Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Roots of unity"

(→‎Properties of roots of unity)
m (proofreading)
Line 7: Line 7:
 
First, we note that since we have an '''n'''th degree polynomial, there will be '''n''' complex roots.
 
First, we note that since we have an '''n'''th degree polynomial, there will be '''n''' complex roots.
  
−
Now, we can convert everything to [[polar]] by letting <math> x = re^{i\theta} </math> and noting that <math> 1 = e^{2\pi ik}</math> for <math> k\in \mathbb{Z}</math> to get <math>r^ne^{ni\theta} = e^{2\pi ik}</math>. The magnitude of the RHS is 1 making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number).  This leaves us with <math> e^{ni\theta} = e^{2\pi ik}</math>.
+
Now, we can convert everything to [[polar]] by letting <math> x = re^{i\theta} </math>, and noting that <math> 1 = e^{2\pi ik}</math> for <math> k\in \mathbb{Z}</math>, to get <math>r^ne^{ni\theta} = e^{2\pi ik}</math>. The magnitude of the RHS is 1, making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number).  This leaves us with <math> e^{ni\theta} = e^{2\pi ik}</math>.
  
−
Taking the [[natural logarithm]] of both sides gives us <math> ni\theta = 2\pi ik</math>.  Solving this gives <math> \theta=\frac{2\pi k}n </math>.  Additionally, we note that for each of <math> k=0,1,2,\ldots,n-1 </math> we get a distinct value for  <math> \theta </math> but once we get to <math> k > n-1 </math> we start getting [[coterminal]] angles.
+
Taking the [[natural logarithm]] of both sides gives us <math> ni\theta = 2\pi ik</math>.  Solving this gives <math> \theta=\frac{2\pi k}n </math>.  Additionally, we note that for each of <math> k=0,1,2,\ldots,n-1 </math> we get a distinct value for  <math> \theta </math>, but once we get to <math> k > n-1 </math>, we start getting [[coterminal]] angles.
  
 
Thus, the solutions to <math> x^n=1 </math> are given by <math> x = e^{2\pi k i/n} </math> for <math> k=0,1,2,\ldots,n-1</math>.  We could also express this in trig form as <math>\displaystyle x=\cos\left(\frac{2\pi k}n\right) + i\sin\left(\frac{2\pi k}n\right) = \mathrm{cis }\left(\frac{2\pi k}n\right). </math>
 
Thus, the solutions to <math> x^n=1 </math> are given by <math> x = e^{2\pi k i/n} </math> for <math> k=0,1,2,\ldots,n-1</math>.  We could also express this in trig form as <math>\displaystyle x=\cos\left(\frac{2\pi k}n\right) + i\sin\left(\frac{2\pi k}n\right) = \mathrm{cis }\left(\frac{2\pi k}n\right). </math>
Line 15: Line 15:
 
== Geometry of  the roots of unity ==
 
== Geometry of  the roots of unity ==
  
−
All of the roots of unity lie on the [[unit circle]] in the complex plane.  This can be seen by considering the magnitudes of both sides of the equation <math> x^n = 1</math>.  If we let <math> x = re^{i\theta} </math> we see that <math> r^n = 1</math> since the magnitude of the RHS of <math> x^n=1 </math> is 1 and for two complex numbers to be equal, both their magnitudes and arguments must be equivalent.
+
All of the roots of unity lie on the [[unit circle]] in the complex plane.  This can be seen by considering the magnitudes of both sides of the equation <math> x^n = 1</math>.  If we let <math> x = re^{i\theta} </math>, we see that <math> r^n = 1</math>, since the magnitude of the RHS of <math> x^n=1 </math> is 1, and for two complex numbers to be equal, both their magnitudes and arguments must be equivalent.
  
−
Additionally, we can see that when the '''n'''th roots of unity are connected in order (more technically, we would call this their [[convex hull]]) they form a regular '''n'''-sided polygon.  This becomes even more evident when we look at the arguments of the roots of unity.
+
Additionally, we can see that when the '''n'''th roots of unity are connected in order (more technically, we would call this their [[convex hull]]), they form a regular '''n'''-sided polygon.  This becomes even more evident when we look at the arguments of the roots of unity.
  
 
== Properties of roots of unity ==
 
== Properties of roots of unity ==
Line 23: Line 23:
  
 
* They occupy the vertices of a regular ''n''-gon.
 
