Roots of unity

From Rosetta Code
Task
Roots of unity
You are encouraged to solve this task according to the task description, using any language you may know.

The purpose of this task is to explore working with   complex numbers.


Task

Given   n,   find the   n-th   roots of unity.

Ada[edit]

with Ada.Text_IO;                 use Ada.Text_IO;
with Ada.Float_Text_IO; use Ada.Float_Text_IO;
with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types;
 
procedure Roots_Of_Unity is
Root : Complex;
begin
for N in 2..10 loop
Put_Line ("N =" & Integer'Image (N));
for K in 0..N - 1 loop
Root :=
Compose_From_Polar
( Modulus => 1.0,
Argument => Float (K),
Cycle => Float (N)
);
-- Output
Put (" k =" & Integer'Image (K) & ", ");
if Re (Root) < 0.0 then
Put ("-");
else
Put ("+");
end if;
Put (abs Re (Root), Fore => 1, Exp => 0);
if Im (Root) < 0.0 then
Put ("-");
else
Put ("+");
end if;
Put (abs Im (Root), Fore => 1, Exp => 0);
Put_Line ("i");
end loop;
end loop;
end Roots_Of_Unity;

Ada provides a direct implementation of polar composition of complex numbers x ei y. The function Compose_From_Polar is used to compose roots. The third argument of the function is the cycle. Instead of the standard cycle 2π, N is used. Sample output:

N = 2
   k = 0, +1.00000+0.00000i
   k = 1, -1.00000+0.00000i
N = 3
   k = 0, +1.00000+0.00000i
   k = 1, -0.50000+0.86603i
   k = 2, -0.50000-0.86603i
N = 4
   k = 0, +1.00000+0.00000i
   k = 1, +0.00000+1.00000i
   k = 2, -1.00000+0.00000i
   k = 3, +0.00000-1.00000i
N = 5
   k = 0, +1.00000+0.00000i
   k = 1, +0.30902+0.95106i
   k = 2, -0.80902+0.58779i
   k = 3, -0.80902-0.58779i
   k = 4, +0.30902-0.95106i
N = 6
   k = 0, +1.00000+0.00000i
   k = 1, +0.50000+0.86603i
   k = 2, -0.50000+0.86603i
   k = 3, -1.00000+0.00000i
   k = 4, -0.50000-0.86603i
   k = 5, +0.50000-0.86603i
N = 7
   k = 0, +1.00000+0.00000i
   k = 1, +0.62349+0.78183i
   k = 2, -0.22252+0.97493i
   k = 3, -0.90097+0.43388i
   k = 4, -0.90097-0.43388i
   k = 5, -0.22252-0.97493i
   k = 6, +0.62349-0.78183i
N = 8
   k = 0, +1.00000+0.00000i
   k = 1, +0.70711+0.70711i
   k = 2, +0.00000+1.00000i
   k = 3, -0.70711+0.70711i
   k = 4, -1.00000+0.00000i
   k = 5, -0.70711-0.70711i
   k = 6, +0.00000-1.00000i
   k = 7, +0.70711-0.70711i
N = 9
   k = 0, +1.00000+0.00000i
   k = 1, +0.76604+0.64279i
   k = 2, +0.17365+0.98481i
   k = 3, -0.50000+0.86603i
   k = 4, -0.93969+0.34202i
   k = 5, -0.93969-0.34202i
   k = 6, -0.50000-0.86603i
   k = 7, +0.17365-0.98481i
   k = 8, +0.76604-0.64279i
N = 10
   k = 0, +1.00000+0.00000i
   k = 1, +0.80902+0.58779i
   k = 2, +0.30902+0.95106i
   k = 3, -0.30902+0.95106i
   k = 4, -0.80902+0.58779i
   k = 5, -1.00000+0.00000i
   k = 6, -0.80902-0.58779i
   k = 7, -0.30902-0.95106i
   k = 8, +0.30902-0.95106i
   k = 9, +0.80902-0.58779i

ALGOL 68[edit]

Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
FORMAT complex fmt=$g(-6,4)"⊥"g(-6,4)$;
FOR root FROM 2 TO 10 DO
printf(($g(4)$,root));
FOR n FROM 0 TO root-1 DO
printf(($xf(complex fmt)$,complex exp( 0 I 2*pi*n/root)))
OD;
printf($l$)
OD

