Complex Number Simplified(part 1)
1. Introduction
After the discovery of various formulae and theorems, mathematics was brimming with the hope of saturation and achieving an end point. But one day even the smallest of the equation proved too much for the sharpest of the brains and that was,
X2 + 1 = 0
and all the efforts to provide a solution proved to be fatal. Suddenly all the quarters and sections of the mathematics were buzzing with furious discussions.
At last a solution was given to the equation and a whole new stream in mathematics opened before the world and that was Complex Numbers.
Complex numbers denotes the world of mathematics that deals with the imaginary planes, axes and numbers. It form an important part in Algebra and plays a vital role in the core mathematics.
Now coming to that equation
This, 
is defined as “iota” which refers to an imaginary unit. It is denoted by i.
If a, b are natural numbers such that then the equation 
is not solvable in N, the set of natural numbers i.e. there is no natural number satisfying the equation 
. So, the set of natural number is extended to form the set I of integers in which every equation of the form 
is solvable. But equations of the form 
where 
are not solvable in I also. Therefore the set I of integers extended to obtain the set Q of all rational numbers in which every equation of the form 
is uniquely solvable. The equations of the form 
etc. are not solvable in Q because there is no rational number whose square is 2. Such numbers are known as irrigational numbers. The set Q of all rational numbers is extended to obtain the set T which includes both rational and irrational numbers. This set is known as the set of real numbers. The equations of the form 
etc. are not solvable in R i.e. there is no real numbers whose square is a negative real number. Euler was the first mathematician to introduce the symbol for the square root of -1 with the property 
. He also called this symbol as the imaginary unit.
Now, a question arises that if iota is imaginary, how come we are using it in the mathematics problems?
It has a very simple answer that by imagining numbers, we can differentiate and solve each and every problem in complex numbers.
This plane which is used in complex numbers is called Argand Plane.
Defining Complex numbers :-
An ordered pair of real numbers and of form a + ib , where a and b are real numbers and 
is called complex number. It is denoted by Z.
It may also be denoted by the symbol (a, b)
z = a + i b, where
Complex Number z = a [Real part] + i b [imaginary Part]
Some Properties of Complex number:
1. The complex numbers do not posses the property of order for example :
The property of order is not defined in Complex number
2. The set of Real numbers is a proper subset of the complex number 
Why ?
Because every real number R can be written as
R = Z
where Z = R + i(0)
So Z is complex number where real part is R and imaginary part is zero.
3. A complex number, z, is said to be
Purely Real if Imaginary part is zero
Purely Imaginary if Real part is zero.
The complex number 0 + 0 + i 0 is both purely real and purely imaginary.
4. As 
Squaring both sides we get
Again Multiplying both sides with i we get
Now multiply both sides with I again,
In general 
for any integer n.
5. As you all might have observed upto now, one of the important property is.
= 
= 
= 
6. We have 
So 
-
Misconception among students !!!!!
[which is wrong]
This creates a confusion in mind but clarifying the misconception needs to change our approach towards the solution.
-
Correction
As the property 
holds good only and only if at least one of 
and 
is real. If both are imaginary equation falters
2. Conjugate Complex Number
As every dusk comes hand in hand with a dawn, complex numbers, like all other things in nature, have conjugates
A conjugate of complex numbers can be obtained only by reversing the sign of its imaginary part.
For example.
If 
then
where
is the complex conjugate of the z. 
is alos the mirror image of z along real axis.
Sum of complex number (z) and its conjugate will be real and double of its real part.
Product of Complex number (z) and its conjugate will always be real and sum of the square of its real and imaginary part.
-
Modulus of z
So from previous property we can have
-
Properties of Conjugate
As 
is the mirror image of z along real axis, it exhibits amazing symmetry with z and properties which are very useful is solving equations.
Some of the important properties are listed below.
1. 
2. 
if purely real and
if purely imaginary
3. 
4. 
5. 
[simple addition]
6. 
[simple substractioin]
7. 
[simple multiplication]
8. 
[simple division]
9. 
10. 
11. If 
So we can see that all the properties are simply the replica of normal properties of algebra and nothing is different from it.
3. Algebraic operations with complex numbers
-
Addition : (a + ib) + (c + id) = (a + c) + i(b + d)
Let 
and 
be two complex numbers. Then their sum 
is defined as the complex number 
.
It follows from this definition that the sum 
is a complex number such that
and
Example If 
and 
then 
-
Properties of Addition of Complex Numbers
(i) Addition is commutative For any two complex numbers 
we have 
(ii) Addition is associative For any three complex numbers 
we have 
.
(iii) Existence of additive identity The complex number 
is the identity element for addition i.e. 
for all 
.
