团结之源

给定一个小整数n，最多打印6个有效数字的所有第n个单位根。我们基本上需要找到方程x ⁿ – 1的所有根。
例子：

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 

如果任何复数提高到一定的幂时给出1，则称其为单位根。
整数的第n个根是任何复数，因此加到n的幂为1。

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

我们可以在这里使用De Moivre的公式，

( 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

利用以上事实，我们可以轻松地打印出所有的第n个统一根！
以下是相同的程序。

C++

// C++ program to print n'th roots of unity
#include 
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= 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;
}

Java

// 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= 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

Python3

# 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

C#

// 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= 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

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
?>

Javascript

// javascript program to prvar n'th roots of unity
 
// This function receives an integer n , and prints
// all the nth roots of unity
function printRoots(n)
{
    // theta = 2*pi/n
    var theta = (3.14*2/n);
 
    // prvar all nth roots with 6 significant digits
    for(k = 0; k < n; k++)
    {
        // calculate the real and imaginary part of root
        var real = Math.cos(k*theta);
        var img = Math.sin(k*theta);
 
        // Prvar real and imaginary parts
        document.write(real.toFixed(6));
        if (img >= 0)
            document.write(" + i ");
        else
            document.write(" - i ");
        document.write(Math.abs(img).toFixed(6)+'<br>');
    }
}
 
// Driver function to check the program
printRoots(1);
 
//document.write('<br>');
printRoots(2);
 
//document.write('<br>');
printRoots(3);
 
// This code is contributed by shikhasingrajput

输出：

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