Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

De Moivre's Theorem and nth Roots of Unity for JEE

ffImage
Last updated date: 17th May 2024
Total views: 71.4k
Views today: 0.71k
hightlight icon
highlight icon
highlight icon
share icon
copy icon

Complex Numbers: De Moivre's Theorem and nth Roots of Unity

Roots of unity are an important topic in mathematics. They are especially important in number theory, and DFT or discrete Fourier transform. A root of unity also called De moivre's number or De ulation of an nth root of unity is depicted by the equation zn = 1, where n is a positive integer. Let us further discuss the sum of nth roots of unity and the primitive nth root of unity.


De Moivre's Theorem

De moivre's Number or De moivre's Theorem states the relationship between any real number x and an integer n. The equation that related the two is denoted as:

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

Here I denote the imaginary unit i,e i2 =1. It is an important formula in mathematics because it connects complex numbers and trigonometry. If we expand the left-hand side and then compare it with the real and imaginary parts, we get expressions of cos(nx) and sin(nx) in terms of cosx and sinx.

It must be noted that the De moivre's Number or De moivre's Theorem is not valid for non-integer valThus in solving, we get the solution as 


Roots of Unity

A root of unity also called De moivre's number is a complex number when raised to some positive integer power n yields 1. Unless otherwise specified the roots of unity are defined or taken as a complex number. The statement holds true even for both 1 and -1.


Pictorial Representation of nth Root of Unity

An nth root of unity is defined by the equation zn = 1, where n is a positive integer. It is also the nth root of the unity formula. The equation can also be written as $\sqrt[n]{1}=z$. It must be noted that all the n roots of nth roots of unity are in Geometric Progression.

Pictorial Representation of the nth Root of Unity


Pictorial Representation of the nth Root of Unity


nth Roots of Unity Formula

As stated in the previous section, the nth root of the unity formula is given as zn = 1, or $\sqrt[n]{1}=z$.


nth Root of Complex Number

De moivre's number r theorem can also be used to find the nth roots of the complex number or equivalently the $\dfrac{1}{n}$ power.

If z is a complex number and it is written in polar form as:  $z=r(\cos (\theta)+i \sin (\theta))$

For this the nth root is $r(\cos (\theta+2 k \pi)+i \sin (\theta+2 k \pi))$

Where k is an integer whose value will vary between 0 and n-1.


How to Find the nth Root of Unity

The nth root of unity can be found below. We know that the nth root of unity is denoted as 

Zn = 1, where n is a positive integer and is the nth root of unity. 

If we rewrite the above equation in polar form

Zn = cos 0 + i sin 0

From the above, we know that the nth root of a complex number can be denoted as,

$Z^{n}=\cos (0+2 k \pi)+i(0+\sin 2 k \pi)$, where k is an integer

Taking nth root on both the sides, we will get. 

$Z=(\cos 2 k \pi+i \sin 2 k \pi)^{1 / n} $


Root of Unity Examples

1. What will be the value of the fifth root of unity?

Solution: Let z5 = 1

Writing the above equation in polar form we get the below equation

$Z^{5}=\cos (0+2 k \pi)+i \sin (0+2 k \pi)=e^{i 2 k \pi}$ , where k = 0, 1, 2,…n-1


Hence, using De Moivre's theorem or formula we get

$Z^{\dfrac{1}{5}}=\cos \left(\dfrac{2 \pi \mathrm{k}}{\mathrm{n}}\right)+i \sin \left(\dfrac{2 \pi \mathrm{k}}{\mathrm{n}}\right)=e^{\dfrac{2 \pi \mathrm{k}}{\mathrm{n}} i}$ ; k is an integer and has values ranging from 0, 1, 2,…n-1


To get the roots, we will substitute the value of k in the formula and get the results


For, k = 0

Z = cos 0 + i sin 0 = 1


For k = 1

$Z=\cos \dfrac{2 \pi}{5}+i \sin \dfrac{2 \pi}{5}=e^{\dfrac{2 \pi}{5} i}$


For k = 2

$Z=\cos \dfrac{4 \pi}{5}+i \sin \dfrac{4 \pi}{5}=e^{\dfrac{4 \pi}{5} i}$


For k = 3

$\begin{align} &\mathrm{Z}=\cos \left(\dfrac{6 \pi}{5}\right)+i \sin \left(\dfrac{6 \pi}{5}\right)=e^{\dfrac{-4 \pi}{5} i} \\ &\because \dfrac{6 \pi}{5}-2 \pi=-\dfrac{4 \pi}{5} \end{align}$


For k = 4

$\begin{align} &z=\cos \left(\dfrac{8 \pi}{5}\right)+i \sin \left(\dfrac{8 \pi}{5}\right)=e^{\dfrac{-2 \pi}{5} i} \\ &\because \dfrac{8 \pi}{5}-2 \pi=-\dfrac{2 \pi}{5} \end{align}$

Therefore, the fourth roots of unity are $1,~e^{\dfrac{2 \pi}{5} i},~e^{\dfrac{4 \pi}{5} i},~e^{\dfrac{-4 \pi}{5} i},~e^{\dfrac{-2 \pi}{5} i}$


2. Calculate the Fourth root of unity by taking an example

Solution: Let, $Z = 1^{\dfrac{1}{4}}$

Z4 = 1


Writing the above equation in polar form we get the below equation

$z^{4}=\cos (0+2 k \pi)+i \sin (0+2 k \pi)=e^{i 2 k \pi}$ , where k = 0, 1, 2,…


Hence, using De Moibstitute the value of k in the formula and get the results

For, k = 0; Z = cos 0 + i ${\pi} { 2}+i \sin \dfrac{3 \pi} { 2}=-\cos \dfrac{\pi}{ 2}-i \sin \dfrac{\pi}{ 2}=-i$


Therefore, the fourth roots of unity are 1, i, -1 and -i.


