The n-th root of unity is a special type of complex number that gives the result 1 when raised to a positive whole number power n. For example, if a number Z satisfies the equation Zⁿ = 1, then Z is called an n-th root of unity. These roots are not just the number 1—there are usually several such complex numbers, equally spaced around a circle on the complex plane. All these roots lie on the unit circle, which means they have a distance of 1 from the origin.

Roots of unity are useful in many branches of mathematics, including algebra, number theory, and geometry. They help in solving polynomial equations, simplifying trigonometric expressions, and performing operations like Fast Fourier Transforms (FFT) in computer science. Another name for roots of unity is de Moivre numbers, named after the French mathematician Abraham de Moivre.

In this maths article we will learn about nth roots of unity in detail

What is Nth Root of Unity?

In mathematics, if n is a positive integer, then a number x is called an n-th root of unity if it satisfies the equation xⁿ = 1. This means that when you multiply x by itself n times, the result is 1. There are exactly n different complex numbers that can satisfy this condition. These are called the n-th roots of unity.

To show roots like this, we often use the radical symbol, written as √, which is used for square roots. When we write √[n]{x}, it means the n-th root of x. In this expression, the small number n is called the index, and the value x inside the root is called the radicand. When n = 2, it becomes the familiar square root, because we are finding a number that gives x when squared.

Note:

• The nth root of \(0\) is \(0\) for all positive integers \(n\), as \(0^{n}\) is equal to \(0\).
• The nth root of \(1\) is known as the root of unity and plays an important role in different areas of Mathematics such as number theory, the theory of equation, etc.

Demoivre’s Theorem for Nth Root of Unity

For a complex number z = r(cos(\theta)+isin(\theta)), then the nth roots of z given by

\(\sqrt[n]{r}\left(cos\left(\frac{\theta+2\pi k}{n}\right)+isin\left(\frac{\theta+2\pi k}{n}\right)\right)\), for \(k = 0, 1, 2, ……, (n-1)\).

Learn more about Demoivre’s Theorem

According to the definition of the nth root of unity:

Step 1: If \(x\) is an nth root of unity, then it satisfies the relation \(x^{n} = 1\).

Step 2: Now \(1\) can also be written as \(\cos(0)+i\sin(0)\).

Step 3: We have \(x^{n} = 1\)

\(\Rightarrow\) \(x^{n}=\cos(0)+i\sin(0)\)

\(\Rightarrow\) \(x^{n}=\cos(2k\pi)+i\sin(2k\pi)\), \(k\) is an integer.

Step 4: Taking the nth root on both sides, we get

\(x=(\cos(2k\pi)+i\sin(2k\pi))^{\frac{1}{n}}\)

Using Demoivre’s Theorem, we get

\(x=\left(\cos\left(\frac{2k\pi}{n}\right)+i\sin\left(\frac{2k\pi}{n}\right)\right)\), where \(k=0,1,2,3,4,……,(n-1)\).

Step 5: So each root on unity is given as:

\(x=\left(\cos\left(\frac{2k\pi}{n}\right)+i\sin\left(\frac{2k\pi}{n}\right)\right)\), where \(0\leq k\leq (n-1)\).

Note: If a complex number is represented by \(\omega\), then

\(\omega=e^{\frac{2\pi}{n}i}=\cos\left(\frac{2k\pi}{n}\right)+i\sin\left(\frac{2k\pi}{n}\right)\)

\(\Rightarrow\) \(\omega^{n}=(e^{\frac{2\pi}{n}i})^{n}=e^{2\pi i}=1\)

Therefore, \(\omega\) is the nth root of unity.

Using de Moivre’s theorem, the complex numbers \(1,\omega, \omega^{2},…..,\omega^{n-1}\) are the nth roots of unity. Thus, we can say that all the complex numbers \(1,\omega, \omega^{2},…..,\omega^{n-1}\) are the points in a plane and vertices of a regular \(n\)-sided polygons, inscribed in a unit circle.

Nth Root of Unity in Complex Numbers

A complex number is generally written in the form x + iy, where:

• x is the real part
• iy is the imaginary part
• i is the square root of -1

Complex numbers are usually shown on a graph called the Argand plane or complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Now, when we talk about roots of unity, we're looking at complex numbers that satisfy the equation:

ωⁿ = 1

This means when the number ω is raised to the power n, the result is 1. One of the general forms of such a number is:

ω = cos(2πk/n) + i sin(2πk/n), where k = 0, 1, 2, ..., n−1

In this case:

• x = cos(2πk/n)
• y = sin(2πk/n)

If we square both and add:
x² + y² = cos²(2πk/n) + sin²(2πk/n) = 1

This shows that each root of unity lies on a circle of radius 1 centered at the origin (0, 0).

