Imaginary Number

Hero of Alexandria, a Greek mathematician, and engineer, is credited with being the first to present a computation involving the square root of a negative number, although it was Rafael Bombelli who first established the rules for multiplication of complex numbers in 1572. William Rowan Hamilton extended the concept of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries in which three of the dimensions are equivalent to the imaginary numbers in the complex field in 1843.

Imaginary Number: Definition

Imaginary numbers are the numbers that provide a negative result when squaring the given number. It’s usually expressed as a series of real values multiplied by the imaginary unit “i” which is pronounced as ‘iota’ and the value of i is √-1 or i² = -1.

Suppose that we have the imaginary number 9i, where

The imaginary unit is I whereas the real number is 9.

When we square the integer 9i, we get a negative result of

 -81. Because i² is equal to -1. This implies that -1 Equals i².

All imaginary numbers are built on the foundation of the letter “i.” the complex number is a  solution that is written using the imaginary number in the form a+bi. A complex number, in other terms, is one that contains both real and imaginary numbers.

Imaginary Number Rules

Consider the number a + ib, which is a complex number. The conjugate pair for a + ib is a – ib. When the complex roots are multiplied then the equations become that type of equations having real coefficients.

Consider the quadratic equation x² = a, where ‘a’ represents a known number. The solution could be written as x = √a. As a result, the following are the rules for various imaginary numbers : 

i = √-1

i² = -1

i³ = i² . i = -1 . i = -i

i⁴ = i² = (-1)² = 1

i⁴n = 1

i⁴n+1 = i⁴n . i = 1 .i = i

Operation related to Imaginary Number

Addition, subtraction, multiplication, and division are the four most basic arithmetic operations in mathematics. Let’s have a look at these operations using imaginary numbers:-

Suppose that we have two complex numbers: a + ib and c + id

Addition of Imaginary Number

When two imaginary numbers, a + bi, and c + di, are added, the real and imaginary components are added and simplified individually. (a+c) + i(b+d) will be the correct answer here.

( a + bi ) + ( c + di ) = (a + c ) + i ( b + d )

Subtraction of Imaginary Numbers 

When we subtract c+di from a+bi then the result is computed as if it were an addition problem. It entails grouping all real and imaginary terms separately and performing simplification.  in that case

 ( a + bi ) -(c + di) = (a-c) + (b-d)i .

Multiplication of Imaginary Number 

Suppose we have two imaginary numbers (a+bi) and (c+di)

When we multiply both the imaginary numbers then we get

(a+bi) (c+di) = (a+bi)c + (a+bi)di

                      = ac+bci+adi+bdi2

                      = (ac-bd) +  i( bc+ad)

This is the result when we multiply two imaginary numbers and similarly, we can multiply many more imaginary numbers.

Division of Imaginary Numbers

Suppose we have two imaginary numbers (a+bi) and (c+di) and When we divide  the first  imaginary number by the second then we get

(a+bi) / ( c+di)

Multiplying both the numerator as well as the denominator by the conjugate pair of the denominator and make it real. So, it becomes

(a+bi) / ( c+di) = (a+bi) (c-di) / ( c+di) (c-di) 

                           = [(ac+bd)+ i(bc-ad)] / c² +d² .

Conclusion

Imaginary numbers are the numbers that provide a negative outcome when we square. The imaginary numbers, on the other hand, are defined as the square root of negative numbers without definite value. It’s usually expressed as a series of real values multiplied by the imaginary unit “i.”

These are a type of complex number, which is made up of a real number and an imaginary number added together. A complex number has the form a + ib, where a and b are both real values and bi is an imaginary number. All the imaginary numbers are built on the foundation of the letter “i.” A complex number is a solution that is written using this imaginary number which is in the form a+bi.