Complex.numbers

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).


Complex Numbers


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


Complex Numbers


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


Complex Numbers


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 .


Complex Numbers



Complex Numbers


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)


Complex Numbers


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)


Complex Numbers


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


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


Complex Numbers


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).


Complex Numbers



Complex Numbers


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.


Complex Numbers


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.


Complex Numbers


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))


Complex Numbers


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.


Complex Numbers


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)


Complex Numbers


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)


Complex Numbers


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.


Complex Numbers


Further Reading

HELM modules 10.1, 10.2 and 10.3.


Complex.numbers

Documents

Transcript of Complex.numbers