Using Euler’s formula, we can also write:
ω = e^(2πi/n)

When raised to the power n, we get:
ωⁿ = (e^(2πi/n))ⁿ = e^(2πi) = 1

So, ω is called the nth root of unity.

According to De Moivre’s Theorem, the full set of nth roots of unity is:
1, ω, ω², ω³, ..., ωⁿ⁻¹

These are evenly spaced points on the unit circle and form the vertices of a regular n-sided polygon (like a triangle, square, pentagon, etc.) inside the circle.

Nth Root of Unity on Argand Plane

The complex numbers are in the form \(x+iy\) and are plotted on the argand or the complex plane. Also, since the roots of unity are in the form

\(\left(\cos\left(\frac{2k\pi}{n}\right)+i\sin\left(\frac{2k\pi}{n}\right)\right)\).

So comparing it with the general form of complex number, we obtain the real and imaginary parts as:

\(x=\cos\left(\frac{2k\pi}{n}\right)\), \(y=\sin\left(\frac{2k\pi}{n}\right)\)

Now, we see that the values of \(x\) and \(y\) satisfies the equation of unit circle with centre \((0, 0)\) i.e. \(x^{2}+y^{2}=1\).

The argument of nth root of unity increase in an arithmetic sequence by \(\frac{2\pi}{n}\) radians. In an argand diagram, this means that we can plot the nth roots of unity by starting with \(1\) and rotating counterclockwise on the unit circle by \(\frac{2\pi}{n}\) consecutively. If we connect consecutive nth roots of unity with line segments, we will obtain a regular polygon inscribed in the unit circle. Let us observe this pattern in the diagram below for different values on \(n\).

1. For \(n=2\), the argand diagram for \(2\)th root of unity is given below:

2. For \(n=3\), the argand diagram for \(3\)th root of unity is given below:

3. For \(n=4\), the argand diagram for \(4\)th root of unity is given below:

4. For \(n=5\), the argand diagram for \(5\)th root of unity is given below:

5. For \(n=6\), the argand diagram for \(6\)th root of unity is given below:

6. For \(n=7\), the argand diagram for \(7\)th root of unity is given below:

Note: The arguments of the nth roots of unity do not all lie in the standard range, which is \(]-\pi, \pi]\). In particular, the cube roots of unity are labeled in the argand diagram above by \(1\), \(e^{\frac{2\pi}{3}i}\), and \(e^{\frac{4\pi}{3}i}\). The last cube root of unity has argument \(\frac{4\pi}{3}\), which is outside the standard range. Since this argument is over the upper bound \(\pi\), we can obtain an equivalent argument by subtracting the full revolution of \(2\pi\) radians from this value:

\(\frac{4\pi}{3}-2\pi=\frac{-2\pi}{3}\)

Here \(\frac{-2\pi}{3}\) lies in the standard range, so we can use this argument to write the third root of unity as \(e^{\frac{-2\pi}{3}i}\). Similarly, for other roots of unity, we can find their argument in the standard range.

Primitive nth Root of Unity

A primitive nth root of unity is a complex number \(\omega\) for which \(k=n\) is the smallest positive integer satisfying \(\omega^{k}=1\). From the table below, check the primitive nth roots of unity for \(n=1,2,3,…..,7\):

Properties of primitive nth root of unity

• Any complex number of the form \(\omega=e^{\frac{2\pi}{n}i}\) for positive integer \(n\) is a primitive nth root of unity.
• If \(\omega\) is a primitive nth root of unity, then all nth roots of unity are given by \(1,\omega, \omega^{2},…..,\omega^{n-1}\).
• Let \(\omega\) be a primitive nth root of unity for \(n\geq 2\), then \(1+\omega+\omega^{2}+…..+\omega^{n-1}=0\).

Sum of Nth Root of Unity

The sum of all the nth roots of unity is zeros, i.e., \(1+\omega+\omega^{2}+……….+\omega^{n-1} = 0\).

Proof: \(1+\omega+\omega^{2}+……….+\omega^{n-1} = \frac{1-\omega^{n}}{1-\omega}\)

Since \(\omega^{n} = 1\) and \(\omega \neq 1\), then

\(1+\omega+\omega^{2}+……….+\omega^{n-1} = 0\).

Product of nth root of unity

The sum of all the nth roots of unity is given as \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=(-1)^{n-1}\).

Proof: \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=\omega^{0+1+2+3+…..+(n-1)}\)

\(\Rightarrow\) \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=\omega^{\frac{n(n-1)}{n}}\)

\(\Rightarrow\) \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=(\omega^{n})^{\frac{(n-1)}{2}}\)

\(\Rightarrow\) \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=(e^{i2\pi})^{\frac{(n-1)}{2}}\)

\(\Rightarrow\) \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=(e^{i\pi})^{n-1}\)

\(\Rightarrow\) \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=(-1)^{n-1}\).