-
Substraction : (a + ib) – (c + id) = (a – c) + i(b – d)
Let 
and 
be two complex numbers. Then the subtraction of 
from 
is denoted by 
and is defined as the addition of 
Thus, 
Example If 
then
-
Multiplication of Complex numbers :
Let 
be two complex numbers. Then the multiplication of 
with 
and is defined as the complex number 
Thus, 
Example If 
then
Note The product 
can also be obtained if we actually carry out the multiplication 
as given below :
-
Properties of Multiplication
(i) Multiplication is commutative For any two complex numbers 
we have 
.
ii)
Multiplication is associative For any three complex numbers 
we have 
.
(iii) Existence of identity element for multiplaction The complex number 
is the identity element for multiplication i.e. for every complex number z, we have 
.
(iv) Existence of multiplicative inverse Corresponding to every non-zero complex number 
there exist a complex number 
such that
Example Find the multiplicative inverse of 
.
Solution 
(v) Multiplication of complex numbers is distributive over addition of complex numbers For any three complex numbers 
we have
(i) 
(Left distributivity)
(ii) 
(Right distributivity)
d. Division of Complex numbers :
The division of a complex number 
by a non-zero complex number 
is defined as the multiplication of 
is defined as the multiplication of 
by the multiplicative inverse of 
and is denoted by 
.
Thus, 
Let, 
= 
[By def. of multplication]
Example If 
then
4. Argument and Modulus of complex Number
While denoting a complex number, we use 
and 
normally but if we want to show it on the Argand plane by a point then it will be having some distance form the origin. This distance from origin is the Modulus of the complex number.
The angle, which line joining complex number and origin makes with the Real axis is called Argument of complex Number.
Let z = a + ib be any complex number, then dividing and multiplying the whole equation with 
we get
where 
and 
Now as we can compare this equation with the diagram
So r is the modulus and 
is the Argument of complex number.
Modulus of A complex number
The modulus of a complex number 
is denoted by 
and is defined as
.
Clearly, 
for all 
.
Example If 
then
and 
.
Remark In the set C of all complex numbers, the order relation is not defined. As such 
has no meaning but 
has its meaning since 
are real numbers.
Properties of modulus
If 
then
(i) 
i.e. 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
where a, 
.
5. Integral Powers of IOTA (i)
We have 
and 
. So 
Note that 
is defined as 1.
To find the values of 
we first divided n by 4. Let m be the quotient and r be the remainder. Then 
where 
.
Thus, if 
then 
where r is the remainder when n is divided by 4.
The values of the negative integral powers of i are found as given below :
To find 
where n is greater than 4, we proceed as discussed above.
Example 1. Evaluate the following :
(i) 
(ii) 
(iii) 
(iv) 
Solution (i) 135 leaves remainder as 3 when it is divided by 4. Therefore 
(ii) The remainder is 3 when 19 is divided by 4, therefore 
(iii) We have, 
. On dividing 999 by 4, therefore 
So, 
(iv) We have, 
Example 2. Show that
(i) 
(ii) 
(iii) 
, for all 
.
Solution (i) We have,
(ii) We have,
(iii) We have,
6. Imaginary Quantities
The square root of a negative real number is called an imaginary quantitiy or an imaginary number.
Example 1. 
are imaginary quantities.
A Useful Result If a, b are positive real numbers, then 
Proof We have 
and 
Therefore, 
Note 1 For any two real numbers 
is true only when at least one of a and b is either positive or zero. In other words, 
is not valid if a and b both are negative.
Note 2. For any positive real number a, we have 
Example 2. Compute the following
(i) 
(ii) 
(iii) 
Solution (i) 
(ii) 
(iii) 
.
Example 3 A student writes the formula 
Then he substitutes 
and 
and funds 
. Explain where he is wrong ?
Solution (i) Since a and b both are negative, therefore 
cannot be written as 
n fact for a and b both negative, we have 
.
Example 4. Is the following computation correct ? If not give the correct computation :
Solution The said computation is not correct, because -2 and -3 both are negative and 
is true when at least one of a and b is positive or zero. The correct computation is
Rest things on Complex Numbers will be continued ..
Reference : Header image taken from http://www.physics.rutgers.edu/ugrad/301/PHYSICS%20301%20HOMEWORK.htm
www.wikipedia.org
Btech in Electronics Engineering from NSIT, Delhi . A die hard fan of mathematics
[...] Fandas , This article is in Continuation with the previous article on Complex Number .Plz read the previous one before reading this . I will be posting questions & solutions on the Algebra [...]
sir this article is so awesome that i am really impressed with the technique and methodology used……….
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