The nth Roots of Unity appear when we consider the complex roots of an equation of the form:
Solving the Equation
As we have an nth degree polynomial, we will have n complex roots. By converting this to the polar form (by letting and noting that for ), we get the expression:
As the magnitude of the right hand side is 1, we can deduce that r = 1, leaving us with . Quick algebraic manipulation gives us:
Hence, we can conclude that the solutions the polynomial are given by which can be converted to the trigonometric equation using Euler’s formula:
All roots of unity lie on the unit circle in the complex plane, as all roots have a magnitude of 1.
Additionally, the nth roots of unity are connected in order, they form a regular n sided polygon. This can easily be seen by analysing the arguments of the roots.
- The sum of the nth roots of unity is 0.
- If is a primitive nth root of unity, then the roots of unity can be expressed as .
- A primitive nth root of unity is such that for .
- This sequence of powers is n periodic because z j + n = z j⋅z n = z j⋅1 = z j for all values of j.
- For each nth root of unity, , we have that . Although obvious, this property should not be forgotten as, for example, it can aid with algebraic manipulation.
Representations of Roots of Unity
Pretty huh? M x