Complex number

1
Complex number

• reminding

2
A.1 Introduction
Real number
Complex number
3
j value and quadratic equation
4
??????????? ????????????
Complex number
x(t)
1D Analog Sine Signal
1D Analog Cos Signal
x(t)
A
0
0
5
?????????????????????? Complex

• Simple algebraic rules ?????????????????????????
  ?????? Real ???????????????????? j2 ??????? -1
• Elimination trigonometry ????????? Eulers
  formula ( )
  ?????????????? ?????????? ????????????????????????
  ?
• Representation by vector ????????????????????????
  ???????????????????????????????????????

6
A.2 Notation for Complex number
y (Imaginary)

• Rectangular form

2j5
5.3 68
r
0j3

• Polar form

-4j0
x (Real)
4
4
4-j4
7
Complex plane
imaginary
( j )
? 90
x(t) 3j4 5 53.1

5
4
real
?
? 180
-

? 0
3
-
( -j )
? 270
Complex plane
8
Conversion Rect Polarform

• ExampleA.1

9
A.2.4 Difficulty in 2,3,4 quadrant
imaginary
? 90
( j )
2
1
? (p) 180
real
?
?
?
?
? 0
3
4
( -j )
? 270
10
A.3 Eulers Formula
imaginary
( j )
? 90
1
1
real
?
? 180

• Rectangular form

? 0
1
( -j )
? 270

• Polar form

x(t) 1j1
Exponential complex form
11
Exponential , Rectangular and Polar form
Rad Degree
Ex 3.14159 rad ? Degree note
3.14159
12
Inverse Euler Formulas
r 1
13
A.4 Algebraic rules
14
Polar form (MultiplicationDivision)
15
A.5 Geometric views
A.5.1 Geometric View of Addition
Im
6j2
2
Re
6
4
3
16
z5 (1j) (-1j) (-1-j) (1-j)
Im
Re
Z5 0
17
A.5.2 Geometric View of Subtraction
Im
-2j8
8
Re
-2
4
3
18
A.5.3 Geometric View of Multiplication
Im
Im
Re
Re
19
A.5.4 Geometric View of Division
Im
Re
20
A.5.5 Geometric View of Inverse
Im
Im
Re
Re
21
A.5.6 Geometric View of Conjugate
Im
Im
Re
Re
22
????????????????? Complex number

• Summing

23
????????????????? Complex number
2) Power
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????????????????? Complex number
3) Power
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????????????????? Complex number
4) Power