Output:

  +2 1.0000⊥0.0000 -1.000⊥0.0000
  +3 1.0000⊥0.0000 -.5000⊥0.8660 -.5000⊥-.8660
  +4 1.0000⊥0.0000 0.0000⊥1.0000 -1.000⊥0.0000 -.0000⊥-1.000
  +5 1.0000⊥0.0000 0.3090⊥0.9511 -.8090⊥0.5878 -.8090⊥-.5878 0.3090⊥-.9511
  +6 1.0000⊥0.0000 0.5000⊥0.8660 -.5000⊥0.8660 -1.000⊥0.0000 -.5000⊥-.8660 0.5000⊥-.8660
  +7 1.0000⊥0.0000 0.6235⊥0.7818 -.2225⊥0.9749 -.9010⊥0.4339 -.9010⊥-.4339 -.2225⊥-.9749 0.6235⊥-.7818
  +8 1.0000⊥0.0000 0.7071⊥0.7071 0.0000⊥1.0000 -.7071⊥0.7071 -1.000⊥0.0000 -.7071⊥-.7071 -.0000⊥-1.000 0.7071⊥-.7071
  +9 1.0000⊥0.0000 0.7660⊥0.6428 0.1736⊥0.9848 -.5000⊥0.8660 -.9397⊥0.3420 -.9397⊥-.3420 -.5000⊥-.8660 0.1736⊥-.9848 0.7660⊥-.6428
 +10 1.0000⊥0.0000 0.8090⊥0.5878 0.3090⊥0.9511 -.3090⊥0.9511 -.8090⊥0.5878 -1.000⊥0.0000 -.8090⊥-.5878 -.3090⊥-.9511 0.3090⊥-.9511 0.8090⊥-.5878

AutoHotkey[edit]

ahk forum: discussion

n := 8, a := 8*atan(1)/n
Loop %n%
i := A_Index-1, t .= cos(a*i) ((s:=sin(a*i))<0 ? " - i*" . -s : " + i*" . s) "`n"
Msgbox % t

AWK[edit]

 
# syntax: GAWK -f ROOTS_OF_UNITY.AWK
BEGIN {
pi = 3.1415926
for (n=2; n<=5; n++) {
printf("%d: ",n)
for (root=0; root<=n-1; root++) {
real = cos(2 * pi * root / n)
imag = sin(2 * pi * root / n)
printf("%8.5f %8.5fi",real,imag)
if (root != n-1) { printf(", ") }
}
printf("\n")
}
exit(0)
}
 
Output:
2:  1.00000  0.00000i, -1.00000  0.00000i
3:  1.00000  0.00000i, -0.50000  0.86603i, -0.50000 -0.86603i
4:  1.00000  0.00000i,  0.00000  1.00000i, -1.00000  0.00000i, -0.00000 -1.00000i
5:  1.00000  0.00000i,  0.30902  0.95106i, -0.80902  0.58779i, -0.80902 -0.58779i,  0.30902 -0.95106i

BASIC[edit]

Works with: QuickBasic version 4.5
Translation of: Java

For high n's, this may repeat the root of 1 + 0*i.

 CLS
PI = 3.1415926#
n = 5 'this can be changed for any desired n
angle = 0 'start at angle 0
DO
real = COS(angle) 'real axis is the x axis
IF (ABS(real) < 10 ^ -5) THEN real = 0 'get rid of annoying sci notation
imag = SIN(angle) 'imaginary axis is the y axis
IF (ABS(imag) < 10 ^ -5) THEN imag = 0 'get rid of annoying sci notation
PRINT real; "+"; imag; "i" 'answer on every line
angle = angle + (2 * PI) / n
'all the way around the circle at even intervals
LOOP WHILE angle < 2 * PI

BBC BASIC[edit]

      @% = &20408
FOR n% = 2 TO 5
PRINT STR$(n%) ": " ;
FOR root% = 0 TO n%-1
real = COS(2*PI * root% / n%)
imag = SIN(2*PI * root% / n%)
PRINT real imag "i" ;
IF root% <> n%-1 PRINT "," ;
NEXT
PRINT
NEXT n%

Output:

