA OA OB+ =

= −uur uuur uuur


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Argand 7iagrams

x

y

$ & (

$

&

(

9

A

:

C

$& z i OA= + =

&& ( z i OB= + =

() ) z i OC = + =

$ &

OB BA OA

BA OA OB

z z

+ == −= −

uur uuur uuur


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The complex conjugate of a complexnumber, z x iy, denoted by z , is given

by z x - iy#


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7eveloping useful rules

( )

( )

( )

&

$

&

$

&$&$

&$&$

&$&$

&$

#)

#(

#&

#$

z

z

z

z

z z z z

z z z z

z z z z dic z and bia z Consider

=

=

−=−

+=++=+=


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Polar Coordinates

With

z ta.es the polar form8

r is called the absolute value or modulus ormagnitude of z and is denoted by ; z ;#

Note that 8

sin y r θ =cos , x r θ =

z z y xr z =+== &&

&&

3322

y x

iy xiy x z z

+=

−+=

3sin2cos θ θ ir z +=


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Complex plane, polar form of a complex number


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i


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Euler Formula – Exponential Form

The polar form of a complex number can be

rewritten as 8

7erive

θ

θ θ

ire

iy xir z

=

+=+= 3sin2cos

( )

( )θ θ

θ θ

θ

θ

ii

ii

eei

ee

−

−

−=

+=

&

$sin

&

$cos


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x

θ #

x

z #

z $

m

e'θ $

r #

r $

$

$$

θ ier z =

&

&&

θ ier z

−=",,, &$&$ >θ θ r r


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A complex number, z $ i , has a magnitude

Example

and argument 8

Bence its principal argument is 8 rad

Bence in polar form 8

&3$$2;; && =+= z

rad&

)

&

$

$tan $

+=+

=∠ − π π

π nn z

Arg ) z π =

)&)

sin)

cos&π

π π iei z =

+=


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A complex number, z $ ' i , has a magnitude

Example

and argument 8

Bence its principal argument is 8 rad

Bence in polar form 8

n what way does the polar form help in manipulating

complex numbers?

&3$$2;; && =+= z

rad&)

&$

$tan $

+−=+

−=∠ − π

π π nn z

)

π −= z Arg

−== −

)sin

)cos&& ) π π

π

ie z i


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What about z #%"+i, z $%"&i, z '%$+i", z (%&$?

Other Examples

π

π

1#"$

$

"

1#"

$

∠=

=

+=ie

i z

π

π

1#"$

$

$"

1#"

&

−∠=

=

−=− i

e

i z

"&

&

"&

"

(

∠==

+=ie

i z

π

π

∠==

+−=

&

&

"&)ie

i z


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D

D

D

m

e

z $ i

& ' i

( &) '&

D

π 1#"


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Arithmetic Operations in Polar Form

The representation of z by its real and imaginary

parts is useful for addition and subtraction#

=or multiplication and division, representation by

the polar form reduces simplification#


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%uppose we have & complex numbers, z # and z $ given by 8

/asier with normal

form than polar form

/asier with polar formthan normal form

magnitudes

multi l !

phases add!

&

$

&&&&

$$$$

θ

θ

)

)

er )y x z

er )y x z

−=−=

=+=

( ) ( )

( ) ( )&$&$

&&$$&$

y y ) x x

)y x )y x z z

−++=

−++=+

( )( )3322&$

&$&$

&$

&$

θ θ

θ θ

−+

−

== )

) )

er r

er er z z


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=or a complex number z & > ",

magnitudesdivide!

phases subtract!

32

&

$3322

&

$

&

$

&

$ &$&$

&

$

θ θ θ θ θ

θ

+−− === ) ) ) )

er r e

r r

er er

z z

&

$

&

$

r

r

z

z =

&$&$

32 θ θ θ θ

+=−−=∠ z


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Example


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Example

i z += (1


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Example

"$& )

=+− i z


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9ne amaing result

What if told you that ii is a realnumber ?


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π π


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&

now cos sin& &

but cos sin

so cos sin& &

i

i

i i

e i

e i i

θ

π

π π

θ θ

π π

= +

= += + =

now cos sini iπ π = +


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&

&

now cos sin& &

but cos sin

so cos sin& &

i

i

i

i i

i i

e i

e i i

e i

θ

π

π

θ θ

π π

= +

= +

= + =

= ÷

now cos sini iπ π = +


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ii "#&"060510*(1"0*$5"611

&

&

&

&

&

now cos sin& &

but cos sin

so cos sin& &

i

i

i

i i

ii

i

i i

e i

e i i

e i

e i

e i

θ

π

π

π

π

θ θ

π π

−

= +

= +

= + =

= ÷

==

( (now cos sini i

π π − −= +


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ii $$$#($0006)6561*&&*"(

&

1

&

(

&

(

&

(

&

now cos sin& &

but cos sin

( (so cos sin

& &

i

i

i

i i

ii

i

i i

e i

e i i

e i

e i

e i

θ

π

π

π

π

θ θ

π π

−

−

= +

= +

− −= + =

= ÷

==


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%o ii is an infinite number ofreal numbers

engineering mathematics complex numbers 2

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