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8/18/2019 Engineering Mathematics Complex Numbers 2
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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
n the beginning there were counting
numbers
And then we needed integers$
&
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%ome observations
n the beginning there were counting
numbers
And then we needed integers$
&'$ '(
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%ome observations
n the beginning there were counting
numbers
And then we needed integers And rationals $
&'$ '(
"#)$
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%ome observations
n the beginning there were counting
numbers
And then we needed integers And rationals
And irrationals$
&'$ '(
"#)$
&
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%ome observations
n the beginning there were counting
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