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- We will cover 5 topics today
- 1. Complex Numbers Definitions and Rules
- 2. The Argand Diagram
- 3. Complex Numbers and Polar Coordinates
- 4. Complex Numbers in Exponential Form
- 5. Applications of Complex Numbers

Equality

if

and

A complex number is defined as

Sum

What is i3, i4, i5 and i6?

Subtraction

Multiplication

Conjugate

The standard form of a complex number is

Division

Real part

Imaginary part

The polar angle θ is called the argument of z and is written ‘arg z’. Polar angles differing by an angle of 2π are equivalent

A complex number in the form z = x + i.ycan be represented by a pair of real numbers (x,y) known as an ordered pair. This pair of numbers can be expressed on a Cartesian axis and this is called an Argand diagram.

r is called the modulus of z (or mod z) and is written |z|

Imaginary Axis, y

Properties of the modulus

r

Real axis, x

Imaginary Axis, y

b (0,2)

Let z = 1 + i. Plot the following complex numbers on an Argand Diagram

a)

b)

c)

d)

d (0,1)

Real axis, x

a (1,-1)

c (0,-2)

Complex Numbers and Polar Coordinates

Recall the following diagram

i.e. r is equal to the modulus of z

Imaginary Axis, y

θ = Arg z

The Principal Value of the Argument

r

The pair of equations

Real axis, x

has exactly one solution for θ within this range. By substitution we can say

The complex number can be specified in terms of the polar coordinates ‘r’ and ‘θ’.

Complex Numbers in Exponential Form

Consider the function

Hence, we can conclude that

If we take the Taylor expansion of both terms we find that

and

Where |z| = r and θ = Arg(z)

We also know that

Complex Numbers in Exponential Form

If

Then

Then the conjugate is written

This is called De Moivre’s Theorem

Hence, we can deduce that

Also,

Hence,

We can also deduce that because

Complex Numbers in Exponential Form

Express the following complex numbers in exponential form using principal values of the arguments

(2) -5i,

Thus

(1) i,

In each case put r.Cos(θ) equal to the real part and r.Sin(θ) equal to the imaginary part.

(3) -3,

Thus

Now we use an Argand diagram to calculate the principal value of the angle θ

thus

Find all of the solutions to the equation

Therefore

We first express 4 - 4.i in polar form, thus

i.e.

Hence,

Five successive values of n give distinct solutions; other values of n merely duplicate existing solutions.

And by using an Argand diagram

The five solutions are

Let hence

Expand Cos6(θ) in terms of θ

We know n = 1 hence, (using a Binomial Expansion)

By De Moivre’s theorem

and

Hence

By adding these together we get

Use the binomial theorem, or otherwise, to expand the expression

De Moivre’s theorem gives that

for any positive integer n. Taking n = 4 use the above to show that

And obtain an analogous expression for Sin(4θ)

We know that

Now we can collect the real and imaginary terms together

Equate real parts to give

Equate imaginary parts to give

Hence

It is given that the polynomial

can be written in the form

where q(x) is a quadratic

a) Obtain q(x)

b) Calculate the roots of q(x)

c) Plot the roots of q(x) on an Argand diagram

a) Obtain q(x)

If then

Equate coefficients of x

x4 : 1 = a i.e. a = 1

x3 : -2 = b i.e. b = -2

x2 : -3 = c – 9a i.e. c = 9a – 3 = 6

Hence

b) The roots of q(x) are given by the quadratic formula

c) Argand Diagram

Imaginary Axis, y

(1,√5)

Real axis, x

(1,-√5)

Obtain the modulus and argument (in radians) of the complex numbers z1 = 2 + 3i and z2 = -1 – i and write z1 and z2 in the polar form r.ei.θ

Hence or otherwise, determine the polar form of the complex numbers

a)

b)

Given

With

Hence

With

However from the Argand diagram we can determine that

Hence the correct root is -3π/4

Therefore

Today we have looked at

1. Complex Numbers Definitions and Rules

2. The Argand Diagram

3. Complex Numbers and Polar Coordinates

4. Complex Numbers in Exponential Form

5. Applications of Complex Numbers

- Essential reading for next week
- HELM Workbook 10.1: Complex Arithmetic
- HELM Workbook 10.2: Argand Diagrams and the Polar Form
- HELM Workbook 10.3: The Exponential Form of a Complex Number
- HELM Workbook 10.4: De Moivre’s Theorem
- OR
- CHAPTER 11 ofMathematics for Engineers (Croft & Davidson)

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