3rd roots of unity The solutions of the equation \(z^3 = 1\) are the numbers
\[\omega_{3} = \cos\dfrac{2\pi}{3} + \sin\dfrac{2\pi}{3}\cdot i,\ \omega_{3}^{2},\ \omega_{3}^{3}=1\]These are the third roots of unity.
For \(n\geq 1\), every solution of the equation \(z^n=1\) is called an \(n\)-th root of unity. There are exactly \(n\) such solutions, denoted \(\omega_{n},\dots ,\omega_{n}^{n}=1\). They lie on the unit circle centered at the origin and form a regular \(n\)-gon one of whose vertices is at \(\omega_{n}^{n}=1\).
Below is an illustration of \(n\)-th roots of unity for \(n=3,4,5\).
2nd roots of unity The solutions of the equation \(z^2 = 1\) are the numbers
\[\omega_{2} = -1\quad \text{and}\quad \omega_{2}^{2} = 1\]These are the second roots of unity. Further, \(\omega_2 = \cos\pi + \sin\pi\cdot i\).
3rd roots of unity The solutions of the equation \(z^3 = 1\) are the numbers
\[\omega_{3} = \cos\dfrac{2\pi}{3} + \sin\dfrac{2\pi}{3}\cdot i,\ \omega_{3}^{2},\ \omega_{3}^{3}=1\]These are the third roots of unity.
4th roots of unity The solutions of the equation \(z^4 = 1\) are the numbers
\[\omega_{4} = i = \cos\dfrac{\pi}{2} + \sin\dfrac{\pi}{2}\cdot i,\ \omega_{4}^{2} = -1,\ \omega_{4}^{3}=-i, \omega_{4}^{4}=1\]These are the fourth roots of unity.
5th roots of unity The solutions of the equation \(z^5 = 1\) are the numbers
\[\omega_{5} = \cos\dfrac{2\pi}{5} + \sin\dfrac{2\pi}{5}\cdot i,\ \omega_{5}^{2},\ \omega_{5}^{3},\ \omega_{5}^{4},\ \omega_{5}^{5}=1\]These are the third roots of unity.
