More on Complex Numbers

The earlier chapter on Complex Numbers explained what we mean by a complex number together with a few properties of complex numbers. Basic mathematical operations of complex numbers were introduced. So, the previous chapter may be considered as the preliminary base on which this chapter aims to build further. In this chapter we shall cover a few advanced topics under complex numbers.

 Amplitude(or Argument) of a complex number:

Let where are real, and ; then the value of for which the equations:

…(1) and

…(2)

are simultaneously satisfied is called the Argument(or Amplitude) of and is denoted by .

Clearly, equations (1) and (2) are satisfied for infinite values of ; any of these values of is the value of . However, the unique value of lying in the interval and satisfying equations (1) and (2) is called the principal value of and we denote this principal value by or .Unless otherwise mentioned, by argument of a complex number we mean its principal value.

Since, and (where n=any integer), it follows that,

where

Geometrical representation of modulus and amplitude:

Let us assume that the point represents the complex number . We draw perpendicular on and join . If and , then from the right-angled triangle we get,

and

where,

and

This form of representation, where, and of the complex number , is called the polar form or the modulus-amplitude form of z.

If the point represents the complex number and then:

a) when P lies in the first quadrant

b) when P lies in the second quadrant

c) when P lies in the third quadrant

d) when P lies in the fourth quadrant

Note:

1) The horizontal axis or the -axis is also called the real axis. The vertical axis or the -axis is called the imaginary axis.

In particular,

when P lies on the positive real axis, principal value of is 0.

when P lies on the negative real axis, principal value of is .

when P lies on the positive imaginary axis, principal value of is .

when P lies on the negative imaginary axis, principal value of is .

2) The argument of i.e., i.e., the origin O, is not defined.

3) It must be kept in mind that is the principal value of while is just one of the many values of which satisfy the equations,

…(1) and

…(2)

Example: Find the amplitude of

Solution:

Clearly, in the z-plane, the point lies in the second quadrant.

Hence, if then, where,

the reqiured amplitude of

Example: Find the amplitude of

Solution:

Clearly, in the z-plane, the point lies in the fourth quadrant.

Hence, if then,

where,

, i.e., the required amplitude of z is

Algebra of Complex Numbers:

Let  and

Now, in the earlier chapter “Complex Numbers” we have shown that:

For addition: where:

and and both are real.

For subtraction: where:

A=a-c $ and and both are real.

For multiplication: where:

and and both are real.

Note: Proceeding similarly, the product of more than two complex numbers can be expressed in the form

For division: where:

and and both are real.

So, it is very clearly that when the four fundamental mathematical operations, viz., addition, subtraction, multiplication and division carried out between two complex numbers, the result is also a complex number of the form where both and are real.

Now, we consider two more algebraic operations:

1) Any integral power of a complex number is a complex number:

Let be a complex number where are real and let be any integer.

Case I: If is a positive integer then,

[  the product of more than two complex numbers can be expressed in the form ] where, are real.

Case II: If is a negative integer w eassume where m is taken to be any positive integer.

Then, where are real.

where, and

any integral power of a complex number is a complex number.

2) Any root of a complex number is a complex number:

Let be a complex number where are real and let be any positive integer.

If the root of z be then,

By hypothesis,  Hence, is an imaginary quantity.

Now, can be an imaginary quantity if and only if is an imaginary number and equation (1) will be satisfies if and only if is an imaginary number of the form where are real.

any root of a complex number is a complex number.

Properties of Complex Numbers:

1) If are real and then

2) If are real and then and

3) The set of complex numbers satisfies commutative, associative and distributive laws, i.e., if be three complex numbers then,

i) [commutative law for addition] and

[commutative law for multiplication].

ii) [associative law for addition] and

[associative law for multiplication].

iii) [distributive law]

4) The sum and product two conjugate complex numbers are both real.

Proof: Let be a complex number where are real.

Then, conjugate of

Now, which is real.

Again, which is real.

Note: If then,

Again,

Hence,

modulus of a complex quantity =the positive square root of the product of the complex quantity and its conjugate.

5) If the sum and product of two complex numbers are both real, then the complex numbers are conjugate to each other.

Proof: Let and be two complex numbers where are al real and

It is given that the sum of and is real.

is real.

Also, is real.

[ ]

[ ]

which shows that and are conjugate to each other.

6) For two complex numbers and ,

Proof: Let and

and 

Again,

Now,

Note:

1) For three complex numbers we have,

Proof:

In general, for n complex numbers we have,

2) For two complex numbers and , we have,

i)

ii)

Proof:

i)

ii)

Similarly,

From (1) and (2) we get,

Exercise:

1) Express in the form where are real:

2) Find the amplitude of the following complex numbers:

3) If and , show that,

4) Express in modulus-amplitude form.

5) If