2:   1.0000  0.0000i, -1.0000  0.0000i
3:   1.0000  0.0000i, -0.5000  0.8660i, -0.5000 -0.8660i
4:   1.0000  0.0000i,  0.0000  1.0000i, -1.0000  0.0000i, -0.0000 -1.0000i
5:   1.0000  0.0000i,  0.3090  0.9511i, -0.8090  0.5878i, -0.8090 -0.5878i,  0.3090 -0.9511i

C[edit]

#include <stdio.h>
#include <math.h>
 
int main()
{
double a, c, s, PI2 = atan2(1, 1) * 8;
int n, i;
 
for (n = 1; n < 10; n++) for (i = 0; i < n; i++) {
c = s = 0;
if (!i ) c = 1;
else if(n == 4 * i) s = 1;
else if(n == 2 * i) c = -1;
else if(3 * n == 4 * i) s = -1;
else
a = i * PI2 / n, c = cos(a), s = sin(a);
 
if (c) printf("%.2g", c);
printf(s == 1 ? "i" : s == -1 ? "-i" : s ? "%+.2gi" : "", s);
printf(i == n - 1 ?"\n":", ");
}
 
return 0;
}

C#[edit]

using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
 
class Program
{
static IEnumerable<Complex> RootsOfUnity(int degree)
{
return Enumerable
.Range(0, degree)
.Select(element => Complex.FromPolarCoordinates(1, 2 * Math.PI * element / degree));
}
 
static void Main()
{
var degree = 3;
foreach (var root in RootsOfUnity(degree))
{
Console.WriteLine(root);
}
}
}

Output:

(1, 0)
(-0,5, 0,866025403784439)
(-0,5, -0,866025403784438)

C++[edit]

#include <complex>
#include <cmath>
#include <iostream>
 
double const pi = 4 * std::atan(1);
 
int main()
{
for (int n = 2; n <= 10; ++n)
{
std::cout << n << ": ";
for (int k = 0; k < n; ++k)
std::cout << std::polar(1, 2*pi*k/n) << " ";
std::cout << std::endl;
}
}

CoffeeScript[edit]

Most of the effort here is in formatting the results, and the output is still a bit clumsy.

# Find the n nth-roots of 1
nth_roots_of_unity = (n) ->
(complex_unit_vector(2*Math.PI*i/n) for i in [1..n])
 
complex_unit_vector = (rad) ->
new Complex(Math.cos(rad), Math.sin(rad))
 
class Complex
constructor: (@real, @imag) ->
toString: ->
round_z = (n) ->
if Math.abs(n) < 0.00005 then 0 else n
fmt = (n) -> n.toFixed(3)
real = round_z @real
imag = round_z @imag
s = ''
if real and imag
"#{fmt real}+#{fmt imag}i"
else if real or !imag
"#{fmt real}"
else
"#{fmt imag}i"
 
do ->
for n in [2..5]
console.log "---1 to the 1/#{n}"
for root in nth_roots_of_unity n
console.log root.toString()

output

> coffee nth_roots.coffee 
---1 to the 1/2
-1.000
1.000
---1 to the 1/3
-0.500+0.866i
-0.500+-0.866i
1.000
---1 to the 1/4
1.000i
-1.000
-1.000i
1.000
---1 to the 1/5
0.309+0.951i
-0.809+0.588i
-0.809+-0.588i
0.309+-0.951i
1.000

Common Lisp[edit]

(defun roots-of-unity (n)
(loop for i below n
collect (cis (* pi (/ (* 2 i) n)))))

The expression is slightly more complicated than necessary in order to preserve exact rational arithmetic until multiplying by pi. The author of this example is not a floating point expert and not sure whether this is actually useful; if not, the simpler expression is (cis (/ (* 2 pi i) n)).

