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 a C++ program for the same.

// A 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;
}

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



Practice Tags :
Article Tags :

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

Recommended Posts:



1.6 Average Difficulty : 1.6/5.0
Based on 3 vote(s)






User Actions