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Chapter 4: Complex Numbers
SIE1002 Engineering Mathematics 1
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Learning Outcomes
After studying this chapter, you should
1. Understand how quadratic equations lead to complex numbers and
how to plot complex numbers on an Argand diagram;
2. Be able to do basic arithmetic operations on complex numbers ofthe form a + ib;
3. Understand the polar form [r, ] of a complex number and its
algebra;
4. Understand Eulers relation and the exponential form of a complexnumber rei;
5. Be able to use de Moivres theorem.
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Introduction
The history of complex numbers goes back to the ancient Greeks who
decided that no number existed that satisfies
For example (a problem posed by Cardan in 1545):
Find two numbers, a and b, whose sum is 10 and whose product is 40.
Eliminating b gives,
Solving this quadratic gives,
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Introduction
This shows that there are no real solutions, but if it is agreed to continue
using the numbers
Then equations (1) and (2) are satisfied.
So these are solutions of the original problem but they are not real numbers.
The square root of -1 is denoted by i orj.
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Complex Number
It is a combination of a Real Numberand an Imaginary Number.
Real Numbers are:
8 88.88 -0.168
Imaginary Numbers are:
XWhen squared, they give a negative result.Normally this doesnt happen, because:
- When you square a positive number you get a positive result;
- When you square a negative number, you also get a positive result.
Just imagine there is such a number! And we are going to need it!
The unit imaginary number (like 1 for Real Numbers) is i.
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Example 1
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Interesting Property
It is a combination of a Real Numberand an Imaginary Number.
Observations:
(i) in repeats the pattern i, 1, i, 1 periodically.
(ii) in always resets to 1 when n is a multiple of four.
These observations allow us to infer the values of larger powers ofi.
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Example 2
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Complex Number Property
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The Argand Diagram
Geometrically, complex numbers can be represented as points on an x-y
plane. The graphical representation of the complex number field is called anArgand Diagram, named after the Swiss mathematician Jean Argand (1768-
1822).
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Example 3
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Arithmetic: equality, addition & subtraction
The rules for adding and subtracting complex numbers are verystraightforward:
To add (or subtract) complex numbers, we simply add (or subtract) their
real parts and their imaginary parts separately.
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Example 4
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Arithmetic: multiplication
Example 5
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Arithmetic: division
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Complex conjugate
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Arithmetic: division
Hence, to solve
We multiply the numerator and denominator of the quotient by the complex
conjugate of the denominator.
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Example 6
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Polar form of a complex number
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Polar form of a complex number
The Polar form:
Since the polar coordinates (r, ) and (r, +2) represent the same point, a
convention is used to determine the argument of z uniquely, restricting its
range to the principal value, where
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Example 7
(a)
(b)
(c)
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Discussion
Example 8
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Advantage of Polar Form - Multiplication
Now, we are familiar with converting from Cartesian to Polar form. Lets see
its advantage in multiplication.
212121
212121
sincoscossinsinsinsincoscoscos
Thus, simplifying our multiplication to
21212121 sincos irrzz
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Advantage of Polar Form - Division
For division,
Hence, we realize by now that the chosen form of a complex number does
affect how conveniently certain arithmetic operations are carried out.
Generally, the polar form is suited for multiplication and division, whereas
the Cartesian form is suited for addition and subtraction.
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Example 9
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De Moivres theorem
An important theorem in complex numbers is named after the French
mathematician, Abraham de Moivre (1667-1754). Although born in France,he came to England where he made the acquaintance of Newton and Halley
and became a private teacher of Mathematics.
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De Moivres theorem
Example 10
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Example 11
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Roots of a complex number
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Roots of a complex number
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Example 12
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Example 12
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Eulers formula
Example 13
(a) (b)
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Engineering Application
RZ
Alternating currents in electrical networks
Voltage is in phase with the
current.
C
jZ
Impedance,
Impedance,
LjZ Impedance,
Angular frequency, f 2
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Example 14
Calculate the complex impedance of the elements shown below when an
alternating current of frequency 100 Hz flows.
The complex impedance is the sum of the individual impedances.
LjRZ
)103.41)(1002(15 3 jZ
9.2515 jZ
30Z and 3
1
30Z 3
1[ans: and ]
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Example 15
55
1
1
11
jjZ
55
55
55
1
1
1
1
11
j
j
jj
j
jZ
50
55
2
11 jj
Z
1.05.01.05.01
jZ
4.06.01
jZ
4.06.0
1
jZ
52.0
4.06.0 jZ
7692.01538.1 jZ
4.06.0
4.06.0
4.06.0
1
j
j
jZ
7692.01538.1 jZ
[ans: ]
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References
Modern Engineering Mathematics, 4th edition with MyMathLab, Glyn James,
Pearson. MATLAB for Engineers, 3rd edition, Holly Moore, Pearson.