Calculating the Roots of Unity is very important for other mathematical computations and theories like the Fourier Transform. Analytic number theory, Algebra theory and more. The root of unity is one of the advanced complex number concepts that students will have to deal with. it is advised that before studying the roots of unity, students must have understood what complex number in their rectangular and polar form is all about. Also, they must have a very good idea about De Moivre Theorem and Eulers Formula. All four concepts’ numbers will be needed to fully compute the root of unity.
It is important to understand that you may not need to understand the entire concept of Roots of Unity if you are only interested in knowing what a complex number and a complex plane are. However, pure mathematics and physics students will find it very useful.
What Are Roots of Unity?
The roots of unity are all the complex solutions whose roots are equal to 1. 1 is what is known as unity. What this means in simple terms is that if any value is substituted into the place of the argument in the polar form of a complex number raised to the nth term i.e. [r(Cosϴ + isinϴ)]n equals 1, then that number is one of the roots of unity.
The root of unity is formally called the nth root of unity. So basically, the roots of unity are all the numbers which, when substituted into the nth term of the general polar form of a complex number, will equal 1.
Expressing this statement in a formula will leave the Roots of Unity Formula as;
Zn = 1
Z indicates the polar form of the complex number, which is given as;
Z = r[cosϴ + isinϴ]
So basically, the formula of the nth root of unity which is given as
Zn = 1 is the same as;
r[cosϴ + isinϴ]n = 1
Note: all numbers can be the root of unity once a formula has been established. The aim of this article is to establish that formula.
Now the next section will consider exactly how students can find the nth root of unity to apply it for any purpose they wish. Before we get to that, we will first write out the De Moivre formula, which is given as;
[r(Cosϴ + isinϴ)]n = [rn(Cosnϴ + isinnϴ)]
And the Euler Formula, which is given as
eiϴ = cosϴ + isinϴ
How to Find Roots of Unity?
We will need to follow a step-by-step process to find the nth root of unity.
Zn = 1
Which by De moivre theorem means that
[rn(Cosnϴ + isinnϴ)] = 1
now, if we focus on just the left hand side of the equation and make ϴ = 0, we will have
[rn(Cosn0 + isinn0)] = [rn(Cos0 + isin0)]
Now cos(0) + isin(0) = 1 which can be confirmed if you use a calculator. That we live us with
Now the only way that [rn(1)] = 1 to show that 0 is a root of unity, r will have to be 1,
So if we make r = 1 we will have
1n x 1 = 1
So we can say that 0 is one of the roots of unity if r, the modulus is equal to 1. Actually, to find any root of unity, r will be assumed to be equal to 1. This statement is the first property of Nth roots of unity.
The second property is that Z = 1 is a root of unity. This has been established in the formula already if we consider it actually. Recall that zn = 1 is the root of unity formula. This is the same as saying as z = 11/n. This is eventually z = 1. So the second property is also very correct.
Now, if z = 1 is the root for unity for any number based on the second property of roots of unity. Then it is safe to say that
Z = cos(2π) + isin(2π) will be a root of unity. Where 2π is standing in place of ϴ
Clearly cos(2π) = 1, while sin(2π) = 0
So Z = cos(2π) + isin(2π) = 1 + 0 = 1
cos(2π) + isin(2π) = 1 means 2π is another root of unity like 0
now we have been able to establish two roots of unity which will help us actually establish the formula for the nth roots of unity.
Now cos(2π) + isin(2π) = 1 is the same as cos(2πk) + isin(2πk) = 1 where k = 0, 1, 2, … (n – 1)
If you substitute k with any value in ensure cos(2πk) + isin(2πk), starting from 0 to any natural number, it will still be 1
This second root of unity given as z = cos(2πk) + isin(2πk) = 1 is known as the general polar form of roots of unity.
So since we now have the general polar form of roots of unity as Since [cos(2πk) + isin(2πk)] = 1
We can say the roots of unity formula zn = 1 can be expressed as
zn = [cos(2πk) + isin(2πk)]
z = [cos(2πk) + isin(2πk)]1/n …… form (ii)
By De moivre’s theorem,
[cos(2πk) + isin(2πk)]1/n = cos(2πk(1/n)) + isin(2πk(1/n))
= cos(2πk/n) + isin(2πk/n)
Z = cos(2πk/n) + isin(2πk/n)
Now, if you understand the Euler formula, you will know that
It is what will help us determine the nth root of the formula.
Clearly since k = 0, 1, 2, 3, …, (n – 1)
The 1st root of unity will be
This formula is the nth root of unity and can be used to find the formula of any roots of unity once n is given.
Now, if a student is asked to find the 4 roots of unity; then this is how to solve it.
The 8 roots of units will be based n = 1, 2, …., 8
When n = 1;
With the above, the 4 roots of unity is 1, – 1, 1 and – i
The pattern of finding the 4 roots of unity is the same way you will need to find a root of unity. The process is actually simple once you know how to find the nth roots of unity.
In this article, we have considered everything about the roots of unity as far as it relates to the complex number series. This piece highlighted exactly how people could solve for the nth root of unity and derive its formula if needed. In certain cases, students may be asked to find the nth roots of unity using the general roots of unity formula. We highlighted exactly how this could be achieved by showing practical step-by-step processes. A single example was also considered so that students could understand how to find the roots of unity.