Complex Numbers
11.0 Cube roots of unity
11.0 Cube roots of unity
By putting $n=3$ in the ${n^{th}}$ root of unity, we get the cube roots of unity i.e., ${x^3} = 1$ has three roots which are called the cube roots of unity. $$\begin{equation} \begin{aligned} {x^3} = 1 \\ \Rightarrow {x^3} - 1 = 0 \\ \Rightarrow (x - 1)({x^2} + x + 1) = 0 \\ \therefore x = 1,{\text{ }}x = \frac{{ - 1 + i\sqrt 3 }}{2},{\text{ }}x = \frac{{ - 1 - i\sqrt 3 }}{2} \\\end{aligned} \end{equation} $$
They are generally denoted by $1,{\text{ }}\omega {\text{ and }}{\omega ^2}$ and geometrically represented by the vertices of an equilateral triangle whose circumcentre is the origin and circumradius is unity i.e., $1$.