Primitive nth Root of Unity

Primitive nth roots of unity are the ones whose multiplication order is n. A number r is called an nth root of unity if rn = 1 and a primitive nth root of unity are if, in addition, n is the smallest integer of k = 1….. n for which rK = 1. 


Let us understand the Primitive nth roots of unity using the example below.


The 5th root of unity is $e^{\dfrac{6\pi i}{5}}$. It can also be said as the 10th,15th, 20th or the 100th root of unity. It is because when $e^{\dfrac{6\pi i}{5}}$ is raised to any power of 5 its value is 1.


Sum of n Roots of Unity

The sum of all the nth roots of unity is zero whereas the product of nth roots of unity is (-1)n-1.

The sum of all n roots of unity can be derived as follows:

Let 

$\omega=e^{2 \pi i n}$

$\omega=e^{2 \pi i n}$,  its roots of unity will be of the form (since it is the primitive nth root of unity)

$\begin{align} &1+\omega+\omega^{2}+\omega^{3}+\cdots+\omega^{n-1}=\dfrac{\omega^{n}-1}{\omega-1} \\ &1+\omega+\omega^{2}+\omega^{3}+\cdots+\omega^{n-1}=\dfrac{1-1}{\omega-1} \\&1+\omega+\omega^{2}+\omega^{3}+\cdots+\omega^{n-1}=0 \end{align}$


Therefore, the sum of all n roots of unity is zero.


Conclusion

The article narrates a thorough walkthrough of De Moivre’s theorem and the nth roots of unity. It also discusses the sum of nth roots of unity which is zero and the primitive nth root of unity. The Primitive nth roots of unity are the ones whose multiplication order is n. It must be noted that all the n roots of nth roots unity are in Geometrical Progression or GP. Also, all the n roots of nth roots unity lie on the circle with their centre at the origin and with a radius of magnitude 1. All these divide the circle into equal parts as can be seen from the figure above.

Popular Vedantu Learning Centres Near You
centre-image
Mithanpura, Muzaffarpur
location-imgVedantu Learning Centre, 2nd Floor, Ugra Tara Complex, Club Rd, opposite Grand Mall, Mahammadpur Kazi, Mithanpura, Muzaffarpur, Bihar 842002
Visit Centre
centre-image
Anna Nagar, Chennai
location-imgVedantu Learning Centre, Plot No. Y - 217, Plot No 4617, 2nd Ave, Y Block, Anna Nagar, Chennai, Tamil Nadu 600040
Visit Centre
centre-image
Velachery, Chennai
location-imgVedantu Learning Centre, 3rd Floor, ASV Crown Plaza, No.391, Velachery - Tambaram Main Rd, Velachery, Chennai, Tamil Nadu 600042
Visit Centre
centre-image
Tambaram, Chennai
location-imgShree Gugans School CBSE, 54/5, School road, Selaiyur, Tambaram, Chennai, Tamil Nadu 600073
Visit Centre
centre-image
Avadi, Chennai
location-imgVedantu Learning Centre, Ayyappa Enterprises - No: 308 / A CTH Road Avadi, Chennai - 600054
Visit Centre
centre-image
Deeksha Vidyanagar, Bangalore
location-imgSri Venkateshwara Pre-University College, NH 7, Vidyanagar, Bengaluru International Airport Road, Bengaluru, Karnataka 562157
Visit Centre
View More
Competitive Exams after 12th Science

FAQs on De Moivre's Theorem and nth Roots of Unity for JEE

1. How do we write the primitive 6th root of unity in complex form?

The nth root of are of the form $e^{\dfrac{2 \pi i k} { n}}$, where k is an integer. Therefore, for the 6th root of unity, the value of k will vary between 0 to 5 to calculate the roots. 

$\begin{align} &e^{0 \}{ 3}}=-\dfrac{1}{2}-i \dfrac{\sqrt{3}}{2} \\ &e^{\dfrac{5 \pi i} { 3}}=\dfrac{1}{2}-i \dfrac{\sqrt{3}}{2} \end{align}$

In complex form, these can be written as $\left(\pm 1, \pm \dfrac{1}{2}, \pm i \dfrac{\sqrt{3}}{2}\right)$

2. What is the cube root of unity? Also, prove that the product of all the nth roots of unity is  (-1)n-1.

The cube root of a number is denoted as $x=3\sqrt{3}y$ or $x=y^{\dfrac{1}{3}}$. However, the cube root of unity is denoted as$z^{\dfrac{1}{3}=1}$. On calculating the values of the cube root of unity comes out to be ${1,-\dfrac{1}{2}+i}$ 

The product of n, nth roots of unity is (-1)n-1.


$\begin{align} &1 . \omega \cdot \omega^{2} \cdot \omega^{3} \ldots \ldots \omega^{n-1}=\omega \dfrac{n(n-1)}{2}\\ &\left(\omega^{n}\right)^{\dfrac{(n-1)}{2}}=\left(e^{2 \pi i}\right)^{\frac{(n-1)}{2}}=(-1)^{(n-1)} \end{align}$

Here we have denoted the complex number as $\omega$.