## What are Complex Numbers?

As we already know that the square of a real number is positive, therefore, the solution of the below-mentioned equation is not available in Real numbers.

Therefore, in order to remove this deficiency of real numbers, we require a number whose square must be -1.

Hence, the mathematicians introduced a larger set of number which is called the set of complex numbers that includes real numbers and every number whose square is not positive.

They designed another number i.e. – 1 which is called an imaginary unit. Complex Number is denoted by the a letter which contain the property that .

Leonard Euler was a Swiss Mathematician who first used the symbol for the number .

It is pertinent to mention here that, the is not a real number.

Complex numbers are represented by Z and the set of all complex numbers is represented by C.

## Pure Imaginary Number

It is a square root of a negative real number. For example, .

## Basic Arithmetic Operations on Complex Numbers

### 1. Addition

Consider, are two complex numbers, where a, b, c and d belong to the real numbers then the sum of these two complex numbers will be

**Examples:**

**a – **

**Solution:**

**b – **

**Solution:**

**c – **

**Solution:**

### 2. Subtraction

Consider, are two complex numbers, where a, b, c and d belong to the real numbers then the difference between these two complex numbers will be

**Examples:**

**a – **

**Solution:**

**b – **

**Solution:**

**c – **

**Solution:**

### 3. Multiplication:

Consider, are two complex numbers, where a, b, c and d belong to the real numbers then the product of these two complex numbers will be

Some multiplication examples of real numbers are given here.

### 4. Division

Consider, are two complex numbers, such that, , the division of these complex numbers is

Few division examples of real numbers are given below: –