stroud worked examples and exercises are in the text programme 1 complex numbers 1
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STROUD
Worked examples and exercises are in the text
PROGRAMME 1
COMPLEX NUMBERS 1
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Introduction
Ideas and symbols
Programme 1: Complex numbers 1
The numerals were devised to enable written calculations and records of quantities and measurements. When a grouping of symbols such asoccurs to which there is no corresponding quantity we ask ourselves why such a grouping occurs and can we make anything of it?
In response we carry on manipulating with it to see if anything worthwhile comes to light.
We call an imaginary number to distinguish it from those numbers to which we can associate quantity which we call real numbers.
1
1
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
The symbol j
Quadratic equations
Programme 1: Complex numbers 1
The solutions to the quadratic equation:
are:
The solutions to the quadratic equation:
are:
We avoid the clumsy notation by defining
2 1 0x
1 and 1x x
2 1 0x
1 and 1x x
1 j
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Powers of j
Positive integer powers
Programme 1: Complex numbers 1
1 j Because:
so:
2
3 2
2 24 2
5 4
1
1 1
j
j j j j
j j
j j j j
STROUD
Worked examples and exercises are in the text
Powers of j
Negative integer powers
Programme 1: Complex numbers 1
Because:
and so:
1
1 12 2
3 2 1
2 24 2
1 1
1
1 1
j j
j j
j j j j j
j j
2 111 so j j j
j
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Complex numbers
Programme 1: Complex numbers 1
A complex number is a mixture of a real number and an imaginary number. The symbol z is used to denote a complex number.
In the complex number z = 3 + j5:
the number 3 is called the real part of z and denoted by Re(z)
the number 5 is called the imaginary part of z, denoted by Im(z)
STROUD
Worked examples and exercises are in the text
Complex numbers
Addition and subtraction
Programme 1: Complex numbers 1
The real parts and the imaginary parts are added (subtracted) separately:
and so:(4 5) (3 2)
4 5 3 2
4 3 5 2
7 3
j j
j j
j j
j
STROUD
Worked examples and exercises are in the text
Complex numbers
Multiplication
Programme 1: Complex numbers 1
Complex numbers are multiplied just like any other binomial product:
and so:
2
2
(4 5) (3 2)
4(3 2) 5(3 2)
12 8 15 10
12 8 15 10 because 1
22 7
j j
j j j
j j j
j j j
j
STROUD
Worked examples and exercises are in the text
Complex numbers
Complex conjugate
Programme 1: Complex numbers 1
The complex conjugate of a complex number is obtained by switching the sign of the imaginary part. So that:
Are complex conjugates of each other.
The product of a complex number and its complex conjugate is entirely real:
(5 8) and (5 8)j j
2 2 2
2 2
( ) ( )
( ) ( )
a jb a jb
a a jb jb a jb
a jba jba j b
a b
STROUD
Worked examples and exercises are in the text
Complex numbers
Division
To divide two complex numbers both numerator and denominator are multiplied by the complex conjugate of the denominator:
7 4 4 37 4
4 3 4 3 4 3
7 4 4 3
4 3 4 3
16 37
16 9
16 37
25 25
j jj
j j j
j j
j j
j
j
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Equal complex numbers
Programme 1: Complex numbers 1
If two complex numbers are equal then their respective real parts are equal and their respective imaginary parts are equal.
If then and a jb c jd a c b d
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Graphical representation of a complex number
Programme 1: Complex numbers 1
The complex number z = 1 + jb can be represented by the line joining the origin to the point (a, b) set against Cartesian axes.
This is called the Argrand diagram and the plane of points is called the complex plane.
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Graphical addition of complex numbers
Programme 1: Complex numbers 1
Complex numbers add (subtract) according to the parallelogram rule:
(5 2) (2 3) 7 5j j j
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Polar form of a complex number
Programme 1: Complex numbers 1
A complex number can be expressed in polar coordinates r and .
where:
and:
(cos sin )
z a jb
r j
cos , sina r b r
2 2 2r a b
STROUD
Worked examples and exercises are in the text
Introduction
The symbol j
Powers of j
Complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Exponential form of a complex number
Programme 1: Complex numbers 1
Recall the Maclaurin series:
2 3 4 5
3 5 7
2 4 6
12! 3! 4! 5!
sin3! 5! 7!
cos 12! 4! 6!
x x x x xe x
x x xx x
x x xx
STROUD
Worked examples and exercises are in the text
Exponential form of a complex number
So that:
2 3 4 5
2 3 4 5
2 4 3 5
12! 3! 4! 5!
12! 3! 4! 5!
12! 4! 3! 5!
cos sin
j j j j je j
j j j
j
j
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Exponential form of a complex number
Therefore:
z cos sin jr j re
Programme 1: Complex numbers 1
STROUD
Worked examples and exercises are in the text
Since:
then:
z jre
Programme 1: Complex numbers 1
Exponential form of a complex number
Logarithm of a complex number
ln ln ln lnjz r e r j
STROUD
Worked examples and exercises are in the text
Learning outcomes
Recognise j as standing for and be able to reduce powers of j to or
Recognize that all complex numbers are in the form (real part) + j(imaginary part)
Add, subtract and multiply complex numbers
Find the complex conjugate of a complex number
Divide complex numbers
State the conditions for the equality of two complex numbers
Draw complex numbers and recognize the paralleogram law of addition
Convert a complex number from Cartesian to polar form and vice versa
Write a complex number on its exponential form
Obtain the logarithm of a complex number
Programme 1: Complex numbers 1
1 j 1
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