complexnumbers(summary)
TRANSCRIPT
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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
1. Addition and Subtraction
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
1. Addition and Subtraction
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=