D[edit]

Using std.complex:

import std.stdio, std.range, std.algorithm, std.complex;
import std.math: PI;
 
auto nthRoots(in int n) pure nothrow {
return n.iota.map!(k => expi(PI * 2 * (k + 1) / n));
}
 
void main() {
foreach (immutable i; 1 .. 6)
writefln("#%d: [%(%5.2f, %)]", i, i.nthRoots);
}
Output:
#1: [ 1.00+ 0.00i]
#2: [-1.00+-0.00i,  1.00+ 0.00i]
#3: [-0.50+ 0.87i, -0.50+-0.87i,  1.00+ 0.00i]
#4: [-0.00+ 1.00i, -1.00+-0.00i,  0.00+-1.00i,  1.00+ 0.00i]
#5: [ 0.31+ 0.95i, -0.81+ 0.59i, -0.81+-0.59i,  0.31+-0.95i,  1.00+ 0.00i]

EchoLisp[edit]

 
(define (roots-1 n)
(define theta (// (* 2 PI) n))
(for/list ((i n))
(polar 1. (* theta i))))
 
(roots-1 2)
(1+0i -1+0i)
(roots-1 3)
(1+0i -0.4999999999999998+0.8660254037844388i -0.5000000000000004-0.8660254037844384i)
(roots-1 4)
(1+0i 0+i -1+0i 0-i)
 

ERRE[edit]

 
PROGRAM UNITY_ROOTS
 
!
! for rosettacode.org
!
 
BEGIN
PRINT(CHR$(12);) !CLS
N=5  ! this can be changed for any desired n
ANGLE=0  ! start at ANGLE 0
REPEAT
REAL=COS(ANGLE)  ! real axis is the x axis
IF (ABS(REAL)<10^-5) THEN REAL=0 END IF ! get rid of annoying sci notation
IMAG=SIN(ANGLE)  ! imaginary axis is the y axis
IF (ABS(IMAG)<10^-5) THEN IMAG=0 END IF ! get rid of annoying sci notation
PRINT(REAL;"+";IMAG;"i")  ! answer on every line
ANGLE+=(2*π)/N
 ! all the way around the circle at even intervals
UNTIL ANGLE>=2*π
END PROGRAM
 

Note: Adapted from Qbasic version. π is the predefined constant Greek Pi.

Forth[edit]

Complex numbers are not a native type in Forth, so we calculate the roots by hand.

: f0. ( f -- )
fdup 0e 0.001e f~ if fdrop 0e then f. ;
: .roots ( n -- )
dup 1 do
pi i 2* 0 d>f f* dup 0 d>f f/ ( F: radians )
fsincos cr ." real " f0. ." imag " f0.
loop drop ;
 
3 set-precision
5 .roots

Fortran[edit]

Sin/Cos + Scalar Loop[edit]

Works with: Fortran version ISO Fortran 90 and later
PROGRAM Roots
 
COMPLEX :: root
INTEGER :: i, n
REAL :: angle, pi
 
pi = 4.0 * ATAN(1.0)
DO n = 2, 7
angle = 0.0
WRITE(*,"(I1,A)", ADVANCE="NO") n,": "
DO i = 1, n
root = CMPLX(COS(angle), SIN(angle))
WRITE(*,"(SP,2F7.4,A)", ADVANCE="NO") root, "j "
angle = angle + (2.0*pi / REAL(n))
END DO
WRITE(*,*)
END DO
 
END PROGRAM Roots

Output

2: +1.0000+0.0000j  -1.0000+0.0000j   
3: +1.0000+0.0000j  -0.5000+0.8660j  -0.5000-0.8660j   
4: +1.0000+0.0000j  +0.0000+1.0000j  -1.0000+0.0000j  +0.0000-1.0000j   
5: +1.0000+0.0000j  +0.3090+0.9511j  -0.8090+0.5878j  -0.8090-0.5878j  +0.3090-0.9511j   
6: +1.0000+0.0000j  +0.5000+0.8660j  -0.5000+0.8660j  -1.0000+0.0000j  -0.5000-0.8660j  +0.5000-0.8660j 
7: +1.0000+0.0000j  +0.6235+0.7818j  -0.2225+0.9749j  -0.9010+0.4339j  -0.9010-0.4339j  -0.2225-0.9749j  +0.6235-0.7818j

Exp + Array-valued Statement[edit]

Works with: Fortran version ISO Fortran 90 and later
program unity
real, parameter :: pi = 3.141592653589793
complex, parameter :: i = (0, 1)
complex, dimension(0:7-1) :: unit_circle
integer :: n, j
 
do n = 2, 7
!!!! KEY STEP, does all the calculations in one statement !!!!
unit_circle(0:n-1) = exp(2*i*pi/n * (/ (j, j=0, n-1) /) )
 
write(*,"(i1,a)", advance="no") n, ": "
write(*,"(sp,2f7.4,a)", advance="no") (unit_circle(j), "j ", j = 0, n-1)
write(*,*)
end do
end program unity

