complex.numbers
TRANSCRIPT
Complex Numbers
Mathematical Methods 2CHEN 20041
Dr Jelena Grbic
University of ManchesterSchool of Mathematics
Contact: after class, in tutorials, or by email
Office hours: Monday 1-2pm in AT 2.114
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Definition of Complex Numbers
A complex number is a generalisation of an ordinary realnumber.A complex number is an ordered pair of real numbers, usuallydenoted by z or w , etc.If a, b are real numbers then we designate a complex number
z = a + ib
where i is a symbol obeying the rule
i2 = −1
For simplicity we shall assume we can write
i =√−1
If z = a + ib, then a is called the real part of z , or Re(z) andb is called the imaginary part of z or Im(z).
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
The real numbers can be considered as a subset of thecomplex numbers where the imaginary part is zero.
Two complex numbers z = a + ib and w = c + id are said tobe equal if and only if a = c and b = d .
The modulus of a complex number z = a + ib is denoted by|z | and is defined by
|z | =√a2 + b2.
The conjugate of a complex number z = a + ib is the complexnumber
z = a− ib
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Example
To find the roots of the quadratic equation
x2 − 2x + 5 = 0
we use the formula
x =−b ±
√b2 − 4ac
2a
to obtain x = 2±√−16
2 . This shows that there are no real roots butthere are two complex roots
x = 1± i2
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Argand Diagram
We introduce a geometrical interpretation of a complexnumber.
There is a close connection between complex numbers andtwo-dimensional vectors.
Consider the complex number z = x + iy .
Complex number is specified by two real numbers x , y .
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Addition and subtraction of complex numbers
Let z and w be any two complex numbers
z = a + ib w = c + id
then
z + w = (a + c) + i(b + d) z − w = (a− c) + i(b − d)
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Multiplication of complex numbers
Consider any two complex numbers z = a + ib and w = c + id .Then
zw = (a + ib)(c + id)
= ac + aid + ibc + i2bd
= ac − bd + i(ad + bc)
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Division of complex numbers
Consider any two complex numbers z = a + ib andw = c + id . Then
z
w=
a + ib
c + id
We want to write the result as a complex number in theCartesian rectangular form, that is, to specify what are thereal and imaginary parts.
Rationalise the complex number zw = a+ib
c+id :
z
w=
a + ib
c + id=
a + ib
c + id× c − id
c − id
=
(ac + bd
c2 + d2
)+ i
(bc − ad
c2 + d2
)Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Example
(1 + i2)(3 + i) = 3 + i6 + i + 2i2 = 1 + i7
Example
1
i=
i
i2= −i
Example
Show that zz is always a real number.
zz = (a + ib)(a− ib) = a2 + b2 = |z |2
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
The Polar Form
The Cartesian (rectangular) form is not the most convenientform when we come to consider multiplication and division ofcomplex numbers. A much more convenient form is the polarform which we now introduce.
Using the Argand diagram, the complex number z = a + ibcan be represented by a vector pointing out from the originand ending at a point with Cartesian coordinates (a, b).
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Complex numbers and rotations
When multiplying one complex number by another, the modulimultiply together and the arguments add together.Let
w = t(cosφ+ i sinφ)
andz = cos θ + i sin θ
Their product is
wz = t(cos(θ + φ) + i sin(θ + φ))
Remark: This result would certainly be difficult to obtain had wecontinued to use the Cartesian rectangular form.
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
The effect of multiplying w by z is to rotate the line representingthe complex number w anti-clockwise through an angle θ which isarg(z), and preserving the length.
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Exercise
Using the polar form of complex numbers, prove that ifz = r(cos θ + i sin θ) and w = t(cosφ+ i sinφ), then
z
w=
r
t(cos(θ − φ) + i sin(θ − φ))
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
The exponential form
We introduce a third way of expressing a complex number:the exponential form.
Using the complex number notation, we discover the intimateconnection between the exponential function and thetrigonometric functions.
We show that using the exponential form, multiplication anddivision of complex numbers are easier than when expressed inpolar form.
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Property of the exponential function (Euler’s formula):
e iθ = cos θ + i sin θ
Let z = r(cos θ + i sin θ). Then the exponential form of z isgiven by
z = re iθ
As a special case we arrive at the famous formula relatingnumbers 0, 1, i , e and π:
e iπ + 1 = 0.
Using the Euler formula, we describe the connection betweenthe exponential function and the trigonometric functions:
cos θ =1
2(e iθ + e−iθ) sin θ =
1
2i(e iθ − e−iθ)
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Example
As a special case we arrive at the famous formula relating numbers0, 1, i , e and π:
e iπ + 1 = 0.
Example
Using the Euler formula, we describe the connection between theexponential function and the trigonometric functions:
cos θ =1
2(e iθ + e−iθ) sin θ =
1
2i(e iθ − e−iθ)
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Problem
Using the exponential form of a complex number, show that therules for the multiplication, division and modulus of a complexnumber are as follows
z1z2 = r1eiθ1r2e
iθ2 = r1r2ei(θ1+θ2)
z1
z2=
r1r2e i(θ1−θ2)
|z | = |r iθe | = r
Problem
Express a rule for a complex conjugation in polar form.
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041
Complex Numbers
DefinitionArgand DiagramArithmetic OperationsForms
Further Reading
HELM modules 10.1, 10.2 and 10.3.
Dr Jelena Grbic Mathematical Methods 2 CHEN 20041