Complex Number Simplified(part 1)

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

1. 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 .

1. 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

1. 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