FunL[edit]

FunL has built-in support for complex numbers. i is predefined to represent the imaginary unit.

import math.{exp, Pi}
 
def rootsOfUnity( n ) = {exp( 2Pi i k/n ) | k <- 0:n}
 
println( rootsOfUnity(3) )
Output:
{1.0, -0.4999999999999998+0.8660254037844387i, -0.5000000000000004-0.8660254037844385i}


FutureBasic[edit]

 
include "ConsoleWindow"
 
dim as long n, root
dim as double real, imag
 
for n = 2 to 7
print n;":" ;
for root = 0 to n-1
real = cos( 2 * pi * root / n)
imag = sin( 2 * pi * root / n)
print using "-##.#####"; real;using "-##.#####"; imag; "i";
if root <> n-1 then print ",";
next
print
next
 

Output:

 2:  1.00000  0.00000i, -1.00000  0.00000i
 3:  1.00000  0.00000i, -0.50000  0.86603i, -0.50000 -0.86603i
 4:  1.00000  0.00000i,  0.00000  1.00000i, -1.00000  0.00000i, -0.00000 -1.00000i
 5:  1.00000  0.00000i,  0.30902  0.95106i, -0.80902  0.58779i, -0.80902 -0.58779i,  0.30902 -0.95106i
 6:  1.00000  0.00000i,  0.50000  0.86603i, -0.50000  0.86603i, -1.00000  0.00000i, -0.50000 -0.86603i,  0.50000 -0.86603i
 7:  1.00000  0.00000i,  0.62349  0.78183i, -0.22252  0.97493i, -0.90097  0.43388i, -0.90097 -0.43388i, -0.22252 -0.97493i,  0.62349 -0.78183i

GAP[edit]

roots := n -> List([0 .. n-1], k -> E(n)^k);
 
r:=roots(7);
# [ 1, E(7), E(7)^2, E(7)^3, E(7)^4, E(7)^5, E(7)^6 ]
 
List(r, x -> x^7);
# [ 1, 1, 1, 1, 1, 1, 1 ]

Go[edit]

package main
 
import (
"fmt"
"math"
"math/cmplx"
)
 
func main() {
for n := 2; n <= 5; n++ {
fmt.Printf("%d roots of 1:\n", n)
for _, r := range roots(n) {
fmt.Printf("  %18.15f\n", r)
}
}
}
 
func roots(n int) []complex128 {
r := make([]complex128, n)
for i := 0; i < n; i++ {
r[i] = cmplx.Rect(1, 2*math.Pi*float64(i)/float64(n))
}
return r
}

Output:

2 roots of 1:
  ( 1.000000000000000+0.000000000000000i)
  (-1.000000000000000+0.000000000000000i)
3 roots of 1:
  ( 1.000000000000000+0.000000000000000i)
  (-0.500000000000000+0.866025403784439i)
  (-0.500000000000000-0.866025403784438i)
4 roots of 1:
  ( 1.000000000000000+0.000000000000000i)
  ( 0.000000000000000+1.000000000000000i)
  (-1.000000000000000+0.000000000000000i)
  (-0.000000000000000-1.000000000000000i)
5 roots of 1:
  ( 1.000000000000000+0.000000000000000i)
  ( 0.309016994374948+0.951056516295154i)
  (-0.809016994374947+0.587785252292473i)
  (-0.809016994374947-0.587785252292473i)
  ( 0.309016994374947-0.951056516295154i)

Groovy[edit]

Because the Groovy language does not provide a built-in facility for complex arithmetic, this example relies on the Complex class defined in the Complex numbers example.

