Complex Number

One of the most basic understandings of squares that we have is- the square of a real number cannot be negative. For instance, (3)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
i² = -1
i³ = -1 * i = -i
                  {This is because i² = -1}
i⁴ = (i²)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

Here, the real number is 28 and the imaginary unit is 6. Therefore, the real part is 28 and the imaginary part is 6.

• -6

This can be written as -6 + 0i. Here, the real number is -6 and the imaginary unit is 0. Therefore, the real part is -6 and the imaginary part is 0.

• 17i

This can be written as 0 + 17i. Here, the real number is 0 and the imaginary unit is 17i. Therefore, the real part is 0 and the imaginary part is 17.

• 3i – 3

This can be written as -3 + 3i. Here, the real number is -3 and the imaginary unit is 3. Therefore, the real part is -3 and the imaginary part is 3. From the mentioned complex number examples, it is quite clear what complex numbers are. Let us discuss some of its algebraic operations.

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

Conclusion

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, which is an imaginary number. Imaginary numbers are the numbers that give us the square root of negative numbers. The concept of imaginary numbers is very useful in mathematics. It makes negative calculations simpler. Complex numbers are made when a real number is added to the imaginary number. Complex numbers have two parts, which can be seen in the above-mentioned example- an imaginary part and a real part. the algebraic operations of Complex Numbers include Addition, Subtraction, Multiplication, and Division.