Imaginary unit

Definition

A quantity that represents the square root of negative one is called an imaginary unit.

Introduction

The imagination of quadratic polynomial with no middle term introduced the concept of an imaginary unit in mathematics. We have seen the quadratic expressions, quadratic expressions without constant term and also learned how to solve the quadratic equations. Now, we are going to study about the quadratic polynomials without the middle term.

$x^2+1 \,=\, 0$

This equation is a best example for a quadratic equation without the middle term. The literal $x$ is a variable and it represents a real number. Now, let’s solve it to find its roots or zeros.

$\implies$ $x^2 \,=\, -1$

In fact, there is no negative quantity physically but a negative sign is displayed before the quantities for our convenience. In this case, the value of $x$ can be either negative or positive real number. Hence, the square of variable $x$ should always be a positive quantity and it cannot be a negative in real.

However, we imagined that the square of a variable $x$ is negative one. Now, let’s continue our process of solving the equation.

$\implies$ $x \,=\, \pm\sqrt{-1}$

$\,\,\,\therefore\,\,\,\,\,\,$ $x \,=\, \sqrt{-1}$ or $x \,=\, -\sqrt{-1}$

Therefore, the solution set of the quadratic equation $x^2+1 \,=\, 0$ is $\Big\{-\sqrt{-1}, \sqrt{-1}\Big\}$

Representation

A Swiss mathematician Leonhard Euler proposed iota ($i$) symbol to represent the square root of negative one.

$i \,=\, \sqrt{-1}$

Therefore, the iota ($i$) symbol is called an imaginary unit. The imaginary unit is also called the unit imaginary number alternatively.

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