THE ROOTS OF UNITY
In order to investigate another property of roots of unity,
product of two complex roots is calculated as:
In higher level mathematics course, while covering the
chapter ‘Multiple Perspectives in Mathematics’, we came
across with the part which explores the roots of complex
numbers. We learned that the best method to solve the
nth roots of complex number is to use De Moivre’s theorem
which equates z n = rcisθ. Considering this fact, we thought
that it can be used to investigate some properties of the
cube roots of unity. So, we start by solving cube roots of
unity step by step to figure out those features.
The first step that we followed is to find the cube roots of
the equation:
ε 3 =1
We expected to obtain three complex roots and the roots
are as follows:
Therefore, the property that the product of two imaginary
roots is 1 and the product of roots of unity is 1 is
investigated.
The sum of the three roots are taken as:
Thus, the result that sum of the roots of unity is equal
to 0 is also shown.
MATHEMATICS
DEPARTMENT
If one of the complex roots is considered as:
And its square is taken, then:
As shown in the operation, the result of taking square is
equal to the other complex root of the equation, ε 3 =1.
Therefore, as a result, it can be said that the three roots of
the equation can be written as; 1, ω and ω2.
On the complex plane, the roots of unity are at the vertices
of the regular triangle inscribed in the unit circle:
In this investigation by taking the cube roots of the unity
as reference, we tried to prove some properties of roots
of unity. First, we used De Moivre’s theorem to find the
three roots. When one of the imaginary roots is called as
ω, the other imaginary root is proven to be ω2. Therefore,
the roots can be expressed as 1, ω and ω2. When the roots
are plotted on the Argand’s diagram, it is observed that
the cube roots of unity are at the vertices of the regular
triangle inscribed in the unit circle. After our observation
on the Argand’s diagram, the product of the three roots
is taken and the result is obtained as 1. By this way, we
made the generalization that the product of roots of unity
is equal to 1. In the last part of our study, we take the sum
of three roots and find zero. As the modulus of each root
is 1, when the roots are considered as vectors, the sum of
vectors would also give a resultant as 0.
References:
• http://www.tutorsonnet.com/cube-roots-of-unityhomework-help.php
• http://www.math-only-math.com/the-cube-roots-of-unity.
html
• https://en.wikibooks.org/wiki/Unit_roots/Properties_of_
unit_roots
• Harcet, J.,Heinrichs,L.,Seiler, P.M., & Skoumal, M.T. (n.d.)
Mathematics higher level: Course companion
Hazal YILDIRIM
11-A
Ece AKDAĞ
11-B
THE CLAPPER 2015 - 2016 35