K.J.WILLIAMS

School of Mechanical & Auto Engineering

Faculty of Science, Engineering & Computing

KINGSTON UNIVERSITY

Learning Objectives

• To revise your knowledge of Complex

Numbers in preparation for further work on the application of complex numbers to engineering applications – A.C. electrical/ electronic systems

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Origins of Complex Numbers

• Complex numbers arise naturally from problems in mathematics involving the roots of negative numbers. • eg :– in quadratic equations such as:• ax2 + bx + c = 0, the general solution is:• x = -b +/- √(b2 – 4ac)

2a

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Origins of Complex Numbers

• If b2 happens to be less than 4ac, then the contents of the square root will be negative – eg. √-9

• Normally, we cannot have a solution to this! NOTE :– 3 x 3 = 9 and -3 x -3 = 9

• but no number when multiplied by itself has the answer – 9!

• There is no such number!

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Origins of Complex Numbers

• This problem was known to mathematics for a long time and was a stumbling block for a while.

• Eventually, a solution was found which involved a dodge that separated the minus part from the positive part.

• eg:- -9 = (-1) x 9 so that √-9 = √(-1) x 9

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Origins of Complex Numbers

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•

•

•

•

Now:- -9 = 9 x (-1), so that

√-9 = √(9 x (-1) ),

= √9 x √(-1),

= 3 x √(-1).

Mathematicians gave a special symbol to the √(-1) and they called it i, where:• i = √(-1)

• So the answer to √-9 = i3 or 3i

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Real & Imaginary Numbers

• Numbers such as 3i are called imaginary number, because normally, we don’t encounter them and after all, it is just a dodge to get out of a problem.

• The numbers we normally encounter in our everyday life are called real numbers,

These are numbers such as:- integers, fractions and decimals eg:- 100, 50, 2/3,

0.5 and so on.

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Complex Numbers

• So now, in mathematics, we have two types of numbers:• Real Numbers:– 25, 1034, ¾, 0.6 and so on. • Imaginary Numbers:- 25i, 1034i, 3/4i, 0.6i

• and so on.

• Then, mathematicians considered the possibility of combing the two to form

Complex numbers.

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Complex Numbers

• So Complex numbers are a combination of

Real and Imaginary numbers.

• Real Numbers:- 50, -0.6, 1/8.

• Imaginary numbers:- -50i, 0.6i, 1/8i.

• Complex Numbers:-(3 + 4i),(-2 -5i),(4-0.6i)

• Just as in algebra, we can use x to represent a real number, we can also use letters to represent complex numbers:• eg:- Z = 3 +4i

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i or j

• Mathematicians originally used the letter i to indicate imaginary numbers.

• Later on, however, electrical engineers found that complex numbers could be used to analyse

A.C. electrical circuits. The problem was that engineers were already using the symbol “i” to represent current – so they changed the symbol for √(-1)from i to j. Nowadays, mathematicians still use i for imaginary numbers while electrical engineers prefer to use j to avoid confusion.

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Cartesian (X-Y or Rectangular)

Coordinates

• Complex numbers can be thought of as two dimensional numbers since the real part and imaginary part are separate. We can plot complex numbers graphically in a two dimensional complex plane using

Cartesian (x-y rectangular) coordinates with Real numbers plotted horizontally and

Imaginary numbers plotted vertically. This is also known as an Argand Diagram.

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Cartesian (x-y rectangular)

Coordinates

• Argand Diagram or Complex Plane

Z = 3+4i

Imaginary numbers 4i

0

3

Real numbers

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Polar Coordinates

• In addition to using Cartesian coordinates for complex numbers, it is often convenient to represent them on a Polar

Plot of magnitude (or modulus) vs angle

(or argument). Even though we can do this, the complex number still consists of two distinct parts which can be easily recovered. 13

Modulus & Argument

• Using Cartesian (x/y, or rectangular) coordinates, a complex number is represented as a point on the complex plane (or Argand diagram).

• Using Polar coordinates the same complex number is represented by a vector with a magnetude (modulus) and an

angle(argument)