ComplexNumbers(Summary)

7/31/2019 ComplexNumbers(Summary)

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Complex Numbers (Summary ) MS410Z/MS418Z/MS510Z/MS516Z

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Complex Numbers (Summary)

I Rectangular or Cartesian Form (x + jy )

Every complex number can be written in rectangular or cartesian form as

z = x + jy

wherex andy are real numbers and 1j = .

The numbersx andy are called respectively the real part and imaginary part of z, written as

x = Re(z) and y = Im(z)

Complex Plane ( Argand Diagram )

A complex number z = a +jb can be represented by a point (a, b) in a coordinate plane,

called the complex plane. The horizontal axis is the real axis and the vertical axis is the

imaginary axis.

The complex number z is represented by the point P or the vector OP .

Basic Operations in Rectangular Form

1. Equality

If a + jb = c + jd,

Then a = c and b = d .

2. Addition and Subtraction

(a +jb) + (c +jd) = (a + c) + j (b + d) (a +jb) (c +jd) = (ac) + j (bd)

Imaginary axis

Real axis

P (a, b)

O


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3. Multiplication

(a +jb) (c +jd) = ac +jad+jbc +j2bd = (acbd) + j (ad+ bc) Mutiplying a complex numberz byj effectively rotates the vector representingz on the

complex plane by 90 0 anti-clockwise about the origin.

4. Conjugate complex numbers

Numbers of the form (a +jb) and (ajb) are said to be conjugate complex numbers.Their product is always a real number .

(a + jb)(a jb) = a2

(jb)2

= a2

j2b

2

= a2

+ b2

If z denotes a complex number , then z is the notation for the conjugate ofz .

5. Division

a jb

c jd

+

+=

( )( )

( )( )

a jb c jd

c jd c jd

+

+ =

( ) ( )ac bd j bc ad

c d

+ +

+2 2

II The Polar Form ( r )

z x jy= + rectangular form

= ( )cos sinr j + trigonometric form

= r polar form

we see that : x r= cos and y r= sin

The modulus or absolute value of z is given by z = r = 22 yx +

The angle is called the argument of the complex number z and written as arg z.

i.e. arg z = where tany

x = , < or 180 180 <

.

Imaginary axis

Real axis

P(x,y)

y

x0

r


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Basic Operations in Polar Form


Addition and subtraction of complex numbers can only be done in rectangular form

2.Multiplication 1(r 1 2() r 2 ) = 1 2r r 1 2+

3. Divisionr

r

1

2

1

2

=

r

r

1

2

1 2 , 2 0r

4. De Moivres Theorem ( )n

r = nr n for any integer n

III The Exponential Form of a Complex Number ( re j)

Eulers Formula: cos sinje j = +

Hence z = ( )cos sinr j +

= re j where is the argument expressed in radians

Basic Operations in Exponential Form


Addition and subtraction of complex numbers can only be done in rectangular form .

2. Multiplication( )1 21 2

1 2 1 2

jj jr e r e r r e

+ =

3.Division( )11 1 1 2

2 22

jr e r j

ej r

r e

=

4. Raising to a powe for any integer nn

nj jnre r e

=


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Supplementary Examples

Question 1: Express )1( 42 jj eje in terms of sine only.

Solution 1: )()1( )42(242 = jjjj eejeje

( ) 22 jj eej =

=

j

eejj

jj

2)2(

22

)2(sin2 2 j=

2sin2=

Question 2: Express

+

j

jj

e

ee24

3 in terms of cosine only.

Solution 2: )(33 2)4(24

jjj

j

jj

eee

ee

+=

+

)(3 33 jj ee +=

+=

26

33 jjee

3cos6=

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