Complex Numbers

There are a few sets of numbers which we are already familiar with. These numbers are called real numbers. The aggregate of the sets of rational and irrational numbers is called the set of real numbers. The basic and most important property of any real number is that its square is always positive or non-negative. Because of this property, it was realized that real numbers are not sufficient for solving all algebraic problems. Hence, there was the need to invent a new set of numbers. These numbers are  different from real numbers. This new set of a numbers are called complex numbers or imaginary numbers.

For example, the quadratic equation , where , (where represents the set of real numbers), cannot be solved if , hence the need for complex numbers.

The equation cannot be solved in the real number system. Hence Euler introduced the symbol to represent and he said .

is a solution to the equation .

Thus it has two solutions, where .

This number is called an imaginary number or complex number.

Definition of a Complex Number

If an ordered pair of two real numbers and is represented by the symbol where , then the ordered pair is called a complex or imaginary number.

Thus, a complex number is defined as an ordered pair of real numbers and written as where and

For example, etc. are all complex numbers.

Real and Imaginary parts of Complex Number

If is a complex number, then the real part of , is denoted by and the imaginary part is denoted by

In a complex number  when the real part is zero or when , then the number is said to be purely imaginary.

Similarly, in a complex number, when the imaginary part, i.e., is zero, or when , then the number is said to be purely real.

For example, and here .

So, is a purely imaginary number.

Again, and here .

So is a purely real number.

Negative of Complex Number

If then

is the negative of the complex  number , also called the additive inverse of z.

Powers of i

We already know that . Similarly we can write the higher powers of   as follows:

This shows that the values repeat themselves in cycles of four.

Complex Number Operations

1. Addition:

Let there be two complex numbers, and We add the real parts and imaginary parts separately.

For example: Say,  and

2. Subtraction:

We change one complex number into its additive inverse and then add the two numbers. Let there be two complex numbers,  and

For example: Say,  and

3. Multiplication:

Let there be two complex numbers,  and

For example: Say,  and

4. Division:

Let there be two complex numbers,  and . Division of these two complex numbers is defined as:

We rationalise the denominator,

For example: Say,  and

5. Conjugate of Complex Number:

When two complex numbers only differ in the sign of their complex parts, they are said to be the conjugate of each other. The conjugate is denoted as .

If then

For example, if , the conjugate of is

Properties of Conjugates:

, i.e., conjugate of conjugate gives the original complex number

6. Modulus of a Complex Number:

The absolute value or modulus of a complex number, is denoted by and is defined as:

Here,

For example: If

Properties of Modulus:

only if when

 

7. Equality of Complex Numbers:

Two complex numbers are said to be equal if and only if and

Illustrations:

1. If , then prove that

Solution:

2. Find the real numbers and if is the conjugate of

Solution:

The conjugate of is 

According to the problem,

Solving equations (1) and (2) we get,

and (Answer)

3. Find the values of and if the complex numbers and are equal

Solution:

and

from (1) we have,

Answer:

4. Find the modulus of the following:

1. 
2. 

Solutions:

4a)

modulus of is given by

4b) 

modulus of is given by

5. If and , show that,

Solution:

or,

or,

or,

or,

or,

or,

[ ]

or,

or, [ squaring  both sides ]

or,

or,

Exercise:

1. If , then prove that
2. If , then prove that
3. Express in the form . Also find the conjugate and modulus of it.
4. Find the modulus of
5. Find the absolute value of
6. Find and if
7. Simplify
8. Show that
9. For any complex number, , prove that: and
10. For any two complex numbers and , prove that:
11. Find the real values of and for which
12. If and find