# COMPLEX NUMBER

COMPLEX NUMBER

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The solution of a quadratic equation can be obtained by the formula.

Example

i) If

This is straight forward enough.

ii) If

In fact  cannot be represented by an ordinary number.

Similarly,

Then from the equation that we have been solving it gives that

Since  stand for

Note that

The power of  reduces to one of

It can be deduced that

Numbers of the form ,where a and b are real numbers are called complex numbers.

Note that cannot be combined any further

In such expression

A is called the real part of a complex number

B is called imaginary part of a complex number (NOT )

Complex number = (real part) + (imaginary part)

OPERATION ON COMPLEX NUMBERS

ADDITION AND SUBTRACTION

Examples:

1.

Solution

2.

Solution

So in general

EXERCISE

I.

II.

MULTIPLICATION OF COMPLEX NUMBERS

Example:
1.
Solution

2.

Solution

3.

= 13

Any pair of complex numbers of the form  has a product which is real.

i.e.

Such complex numbers are said to be conjugate

Each is a conjugate of the other.

Hence

i.e.

.if  then the conjugate is

A division will be done by multiplying numerator and denominator the conjugate of the denominator.

Example

For division, the numerator and denominator both will be multiplied by the conjugate of the denominator.

i.e.

NOTE: – The complex number is zero if and only if the real term and the imaginary term are each zero.
– The real term is given first even when is negative

i.e.

Suppose

Thus two complex numbers are equal if and only if the real terms and the imaginary terms are separately equal.

Example:

Find the value of x and y if

a)

Solution

b)

Solution

GRAPHICAL REPRESENTATION OF COMPLEX NUMBERS

Consider the reference line denoted by XX1 and YY1

i) x- axis represents real number (i.e. XX1 is called real axis

ii) y- axis represents imaginary number (YY1 is called imaginary axis

ARGAND DIAGRAM

If  is a complex number this can be represented by the line  where P is the point (x, y)

This graphical representation constitutes an Argand diagram

Example:

Draw an Argand diagram to represent the vectors

i)

ii)

Z is often used to denote a complex

Solution

Using the same XY – plane

GRAPHICAL ADDITION AND SUBTRACTION

Consider Z and Z2 representing an Argand diagram

Taking  and  .The coordinates of C are

Hence  represented the complex number
.

MODULUS AND ARGUMENT

Let  be the complex number which suggests that  represents  and A (a, b) is the point

r is the length of OA

Is the angle between the positive x axis and

OA is called the modulus of complex number

i.e.

The angle  is called the argument of  and written arg()

Note:

The position of OA is unique and corresponds to only one value of  in the range

An argument is also known as amplitude

–      To find the argument of  we use  together with quadrant diagram

Example

Find the argument of each of the following complex numbers

a)

b)

c)

d)

Solution
(a) 4+3i

Solution
(b)

Solution
(c)-4-3i

Solution
(d)

EXERCISE

Represent the following complex numbers by lines on Argand diagrams. Determine the modulus and argument of each complex number

a)

b)

SQUARE ROOTS OF COMPLEX NUMBERS

Example

Find

Solution

Example;

Given that is a root of the equation  find the other two roots

Solution

Polynomial has real coefficient  and its conjugate is a root of the polynomial

⇒To find the other factor of z3 – 6z2 + 21z – 26 =0

EXERCISE

1. Solve the following equation

2. Given that   express the complex number   in polynomial form hence find resulting complex when

POLAR FORM OF A COMPLEX NUMBER

If  then Z can be written in polar form i.e. in terms r and

Let OP be a vector
r be the length of the vector

be the angle made with OX

From the diagram (Argand diagram) we can see that

Example

Express  in polar form

Solution

NOTE

If the argument is greater than 90, care must be taken in evaluating the cosine and sine to include the appropriate signs.

E.g.

Express in the form

Solution

Since the vector lies in the 3rd quadrant

i.e.

â—¦
â—¦

CONJUGATES IN POLAR FORM

NB:

Taking the conjugate in polar form changes the sign of its argument

Example

Express  in polar form and then find its conjugate

Solution

Example

If

Find i)     ii)

Solution

i)

ii)

DEMOIVRE’S THEOREM

Demoivre’s theorem is a generalized formula to compute powers of a complex number in its polar form

Consider  from the earlier discussion we can find (Z)(Z)

This brings us to Demoivres theorem

If  and n is a positive integer

Then

Proof demoivre’s theorem by induction.
Test formula to be true for n

Let us show that the formula is true for n = k+1

Since the formula was shown to be true for n = 1, 2 hence its true for integral value of n.

Example

1.   Find

Solution

2.   From Demoivere’s theorem prove that the complex number   is always real and hence find the value of the expression when n = 6

Solution

FINDING THE nth ROOT

Demoivere’s formula is very useful in finding roots of complex numbers.

If n is any positive integer and Z is any complex number we define an nth root of Z to be any complex number ‘w’ which satisfy the equation

Examples

1.   Find all cube roots of -8

Solution

-8 lie on the negative real axis

2.   Solve  giving your solution in polar form

Solution:

EXERCISE

1.   Find all fourth roots of 1.

2.   Evaluate

Proving trigonometric identities using Demoivre’s theorem

Examples

Prove that

i)

ii)

Solution

Note;

To solve such question you should be aware of the binomial theorem

i)

EXERCISE

Show that;

Example

Find an expression for

i)               ii)

Solution

i) We know that

ii)

EXERCISE

Use Demoivre’s theorem to find the following integrals

a)

b)

c)

THE EULER’S FORMULA (THE EXPONENTIAL OF A COMPLEX NUMBER)

Euler’s formula shows a deep relationship between the trigonometric function and complex exponential

Since

Re organizing into real and imaginary terms gives.

Hence if Z is a complex number its exponent form is  in which

Example

1. Write  in polar form and then exponential form

Solution

2. Express  in Cartesian form correct to 2 decimal places

Solution

Note that;

The exponents follow the same laws as real exponents, so that

If

ROOTS

Sometimes you can prefer to find roots of a complex number by using exponential form.

From the general argument

If

Example

Find the cube root of Z = 1

Solution

Example 2

Calculate the fifth root of 32 in exponential form

Solution

LOCI OF THE COMPLEX NUMBERS

Complex number can be used to describe lines and curves areas on an Argand diagram.

Example 01

Find the equation in terms of x and y of the locus represented by |z|=4

Solution

This is the equation of a circle with centre (0.0) radius 4

Example 02

Describe the locus of a complex variable Z such that

Solution

This is the equation of a circle with centre (2,-3), radius 4 in which the point  (x, y) lies on  and out of the circle.

Example 03

If Z is a complex number, find the locus in Cartesian coordinates represented by the equation

Solution

This is the needed locus which is a circle with centre (3, 0) and radius 2

Example 04

If Z is a complex number, find the locus of the following inequality

Solution

We consider in two parts

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