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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