* They occupy the vertices of a regular ''n''-gon.
−
* For <math> n>1 </math> the sum of the ''n''th roots of unity is 0. More generally, if <math>\zeta</math> is a primitive ''n''th root of unity (i.e. <math>\zeta^m\neq 1</math> for <math>1\le m\le n-1</math>), then <math>\sum_{k=0}^{n-1} \zeta^{km}=\begin{cases} n & \mathrm{n\mid m}, \\ 0 & \mathrm{otherwise.}\end{cases} </math>
+
* For <math> n>1 </math>, the sum of the ''n''th roots of unity is 0. More generally, if <math>\zeta</math> is a primitive ''n''th root of unity (i.e. <math>\zeta^m\neq 1</math> for <math>1\le m\le n-1</math>), then <math>\sum_{k=0}^{n-1} \zeta^{km}=\begin{cases} n & \mathrm{n\mid m}, \\ 0 & \mathrm{otherwise.}\end{cases} </math>
 
** This is an immediate result of [[Vieta's formulas]] on the polynomial <math> x^n-1 = 0 </math> and [[Newton sums]].
 
** This is an immediate result of [[Vieta's formulas]] on the polynomial <math> x^n-1 = 0 </math> and [[Newton sums]].
 
* If <math>\zeta</math> is one of the roots of unity (but not 1), then the roots of unity can be expressed as <math> 1, \zeta, \zeta^2,\ldots,\zeta^{n-1}</math>.
 
* If <math>\zeta</math> is one of the roots of unity (but not 1), then the roots of unity can be expressed as <math> 1, \zeta, \zeta^2,\ldots,\zeta^{n-1}</math>.
Line 29: Line 29:
  
 
== Uses of roots of unity ==
 
== Uses of roots of unity ==
−
Roots of unity show up in many, suprising places.  Here, we list a few:
+
Roots of unity show up in many suprising places.  Here, we list a few:
  
−
* Geometry
+
* [[Geometry]]
−
* Factoring
+
* [[Factoring]]
−
* Number theory
+
* [[Number theory]]
  
 
== See also ==
 
== See also ==
 
* [[Complex number]]s
 
* [[Complex number]]s

Revision as of 12:39, 3 July 2006

Roots of unity

Roots of unity come up when we examine the complex roots of the polynomial $x^n=1$.

Solving the equation

First, we note that since we have an nth degree polynomial, there will be n complex roots.

Now, we can convert everything to polar by letting $x = re^{i\theta}$, and noting that $1 = e^{2\pi ik}$ for $k\in \mathbb{Z}$, to get $r^ne^{ni\theta} = e^{2\pi ik}$. The magnitude of the RHS is 1, making $r^n=1\Rightarrow r=1$ (magnitude is always expressed as a positive real number). This leaves us with $e^{ni\theta} = e^{2\pi ik}$.

Taking the natural logarithm of both sides gives us $ni\theta = 2\pi ik$. Solving this gives $\theta=\frac{2\pi k}n$. Additionally, we note that for each of $k=0,1,2,\ldots,n-1$ we get a distinct value for $\theta$, but once we get to $k > n-1$, we start getting coterminal angles.

Thus, the solutions to $x^n=1$ are given by $x = e^{2\pi k i/n}$ for $k=0,1,2,\ldots,n-1$. We could also express this in trig form as $\displaystyle x=\cos\left(\frac{2\pi k}n\right) + i\sin\left(\frac{2\pi k}n\right) = \mathrm{cis }\left(\frac{2\pi k}n\right).$

Geometry of the roots of unity

All of the roots of unity lie on the unit circle in the complex plane. This can be seen by considering the magnitudes of both sides of the equation $x^n = 1$. If we let $x = re^{i\theta}$, we see that $r^n = 1$, since the magnitude of the RHS of $x^n=1$ is 1, and for two complex numbers to be equal, both their magnitudes and arguments must be equivalent.

Additionally, we can see that when the nth roots of unity are connected in order (more technically, we would call this their convex hull), they form a regular n-sided polygon. This becomes even more evident when we look at the arguments of the roots of unity.

Properties of roots of unity

Listed below is a quick summary of important properties of roots of unity.

  • They occupy the vertices of a regular n-gon.
  • For $n>1$, the sum of the nth roots of unity is 0. More generally, if $\zeta$ is a primitive nth root of unity (i.e. $\zeta^m\neq 1$ for $1\le m\le n-1$), then $\sum_{k=0}^{n-1} \zeta^{km}=\begin{cases} n & \mathrm{n\mid m}, \\ 0 & \mathrm{otherwise.}\end{cases}$
  • If $\zeta$ is one of the roots of unity (but not 1), then the roots of unity can be expressed as $1, \zeta, \zeta^2,\ldots,\zeta^{n-1}$.
  • Also, don't overlook the most obvious property of all! For each root of unity, $\zeta$, we have that $\zeta^n=1$

Uses of roots of unity

Roots of unity show up in many suprising places. Here, we list a few:

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Roots_of_unity&oldid=5893"