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Complex number

1. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Number

2. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Learning Outcomes
Attheendofthelecture,studentshouldbeableto:
understand3basicrepresentationofcomplexnumber.
performconversionfromrectangulartopolaror
exponentformorviceversa.
Performmathematicaloperationincomplexnumber
UnderstandthebasicrelationsofEular’sFormula

3. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Numbers
RepresentationofComplexNumbers:(Rectangular,
PolarandExponentialForm)
MathematicsOperations:Addition,subtraction,
multiplication,division
Euler’sFormula

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Representations of Complex Number

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Representation of Complex Numbers :
Rectangular Form
Acomplexnumberzmaybewritteninrectangularform
as
Example: 3 + j 4

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
ECE201 Lect-4 6
•xis the real part
•yis the imaginary part
•zis the magnitude
•qis the phase
q
x
y
real
axis
imaginary
axis
Representation of Complex Numbers :
Rectangular Form

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Representation of Complex Numbers :
Rectangular Form

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Representation of Complex Numbers :
Polar Form
Asecondwayofrepresentingthecomplexnumberzis
byspecifyingitsmagnituderandtheangleitmakes
withtherealaxis
Example: 5 53.1

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Graphical Representation of Complex
Numbers

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Conversion of Rectangular and Polar
Form

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Conversion of Rectangular and Polar
Form

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Conversion of Rectangular and Polar
Form
Inconvertingfromrectangulartopolarform,wemust
exercisecareindeterminingthecorrectvalueofq

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Representation of Complex Numbers :
Exponential Form
Thethirdwayofrepresentingthecomplexzisthe
exponentialform:

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Representation of Complex Numbers :
Summary

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Practice Example

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.1
Expressthefollowingcomplexnumbersinpolarand
exponentialform:

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.1 (Solution)
Expressthefollowingcomplexnumbersinpolarand
exponentialform:

18. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.1 (Solution)
Expressthefollowingcomplexnumbersinpolarand
exponentialform:

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.1 (Solution)
Expressthefollowingcomplexnumbersinpolarand
exponentialform:

20. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.1 (Solution)
Expressthefollowingcomplexnumbersinpolarand
exponentialform:

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.1 (Solution)

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Conversion Using Scientific Calculator:
Rectangular to Polar Form
Step 1: you have to be careful that you have your calculator set correctly in
degrees
Press “mode”4 times until the screen appear DegRad Gra( Choose 1)
Step 2: Change to Complex Number Mode
Press “mode”1 times until the screen appear Comp (1) CMPLX (2)
( Choose 2)
Step 3: insert rectangular form of complex number i.e-6-j8
First press -6-8, then continue by press “i” symbol (purple color) (same as
“ENG”) and finally press ‘=‘. You will see RI symbol appear on the top
right corner.
Step 4: to convert to polar, press “SHIFT” followed by “+” symbol and then
“=“ button. You will see value of magnitude r appear.

23. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Conversion Using Scientific Calculator:
Rectangular to Polar Form
Step 5: In order to display angle or theta value, change from real to imaginary
Press “SHIFT”followed by = button,, the value of angle are display
accordingly.
Take note that the value of angle appear need to be subtract with 360 degin
order to obtained the correct value of theta.
i.e. Quadrant 3: 360deg-126.86deg= 233deg
To confirm your results of conversion, you may recheck with the online
calculator from the following webpage:
https://www.intmath.com/complex-numbers/convert-polar-rectangular-
interactive.php

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Mathematical Operation

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Number Addition and
Subtraction
Perform addition and subtraction of complex numbers in rectangular form
We add/minus (algebraically) real part to real part and imaginary part to
imaginary part
Addition
Subtraction

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Number of Multiplication
the product/multiplication and quotient/division of the two complex
numbers are best done in polar or exponential form
Multiply numbers in polar form, we multiply the magnitudes and add the
angles.
Multiplication

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Number of Division
the product/multiplication and quotient/division of the two complex
numbers are best done in polar or exponential form
To divide numbers in polar form, we divide the magnitudes and subtract the
angle of the divisor from the angle of the dividend
Division

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.3: Mathematical Operation

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.3: Solution
=5.4<68.2
0
×6<−9.4
0
=5.4<68.2
0
×6<−9.4
0
=32.4<58.8
0
@17+??????28

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.3: Solution
=17+28??????

32. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.3: Solution
=5.4<68.2
0
×6<−9.4
0
=5.4<68.2
0
×6<−9.4
0
=32.4<58.8
0
@17+??????28

33. © FTK UTHM 2018
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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Example B.3: Solution

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Conjugate

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Complex Conjugate
The complex conjugate of a complex number is found by replacing every j
by -j
(Notice that we denote the complex conjugate by the symbol *)
-
+

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Euler’s Formula

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Euler’s Formula
Euler’s formula is an important result in complex variables.

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Euler’s Useful Identities
Euler’s formula is an important result in complex variables.

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Practice

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Practice Problem

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Practice Problem

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Practice Problem

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
=−3+??????7−??????−??????×[−3+??????7+??????−??????]
=(−11+??????8)(5+??????6)
=−103−??????26
Answer

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
=
(8+??????)(8+??????)
−3−??????7
=
63+??????16
−3−??????7
×
−3+??????7
−3+??????7
=
−189+441??????−48??????−112
9−21??????+21??????+49
Answer

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
=
2(−31+??????53)
5+??????8
=
2(−31+??????53)
5+??????8
×
5−??????8
5−??????8
=
538+??????1026
89
=
Answer

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Practice Problem

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Answer
=
2.2+??????8
0.414+??????2.414
=3.37−??????0.33
@3.37∠−5.62°

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Complex Number Revisit (Electrical Principles II:BNR10303)
By Dr.LamHong Yin
Answer
=−0.512+??????0.897