^{2}= 9 and (-3)

^{2}= 9. In both these cases, we can say that the value of the square root of a real number is never negative. Now, how should we find the square root of negative numbers? For instance, what is the value of √-9? Is it -3? Or 3? Is there an answer to this question? A famous mathematician from the 18th century, Euler give us the concept of Iota (i). Iota gives the square root of -1, that is, √-1 = i.

Iota is an imaginary number. Imaginary numbers are the numbers that give us the square root of negative numbers. Through this, we can find the value of √-9. The value of √-9 will be √-1. √9 = i3. Therefore, the value of √-9 is i3. Hence, the concept of imaginary numbers is very useful in mathematics. It makes negative calculations simpler and hassle-free.

An important component of Iota is its integral power. This component is very easy to understand. As we know that i = √-1. Let us get a quick understanding of this simple component.

i = √-1

i

^{2}= -1

i

^{3}= -1 * i = -i

{This is because i

^{2}= -1}

i

^{4}= (i

^{2})

^{2}= 1

Now, when we need to find the value of n>4, we just need to divide the value of “n” with 4. After this step, we must write the equation in the form: n=4m + r. Here, “r” is the reminder of the n/4, and “q” is the quotient for the same.

Since we have understood the concept of Iota or Imaginary numbers. Let us discuss what are complex numbers briefly.

## Complex Number

Since we know what imaginary numbers are, it is quite easy to interpret what are complex numbers. Complex numbers are made when a real number is added to the imaginary number. Let’s say “4i” is an imaginary number. If we add 40 to it. It will take the form- 40 + 4i or 4i + 40. This form is commonly known as Complex numbers. Complex numbers have two parts, which can be seen in the above-mentioned example- an imaginary part and a real part. Theoretically, these are the numbers that can be written in a basic form- the form being “x + yi”.Here, “x” is the real part, and “iy” is the imaginary part of the complex number “x + yi”. Let us understand this with the help of some complex number examples.- 28 + 6i

- -6

- 17i

- 3i – 3

## Algebraic Operations

Some of the algebraic operations of Complex Numbers include Addition, Subtraction, Multiplication, and Division. Let the complex numbers be (16 + 6i) and (6 +2i).- Additional operation: The addition operation is such that the real numbers and the imaginary units are added together. For example: The sum of these two numbers is (16 + 6) + (6 + 2)i = 22 + 8i.
- Subtraction operation: The subtraction operation is such that the difference between real number and imaginary number is found. For example: the difference is (16-6) + (6 – 2)i = 10 + 4i.
- Multiplication operation: The multiplication operation is done as

follows (16 + 6i) * (6 + 2i) = 16 (6) + 16 (2i) + 6i (6) + 6i (2i) = 108 + 97i.

- Division operation: The division is done by dividing the complex numbers, as

follows- 16 + 6i/6 + 2i = (16 + 6i) * 1/(6 + 2i) = (16 + 6i) * 6 – 2i/40 = (108 + 97i)/40