2. complex number system (option 1)
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8/6/2019 2. Complex Number System (Option 1)
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COMPLEX NUMBER COMPLEX NUMBER
SYSTEMSYSTEM
1
8/6/2019 2. Complex Number System (Option 1)
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y COMPLEX NUMBER
NUMBER OF THE FORM C= a+Jba = real part of Cb = imaginary part.
2
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Definition of a Complex Number Definition of a Complex Number
If a and b are real numbers, the number a + bi is acomplex number, and it is said to be written instandard form.
If b = 0, the number a + bi = a is a real number.
If a = 0, the number a + bi is called an imaginarynumber.
Wr ite the complex number in standard form
81 !! 81 i !y 241 i 221 i
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R eal Number sImaginary
Number s
Real numbers and imaginary numbers aresubsets of the set of complex numbers.
Complex Number s
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y Conversion between Rectangular and
polar formConvert Between FormC = a + jb (Rectangular Form)C = C<ø ( Polar Form)
C is Magnitudea = C cos ø and b=C sin øwhereC = ¥ a2 + b2
ø = tan-1 b/a
5
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Complex Conjugates and Division Complex Conjugates and Division
Complex conjugates-a pair of complexnumbers of the form a + bi and a ± bi
where a and b are real numbers.
( a + bi )( a ± bi )a 2 ± abi + abi ± b 2 i 2
a 2 ± b 2( -1 )a 2
+b 2
The product of a complex conjugate pair is apositive real number.
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Complex PlaneComplex PlaneA
complex number can be plotted on a plane withtwo perpendicular coordinate axes
The horizontal x -axis, called the real axisThe vertical y -axis, called the imaginary axis
R epresent z = x + jy geometr ically
as the point P ( x,y) in the x-y plane,
or as the vector from the
or igin to P ( x,y). OP
uuuv
The complex
plane x-y plane is also known as
the complex plane.
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Complex plane, polar form of a complex number
¹ º
¸©ª
¨!
x
y1tanU
Geometr ically, | z| is the distance of the point z from the or igin
while is the directed angle from the positive x-axis to OP in
the above f igure.
From the f igure,
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is called the argument of z and is denoted by arg z . Thus,
For z = 0, is undefined.
A complex number z � 0 has infinitely many possiblearguments, each one differing from the rest by somemultiple of 2. In fact, arg z is actually
The value of that lies in the interval (-, ] iscalled the principle argument of z (� 0) and isdenoted by Arg z .
0tanarg 1 {¹ º ¸©
ª¨!! z x y zU
,...
2,1,0,2tan
1!s
¹ º
¸
©ª
¨!
nn x
y
T U
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Consider the quadratic equation x 2 + 1 = 0.
Solving for x , gives x 2 = ± 1
12
! x
1! x
We make the following def inition:
1!i
Complex Number s
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1!i
Complex Number s : power of j
12
!i Note that squar ing both sides yields:
therefore
and
so
and
iiiii !!! *1*13 2
1)1(*)1(*224
!!! iii
iiiii !!! *1*45
1*1*2246
!!! iiii
And so on«
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Addition and Subtraction of Addition and Subtraction of Complex Numbers Complex Numbers
If a + bi and c +di are two complex numberswritten in standard form, their sum and
difference are defined as follows.
i )d b( )ca( )dic( )bia( !
i )d b( )ca( )dic( )bia( !
Sum:
Difference:
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Perform the subtraction and write theanswer in standard form.
( 3 + 2i ) ± ( 6 + 13i )3 + 2i ± 6 ± 13i
±3 ± 11i
234188 i
234298 i i y
234238 i i
4
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Multiplying Complex NumbersMultiplying Complex Numbers
Multiplying complex numbers is similar tomultiplying polynomials and combining liketerms.
Perform the operation and write the result instandard form. ( 6 ± 2i )( 2 ± 3i )
F O I L12 ± 18i ± 4i + 6i 2
12 ± 22i + 6 ( -1 )6 ± 22i
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Consider ( 3 + 2i )( 3 ± 2i )
9 ± 6i + 6i ± 4i 2
9 ± 4( -1 )9 + 4
13
This is a real number. The product of twocomplex numbers can be a real number.
This concept can be used to divide complex number s.
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To find the quotient of two complex numbersmultiply the numerator and denominator
by the conjugate of the denominator.
d i c
bi a
d i c
d i c
d i c
bi a
y
!
22
2
d c
bd i bci ad i ac
!
22
d c
i ad bcbd ac
!
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Perform the operation and write theresult in standard form.
i
i
2
1
76
i
i
i
i
21
21
21
76
y
!
22
2
21
147126
!
i i i
41
5146
!
i
5
520 i !
5
5
5
20 i ! i ! 4
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i i
i
4
31 i
i
i i
i
i
i
y
y
!
4
4
4
31
Perform the operation and write theresult in standard form.
222
2
14312
! i i i i
116
312
11
!
i i
i i 17
3
17
12
1 ! i i 17
3
17
12
1 !
i
17
317
17
1217
! i
17
14
17
5!
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Expressing Complex NumbersExpressing Complex Numbersin Polar Formin Polar Form
Now, any Complex Number can be expressed as:X + Y iThat number can be plotted as on ordered pair
inrectangular form like so«
6
4
2
-2
-4
-6
-5 5
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Expressing Complex NumbersExpressing Complex Numbersin Polar Formin Polar Form
Remember these relationships between polarand
rectangular form: x
y!Utan 222
r y x !
Ucosr x !Usinr
y!
So any complex number, X + Yi, can be wr itten in
polar form: ir r Y i X UU sincos !
)sin(cossincos UUUU ir ir r !
Ur cis
Here is the shorthand way of wr iting polar form:
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Expressing Complex NumbersExpressing Complex Numbersin Polar Formin Polar Form
Rewrite the following complex number in polar form:4 - 2i
R ewr ite the following complex number in
rectangular form: 0307cis
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Expressing Complex NumbersExpressing Complex Numbersin Polar Formin Polar Form
Express the following complex number in
rectangular form:)
3
sin
3
(cos2T T
i
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Expressing Complex NumbersExpressing Complex Numbersin Polar Formin Polar Form
Express the following complex number in
polar form: 5i
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Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form
)sin(cos 111 UU ir
The product of two complex numbers,
and
Can be obtained by using the following formula:
)sin(cos 222 UU ir
)sin(cos*)sin(cos 222111 UUUU ir ir
)]sin()[cos(* 212121 UUUU ! ir r
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Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form
)sin(cos 111 UU ir
The quotient of two complex numbers,
and
Can be obtained by using the following formula:
)sin(cos 222 UU ir
)sin(cos/)sin(cos 222111 UUUU ir ir
)]sin()[cos(/ 212121 UUUU ! ir r
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Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form
Find the product of 5cos30 and ± 2cos120
Next, write that product in rectangular form
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Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form
Find the quotient of 36cos300 divided by
4cis120
Next, write that quotient in rectangular form
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Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form
Find the result of
Leave your answer in polar form.
Based on how you answered this problem,what generalization can we make about
raising a complex number in polar form to
a given power?
4))120sin120(cos5( i
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