complex algebra an introduction

1. 1
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
Complex Numbers: Argand diagram
An Argand diagram uses the real and imaginary parts of a complex number as analogues of x and y in the
Cartesian plane. The area of an Argand diagram is called the complex plane by mathematicians.
The study guide: Basics of Complex Numbers describes the Cartesian form of a complex number z
as: z  a  bi Cartesian form of a complex number Where a is the real part of a complex number, written
Rez, and b is the imaginary part of a complex number, written Imz .
In an Argand diagram the horizontal axis defines the real part of the complex number and the vertical axis
defines the imaginary part. A complex number in Cartesian form has the coordinate a,b in an Argand
diagram. A purely real number is positioned on the horizontal axis of an Argand diagram and a purely
imaginary number is positioned on the vertical axis of an Argand diagram.
The modulus and argument of a complex number As soon as you represent a complex number visually
you can begin to explore other ways of describing it.
For example, an Argand diagram shows the similarity of a complex number with a vector, you can think
of Rez and Imz being the number of steps along the real and imaginary axes respectively required to
‘reach’ z. You can also get to z in a straight line directly from the origin. In order to do this all you need
to know is what direction to face and how far to go. How far you need to go is the length of the line
representing the complex number in an Argand diagram and is called the modulus of the complex
number. It is very common to represent the modulus of a complex number z by the symbol z or simply by
the letter r. The value of the modulus is always positive.
You can use Pythagoras’ theorem to find the modulus of a complex number (see study guide: Pythagoras’
Theorem). Below, the Argand given for z  5  3i from page 1 of the guide has been annotated to reveal a
right-angled triangle:

2. 2
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
The modulus tells you how far to go from the origin to get to a complex number in an Argand diagram.
You also need to know what direction the complex number is. This is known as the argument of a
complex number which is written either as Argz or the Greek letter .
Complex numbers either on the real axis or in the upper half of the complex plane have a positive
argument measured anti-clockwise from the real axis. Complex numbers in the lower half of the complex
plane have a negative argument measured clockwise from the real axis. You can use the inverse tangent
(or arctan) function to find arguments. This can be confusing and the table below is designed to help you.
It is a very good idea to sketch your complex number before trying to calculate its argument.

3. 3
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
The real axis is part of quadrants 1 and 2 (not quadrants 3 and 4). Given this: Positive real numbers are in
quadrant 1 and have an argument of 0. Negative real numbers are in quadrant 2 and have an argument of
. Also: The argument of a purely imaginary number above the real axis (positive b) is  / 2 . The
argument of a purely imaginary number below the real axis (negative b) is  / 2.

4. 4
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala

5. 5
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala

6. 6
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala

7. 7
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
Complex Modulus

8. 8
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
De Moivre's Theorem
A complex number is made up of both real and imaginary components. It can be represented by
an expression of the form (a+bi), where a and b are real numbers and i is imaginary. When
defining i we say that i = √(-1). Along with being able to be represented as a point (a,b) on a
graph, a complex number z = a+bi can also be represented in polar form as written below:
z = r (cos θ + i sinθ)
where r = [z] = √(a2 + b2)
and
θ = tan-1(b/a) or θ = arctan(b/a)
and we also have: a = r cosθ and b = r sinθ
Statement of DeMoivre's Theorem
Let 'n' be any rational number, positive or negative, then
[cos θ + i sin θ ]n = cos nθ + i sin nθ
Basically, in order to find the nth power of a complex number we need to take the nth power of
the absolute value or length and multiply the argument by n.
Let z = r (cos θ + i sinθ) and n be a positive integer. Then z has n distinct nth roots given by:
where k = 0, 1, 2, ... , n-1
,
De Moivre's Theorem states that for any complex number as given below:
z = r ∙ cosθ + i ∙ r ∙ sinθ
the following statement is true:
zn = rn (cosθ + i ∙ sin(nθ)), where n is an integer.
If the imaginary part of the complex number is equal to zero or i = 0, we have:
z = r ∙ cosθ and zn = rn (cosθ)
Exponential form of complex number:
z = reiθ where r is the modulus of z and θ is its argument.
Example 1: Compute the three cube roots of -8.
Solution: Since -8 has the polar form 8 (cos π + i sin π), its three cube roots have the form
3√8 {cos[(π + 2πm)/3] + i sin[(π + 2πm)/3]} for m=0, 1, and 2.

9. 9
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
Thus the roots are
2 (cos π/3 + i sin π/3) = 1 + √3 i,
2 (cos π + i sin π) = -2, and
2 (cos 5π/3 + i sin 5π/3) = 1 - √3 i.
Example 2: Use De Moivre's Theorem to compute (1 + i)12.
Solution: The polar form of 1 + i is √2 (cos π/4 + isin π/4). Thus, by De Moivre's Theorem, we
have:
(1 + i)12 = [√2 (cos π/4 + i sin π/4)]12
= (√2)12(cos π/4 + i sin π/4)12
= 26 (cos 3π + i sin 3π)
= 64(cos π + i sin π)
= 64(-1) = -64.
Example 3: Use De Moivre's Theorem to compute (√3 + i)5.
Solution: It is straightforward to show that the polar form of √3 + i is 2(cos π/6 + i sin π/6). Thus
we have:
(√3 + i)5 = [2(cos π/6 + i sin π/6)]5
= 25(cos π/6 + i sin π/6)5
= 32(cos 5π/6 + i sin 5π/6)
= 32(-√3/2 + 1/2 i)
= -16√3 + 16 i.
Example 4: Find the sixth roots of √3+i
Solution: The modulus of √3+i is 2 and the argument is π/6.
The sixth roots are therefore
Powers And Roots OfComplex Numbers
To find the nth power of a complex number, it is possible, but labor-intensive, to multiply it out. It is
more difficult to find the nth root. A simpler method to find powers and roots of complex numbers is to
convert the function to exponential notation using polar coordinates. Consider the complex

10. 10
Complex algebra - anintroduction
Prof.PavithranPuthiyapurayil,GovernmentCollege of engineering,Kannur, Kerala
number ,where n is an integer and, in exponential form, . The complex number
becomes , which can be calculated. In the case that , , which
leads to DeMoivre's theorem. The complex number also has n roots, called roots of unity,
which are distinct solutions to the function . These roots can be found using the
function , where .