Chapter 4 Complex Numbers

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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.

Documents

Chapter 4 Complex Numbers