Roots of Unity

The nth Roots of Unity appear when we consider the complex roots of an equation of the form:

z^n = 1.

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  Screen Shot 2016-04-21 at 10.05.09 AM.png and noting that $1 = e^{2\pi ik}$ for $k\in \mathbb{Z}$), we get the expression:

$r^ne^{ni\theta} = e^{2\pi ik}$

As the magnitude of the right hand side is 1, we can deduce that r = 1, leaving us with $e^{ni\theta} = e^{2\pi ik}$. Quick algebraic manipulation gives us:

$\theta=\frac{2\pi k}n$

Hence, we can conclude that the solutions the polynomial are given by Screen Shot 2016-04-21 at 10.12.27 AM which can be converted to the trigonometric equation using Euler’s formula:

Screen Shot 2016-04-21 at 10.14.41 AM.png

Geometry

All roots of unity lie on the unit circle in the complex plane, as all roots have a magnitude of 1.

The roots of unity

Source: mathworld.wolfram.com

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.

Properties

  • The sum of the nth roots of unity is 0.
  • If $\zeta$ is a primitive nth root of unity, then the roots of unity can be expressed as $1, \zeta, \zeta^2,\ldots,\zeta^{n-1}$.
    • A primitive nth root of unity is such that $\zeta^m\neq 1$ for $1\le m\le n-1$.
    • This sequence of powers is n periodic because z j + n = z jz n = z j⋅1 = z j for all values of j.
  • For each nth root of unity, $\zeta$, we have that $\zeta^n=1$. Although obvious, this property should not be forgotten as, for example, it can aid with algebraic manipulation.

Representations of Roots of Unity

Colorful "string" art on the 17 th roots of unity

17th roots of unity | Source: nyates314.wordpress.com

Complex Polynomial Functions of degree 3: cubic roots of unity | matematicasVisuales

z^3-1 | Source: matematicasVisuales

8th roots of unity | isaacdooley.com

Pretty huh? M x

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s