2. Complex Number System (Option 1)

8/6/2019 2. Complex Number System (Option 1)

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COMPLEX NUMBER COMPLEX NUMBER

SYSTEMSYSTEM

1



y COMPLEX NUMBER

NUMBER OF THE FORM C= a+Jba = real part of Cb = imaginary part.

2



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



R eal Number sImaginary

Number s

Real numbers and imaginary numbers aresubsets of the set of complex numbers.

Complex Number s



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



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.



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.



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,



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



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



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«



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:



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



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



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.



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

!



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



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!



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




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:




Rewrite the following complex number in polar form:4 - 2i

R ewr ite the following complex number in

rectangular form: 0307cis




Express the following complex number in

rectangular form:)

3

sin

3

(cos2T T

i




Express the following complex number in

polar form: 5i



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




)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




Find the product of 5cos30 and ± 2cos120

Next, write that product in rectangular form




Find the quotient of 36cos300 divided by

4cis120

Next, write that quotient in rectangular 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

2. Complex Number System (Option 1)

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