Group of Nth Root of Unity

In mathematics, the group of nth roots of unity refers to a collection of complex numbers that, when raised to the power of n, result in the value of 1. These roots form a group under multiplication and have interesting properties in algebra and number theory.

The group of nth roots of unity exhibits important algebraic properties. For instance, the product of any two nth roots of unity is also an nth root of unity. Furthermore, the nth roots of unity are closed under multiplication, have an identity element (1), and each element has an inverse within the group.

These properties make the group of
nth roots of unity significant in various areas of mathematics, including complex analysis, number theory, and algebraic geometry. 

Nth Root of Unity Properties

The properties of nth root of unity are listed below:

• The \(n\) roots of the nth roots of unity are found on the perimeter of a circle with a radius of 1 and the origin as its centre (0, 0).
• The sum of all the nth roots of unity is zero.
• The product of all the nth roots of unity is: \(1\cdot\omega\cdot\omega^{2}……….\omega^{n-1}=(-1)^{n-1}\).
• The nth roots of unity are in geometric progression with a common ratio of \(i\left(\frac{2\pi}{n}\right)\).
• \(\sqrt[n]{xy}=\sqrt[n]{x}\sqrt[n]{y}\).
• \(\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}\).

Solved Examples of Nth Root of Unity

Example 1: Find the quintic roots of unity.

Solution: We know that the nth roots of unity are given in polar form as \(e^{\frac{2k\pi}{n}i}\), for \(k = 0, 1, 2, ….., (n-1)\).

Here we have to find the quintic roots of unity, which are the same as the 5th roots of unity. Hence, write the quintic roots of unity by substituting \(n = 5\) and \(k = 0, 1, 2, 3, 4\) into the above formula.

For \(k = 0\): \(e^{0i} = 1\),

For \(k = 1\): \(e^{\frac{2\pi}{5}i}\),

For \(k = 2\): \(e^{\frac{2\pi\times 2}{5}i} = e^{\frac{4\pi}{5}i}\),

For \(k = 3\): \(e^{\frac{2\pi\times 3}{5}i} = e^{\frac{6\pi}{5}i}\),

For \(k = 4\): \(e^{\frac{2\pi\times 4}{5}i} = e^{\frac{8\pi}{5}i}\).

Since, the argument of a complex number, by convention, should lie in the standard range \(]-\pi, \pi[\). The last two quintic roots of unity have arguments \(\frac{6\pi}{5}\) and \(\frac{8\pi}{5}\), which do not lie in this range.

So, these arguments are over the upper bound \(\pi\), we can obtain equivalent arguments by subtracting the full revolution of \(2\pi\) radians from this value.

\(\Rightarrow\) \(\frac{6\pi}{5}-2\pi=\frac{-4\pi}{5}\), and

\(\Rightarrow\) \(\frac{8\pi}{5}-2\pi=\frac{-2\pi}{5}\).

Hence, the quintic roots of unity are \(1\), \(e^{\frac{2\pi}{5}i}\), \(e^{\frac{4\pi}{5}i}\), \(e^{\frac{-4\pi}{5}i}\), \(e^{\frac{-2\pi}{5}i}\).

Example 2: Express \(\omega^{-1}\) in terms of positive powers of \(\omega\).

Solution: We have to express that \(\omega^{-1} = \omega^{k}\), for some positive integer ‘\(k\)’.

Since \(\omega\) is the \(nth\) root of unity, we know that

\(\omega^{n} = 1\)

Multiply both sides of the above equation by \(\omega^{-1}\) and use the rule of exponents, we have

\(\omega^{n} \omega^{-1} = \omega^{-1}\)

\(\Rightarrow\) \(\omega^{n-1} = \omega^{-1}\)

Since \((n-1)\) is a positive number, we have

\(\Rightarrow\) \(\omega^{-1} = \omega^{n-1}\).

Example 3: Find the cube roots of unity (3rd roots of 1).

Solution:

We use the general formula for the n-th roots of unity:

ω = e^(2πik/n), where k = 0, 1, ..., n−1.

Here, n = 3 (since we want cube roots), so we calculate for k = 0, 1, and 2.

• For k = 0: ω₀ = e^(2πi × 0 / 3) = e^0 = 1
• For k = 1: ω₁ = e^(2πi × 1 / 3) = e^(2πi / 3)
• For k = 2: ω₂ = e^(2πi × 2 / 3) = e^(4πi / 3)

Since we prefer all angles between −π and π, we can rewrite:
e^(4πi / 3) = e^(−2πi / 3)

So the final cube roots of unity are:
1, e^(2πi / 3), e^(−2πi / 3)

These three complex numbers lie on the unit circle and form an equilateral triangle.

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