Question

What are the roots of unity?

Answer 1

Hint: We are asked to tell the meaning of roots of unity. The word ‘roots’ means it has something to do with an equation involving the polynomial of some power ‘n’. And ‘of unity’ means that of the number 1. It means that on one side of the equation we have some polynomial and on the other side we have 1. And also, there is no number mentioned so we try to give a general polynomial of ‘n’ degree.

Complete step by step solution:
Let $P\left(x\right)$ be any polynomial. Then, any number $a$ is called the ‘root’ of the polynomial $P\left(x\right)$ if it satisfies the equation:
$P\left(a\right)=0$
The term ‘unity’ means the integer 1. For any natural number $n$, the $n^{th}$ root of unity means the roots of the equation:
$x^n=1$
Consider for example $n=3$, then in real numbers, we only have a single solution viz. $x=1$. But if we extend our domain to the set of complex numbers as well then we have two more solutions.
$\omega=\dfrac{-1+\sqrt{3}i}{2}$
$\gamma=\dfrac{-1-\sqrt{3}i}{2}$
So, we can further modify our definition as:
For a given positive integer $n$, the $n^{th}$ root of unity is any complex number $z$ such that it satisfies the equation:
$x^n=1$
In other words:
$z^n=1$

Note: It is common to not extend our domain of reference, because most of the time only the real numbers are talked about, but we should also consider any possibility of how the equation might be satisfied. Taking only the real numbers would be wrong. Moreover, note that 1 will be a common solution of all the $n^{th}$ roots of unity.