Roots of Unity

Given a small integer n, print all the n’th roots of unity up to 6 significant digits. We basically need to find all roots of equation xn – 1.

Examples:

Input :  n = 1
Output : 1.000000 + i 0.000000
x - 1 = 0 , has only one root i.e., 1

Input :  2
Output : 1.000000 + i 0.000000
    -1.000000 + i 0.000000
x2 - 1 = 0 has 2 distinct roots, i.e., 1 and -1 


Any complex number is said to be root of unity if it gives 1 when raised to some power.

nth root of unity is any complex number such that it gives 1 when raised to the power n.

Mathematically, 
An nth root of unity, where n is a positive integer 
(i.e. n = 1, 2, 3, …) is a number z satisfying the
equation 

z^n  = 1
or , 
z^n - 1 = 0

We can use the De Moivre’s formula here ,

( Cos x + i Sin x )^k = Cos kx + i Sin kx

Setting x = 2*pi/n, we can obtain all the nth roots 
of unity, using the fact that Nth roots are set of 
numbers given by,

Cos (2*pi*k/n) + i Sin(2*pi*k/n)
Where, 0 <= k < n

Using the above fact we can easily print all the nth roots of unity !
Below is the program for the same.

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to print n'th roots of unity
#include <bits/stdc++.h>
using namespace std;
  
// This function receives an integer n , and prints
// all the nth roots of unity
void printRoots(int n)
{
    // theta = 2*pi/n
    double theta = M_PI*2/n;
  
    // print all nth roots with 6 significant digits
    for(int k=0; k<n; k++)
    {
        // calculate the real and imaginary part of root
        double real = cos(k*theta);
        double img = sin(k*theta);
  
        // Print real and imaginary parts
        printf("%.6f", real);
        img >= 0? printf(" + i "): printf(" - i ");
        printf("%.6f\n", abs(img));
    }
}
  
// Driver function to check the program
int main()
{
    printRoots(1);
    cout << endl;
    printRoots(2);
    cout << endl;
    printRoots(3);
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java program to print n'th roots of unity
import java.io.*;
  
class GFG {
  
// This function receives an integer n , and prints
// all the nth roots of unity
static void printRoots(int n)
{
    // theta = 2*pi/n
    double theta = 3.14*2/n;
  
    // print all nth roots with 6 significant digits
    for(int k=0; k<n; k++)
    {
        // calculate the real and imaginary part of root
        double real = Math.cos(k*theta);
        double img = Math.sin(k*theta);
  
        // Print real and imaginary parts
        System.out.println(real);
        if (img >= 0)
            System.out.println(" + i ");
        else
            System.out.println(" - i ");
        System.out.println(Math.abs(img));
    }
}
  
// Driver function to check the program
public static void main (String[] args)
{
    printRoots(1);
    //System.out.println();
    printRoots(2);
    //System.out.println();
    printRoots(3);
}
}
// This code is conributed by Raj

chevron_right


Python3

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python3 program to print n'th roots of unity
  
import math
  
# This function receives an integer n , and prints
# all the nth roots of unity
def printRoots(n):
  
    # theta = 2*pi/n
    theta = math.pi * 2 / n
  
    # print all nth roots with 6 significant digits
    for k in range(0, n):
  
        # calculate the real and imaginary part of root
        real = math.cos(k * theta)
        img = math.sin(k * theta)
  
        # Print real and imaginary parts
        print(real, end=" ")
        if(img >= 0):
            print(" + i ", end=" ")
        else:
            print(" - i ", end=" ")
        print(abs(img))
  
  
# Driver function to check the program
if __name__=='__main__':
    printRoots(1)
    printRoots(2)
    printRoots(3)
  
# This code is contributed by
# Sanjit_Prasad

chevron_right


C#

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program to print n'th roots of unity 
using System;
  
class GFG { 
  
// This function receives an integer n , and prints 
// all the nth roots of unity 
static void printRoots(int n) 
    // theta = 2*pi/n 
    double theta = 3.14*2/n; 
  
    // print all nth roots with 6 significant digits 
    for(int k=0; k<n; k++) 
    
        // calculate the real and imaginary part of root 
        double real = Math.Cos(k*theta); 
        double img = Math.Sin(k*theta); 
  
        // Print real and imaginary parts 
        Console.Write(real); 
        if (img >= 0) 
            Console.Write(" + i "); 
        else
            Console.Write(" - i "); 
        Console.WriteLine(Math.Abs(img)); 
    
  
// Driver function to check the program 
static void Main() 
    printRoots(1); 
       
    printRoots(2); 
       
    printRoots(3); 
// This code is conributed by mits

chevron_right


PHP

= 0 ? print(” + i “): print(” – i “);
printf(round(abs($img), 6) . “\n”);
}
}

// Driver Code
printRoots(1);
printRoots(2);
printRoots(3);

// This code is contributed by mits
?>


Output:

1.000000 + i 0.000000
1.000000 + i 0.000000
-1.000000 + i 0.000000
1.000000 + i 0.000000
-0.500000 + i 0.866025
-0.500000 - i 0.866025

References : Wikipedia

This article is contributed by Ashutosh Kumar .If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



My Personal Notes arrow_drop_up



Article Tags :
Practice Tags :


Be the First to upvote.


Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.