Transcript 01 Complex numbers 1 Powerpoint

PROGRAMME 1
COMPLEX
NUMBERS 1
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Introduction
Ideas and symbols
The numerals were devised to enable written calculations and records of
quantities and measurements. When a grouping of symbols such as 1
occurs 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 1 an imaginary number to distinguish it from those numbers to
which we can associate quantity which we call real numbers.
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
The symbol j
Quadratic equations
The solutions to the quadratic equation:
x2  1  0
are:
x  1 and x  1
The solutions to the quadratic equation:
are:
x2  1  0
x   1 and x   1
We avoid the clumsy notation by defining
STROUD
j  1
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Powers of j
Positive integer powers
Because: j  1
so:
j 2  1
j3  j 2 j   j
j  j
4
2
   1
2
2
1
j5  j 4 j  j
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Powers of j
Negative integer powers
1
j 2  1 so j     j 1
j
Because:
and so:
j 1   j
j 2   j 2    1  1
1
1
j 3   j 2  j 1    1  j   j
j
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4
 j
2
   1
2
2
1
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Complex numbers
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
Programme 1: Complex numbers 1
Complex numbers
Addition and subtraction
The real parts and the imaginary parts are added (subtracted) separately:
and so:
(4  j 5)  (3  j 2)
 4  j5  3  j 2
 4  3  j5  j 2
 7  j3
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Complex numbers
Multiplication
Complex numbers are multiplied just like any other binomial product:
and so:
(4  j 5)  (3  j 2)
 4(3  j 2)  j5(3  j 2)
 12  j8  j15  j 210
 12  j8  j15  10
 22  j 7
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because j 2  1
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Complex numbers
Complex conjugate
The complex conjugate of a complex number is obtained by switching the sign
of the imaginary part. So that:
(5  j8) and (5  j8)
Are complex conjugates of each other.
The product of a complex number and its complex conjugate is entirely real:
(a  jb)  (a  jb)
 a( a  jb)  jb( a  jb)
 a 2  jba  jba  j 2b 2
 a 2  b2
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Complex numbers
Division
To divide two complex numbers both numerator and denominator are
multiplied by the complex conjugate of the denominator:
7  j 4  7  j 4   4  j 3


4  j 3  4  j 3  4  j 3
 7  j 4    4  j3
 4  j 3   4  j 3
16  j 37 


16  9 


STROUD
16
37
j
25
25
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Equal complex numbers
If two complex numbers are equal then their respective real parts are equal
and their respective imaginary parts are equal.
If a  jb  c  jd then a  c and b  d
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Graphical representation of a complex number
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
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Graphical addition of complex numbers
Complex numbers add (subtract) according to the parallelogram rule:
(5  j 2)  (2  j3)  7  j5
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Polar form of a complex number
A complex number can be expressed in
polar coordinates r and .
z  a  jb
 r (cos   j sin  )
where:
a  r cos , b  r sin 
and:
r 2  a 2  b2
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
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
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Exponential form of a complex number
Recall the Maclaurin series:
x 2 x3 x 4 x5
e 1 x 
 
 
2! 3! 4! 5!
x3 x5 x 7
sin x  x    
3! 5! 7!
x2 x4 x6
cos x  1  
 
2! 4! 6!
x
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Exponential form of a complex number
So that:
e j  1 
 j 
j 
 1  j 
2
2!
2
STROUD
3
3!
3

3!
 2 4

 1 

 
2!
4!


 cos   j sin 
2!
j
 j 

4
 j 

4!
4
 j 

5!
5

5
 j 
4!
5!

3 5
j  


3!
5!




Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Exponential form of a complex number
Therefore:
z  r  cos  j sin    re j
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Exponential form of a complex number
Logarithm of a complex number
Since:
z  re j
then:
ln z  ln r  ln e j  ln r  j
STROUD
Worked examples and exercises are in the text
Programme 1: Complex numbers 1
Learning outcomes
Recognise j as standing for 1 and be able to reduce powers of j to  j or 1
 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
STROUD
Worked examples and exercises are in the text