A root of unity is a complex number that, when raised to a positive integer power, results in . Roots of unity have connections to many areas of mathematics, including the geometry of regular polygons, group theory, and number theory.
The following problem, although not seemingly related to complex numbers, is a good demonstration of how roots of unity work:
Brilli the ant stands on vertex 1 of the regular decagon below.
- He starts by hopping 1 space at a time (from 1 to 2, then from 2 to 3, and so on). He performs 10 hops in this way.
- He then hops 2 spaces at a time (from 1 to 3, then from 3 to 5, and so on). He performs 10 hops in this way.
- He continues to increase the hop distance every 10 hops: hopping 3 spaces 10 times, then hopping 4 spaces 10 times, and so on.
- After Brilli has hopped 10 spaces 10 times, he ends his workout.
When Brilli has completed his workout, which vertex will he be standing on?
The "hops" of different distances in the previous problem are analogous to the 10 roots of unity. In fact, the 10 roots of unity, when graphed in the complex plane, form a regular decagon. Peforming the 10 "hops" around the decagon is just like raising a 10 root of unity to the 10 power. In the case of Brilli, it brings him back where he started from. In the case of 10 roots of unity, it results in .
For any positive integer , the roots of unity are the complex solutions to the equation , and there are solutions to the equation.
If is even, there will be 2 real solutions to the equation , which are and if is odd, there will be 1 real solution, which is
What are the 3 roots of unity?
By definition, the 3 roots of unity are the solutions to the equation .
A keen eye would recognize that is one of the solutions to this equation, and hence is a 3 root of unity.
However, there are two more complex solutions to this equation that need to be accounted for. These solutions are and .
Written as a set, the 3 roots of unity are
These solutions can be confirmed using complex number arithmetic. You can try this in the example below.
Euler's formula can be used to find the roots of unity for any positive integer .
Let be a positive integer and be the set of all roots of unity. Then
It is important to note that set of all roots of unity always contains elements. Each of these roots of unity can be found by changing the value of in the expression .
By Euler's formula,
Let be any positive integer. Then
It is given that , with complex and a positive integer. Now we have
Raising each power to the power gives
Note that if , then the angle is co-terminal with .
Therefore, there are distinct solutions to each given by with .
Solve the equation for all
By the theorem above, the solutions are given by
By Euler's formula, these are, respectively,
Roots of unity have a strong relation to geometry. By graphing the roots of unity on the complex plane, they can be used to generate the vertices of a regular polygon.
For any integer , the roots of unity, when graphed on the complex plane, are the vertices of a regular -gon.
An equilateral triangle has its centroid located at the origin and a vertex at . What are the coordinates of the other two vertices?
The point corresponds to the complex number on the complex plane. This number is a 3 root of unity. The other 3 roots of unity will be the remaining vertices of the equilateral triangle on the complex plane:
These complex numbers correspond to the following points on the coordinate plane:
Because these values can be easily calculated or memorized, they are very useful for performing rotations in the complex plane. By extension, they can be used to perform rotations in any two-dimensional or even three-dimensional space.
The point is rotated radians counterclockwise about the origin. What is the resulting image?
Note that the corresponding point in the complex plane is . Also note that is an root of unity and its value is .
The rotation in the complex plane can be achieved by multiplying the complex numbers. This resulting image in the complex plane is
Then the corresponding image in the coordinate plane is
From geometric progressions, we have
This identity, along with the properties of roots of unity, can be used to find the solutions of certain polynomial equations.
Find all complex solutions of the equation .
By the identity above, the equation becomes
The solutions will be the 4 roots of unity with the exception of
Expanding using Euler's formula gives
The last solution is rejected because . Thus, the solutions are and .
Roots of unity have many special properties and applications. These are just some of them:
- If is an root of unity, then so is where is any integer.
- If is an root of unity, then .
- The sum of all roots of unity is always zero for .
- The product of all roots of unity is always .
- and are the only real roots of unity.
- If a number is a root of unity, then so is its complex conjugate.
- The sum of all the power of the roots of unity is for all integers such that is not divisible by
- The sum of the absolute values of all the roots of unity is
- If is an root of unity not equal to then .
If is an root of unity, then so is where is any integer.
By using index rule, we can deduce that . As is an root of unity, , and therefore and . Hence is also an root of unity.
Find the product of all the roots of unity.
By definition, the product of the roots of unity is the same as the product of the roots of the equation
By Vieta's formula, the product of roots is related to the constant term of the polynomial. The degree of the polynomial is even, so the product of roots is the same as the constant term, .
Find the sum of all roots of unity.
We are looking for the sum of roots of
By Vieta's formula, the sum of the roots is the opposite of the coefficient of the first-degree term. The coefficient of the first-degree term is , so the sum of roots is .
Find the sum of all the powers of the roots of unity.
The equation with roots is
This will be true for any power such that is an integer and is not divisible by taking the power sum of the roots of unity
Find the sum of the absolute values of the roots of
If it was just the sum, then we would get zero. But the question asks us about modulus. So we use the fact that each root can be expressed as , where is a real number. What is Recalling Euler's identity, we can rewrite it as
So each root has an absolute value of Hence, the sum of the absolute values of the roots is .
You can generalize for the roots of unity.