/** The following closure creates a list of n evenly-spaced points around the unit circle,
* useful in FFT calculations, among other things */

def rootsOfUnity = { n ->
(0..<n).collect {
Complex.fromPolar(1, 2 * Math.PI * it / n)
}
}

Test program:

def tol = 0.000000001  // tolerance: acceptable "wrongness" to account for rounding error
 
((1..6) + [16]). each { n ->
println "rootsOfUnity(${n}):"
def rou = rootsOfUnity(n)
rou.each { println it }
assert rou[0] == 1
def actual = n > 1 ? rou[Math.floor(n/2) as int] : rou[0]
def expected = n > 1 ? (n%2 == 0) ? -1 : ~rou[Math.ceil(n/2) as int] : rou[0]
def message = n > 1 ? (n%2 == 0) ? 'middle-most root should be -1' : 'two middle-most roots should be conjugates' : ''
assert (actual - expected).abs() < tol : message
assert rou.every { (it.rho - 1) < tol } : 'all roots should have magnitude 1'
println()
}

Output:

rootsOfUnity(1):
1.0

rootsOfUnity(2):
1.0
-1.0 + 1.2246467991473532E-16i

rootsOfUnity(3):
1.0
-0.4999999998186198 + 0.8660254038891585i
-0.5000000003627604 - 0.8660254035749988i

rootsOfUnity(4):
1.0
6.123233995736766E-17 + i
-1.0 + 1.2246467991473532E-16i
-1.8369701987210297E-16 - i

rootsOfUnity(5):
1.0
0.30901699437494745 + 0.9510565162951535i
-0.8090169943749473 + 0.5877852522924732i
-0.8090169943749475 - 0.587785252292473i
0.30901699437494723 - 0.9510565162951536i

rootsOfUnity(6):
1.0
0.4999999998186201 + 0.8660254038891584i
-0.5000000003627598 + 0.8660254035749991i
-1.0 - 6.283181638240517E-10i
-0.4999999992744804 - 0.8660254042033175i
0.5000000009068993 - 0.8660254032608401i

rootsOfUnity(16):
1.0
0.9238795325112867 + 0.3826834323650898i
0.7071067811865476 + 0.7071067811865475i
0.38268343236508984 + 0.9238795325112867i
6.123233995736766E-17 + i
-0.3826834323650897 + 0.9238795325112867i
-0.7071067811865475 + 0.7071067811865476i
-0.9238795325112867 + 0.3826834323650899i
-1.0 + 1.2246467991473532E-16i
-0.9238795325112868 - 0.38268343236508967i
-0.7071067811865477 - 0.7071067811865475i
-0.38268343236509034 - 0.9238795325112865i
-1.8369701987210297E-16 - i
0.38268343236509 - 0.9238795325112866i
0.7071067811865474 - 0.7071067811865477i
0.9238795325112865 - 0.3826834323650904i

Haskell[edit]

import Data.Complex
 
rootsOfUnity n = [cis (2*pi*k/n) | k <- [0..n-1]]

Output:

*Main> rootsOfUnity 3
[1.0 :+ 0.0,
(-0.4999999999999998) :+ 0.8660254037844387,
(-0.5000000000000004) :+ (-0.8660254037844384)]

Icon and Unicon[edit]

procedure main()
roots(10)
end
 
procedure roots(n)
every n := 2 to 10 do
every writes(n | (str_rep((0 to (n-1)) * 2 * &pi / n)) | "\n")
end
 
procedure str_rep(k)
return " " || cos(k) || "+" || sin(k) || "i"
end

Notes:

IDL[edit]

For some example n:

n = 5
print, exp( dcomplex( 0, 2*!dpi/n) ) ^ ( 1 + indgen(n) )

Outputs:

( 0.30901699, 0.95105652)( -0.80901699, 0.58778525)( -0.80901699, -0.58778525)( 0.30901699, -0.95105652)( 1.0000000, -1.1102230e-16)

J[edit]

   rou=: [: ^ 0j2p1 * i. % ]
 
rou 4
1 0j1 _1 0j_1
 
rou 5
1 0.309017j0.951057 _0.809017j0.587785 _0.809017j_0.587785 0.309017j_0.951057

The computation can also be written as a loop, shown here for comparison only.

rou1=: 3 : 0
z=. 0 $ r=. ^ o. 0j2 % y [ e=. 1
for. i.y do.
z=. z,e
e=. e*r
end.
z
)

Java[edit]

Java doesn't have a nice way of dealing with complex numbers, so the real and imaginary parts are calculated separately based on the angle and printed together. There are also checks in this implementation to get rid of extremely small values (< 1.0E-3 where scientific notation sets in for Doubles). Instead, they are simply represented as 0. To remove those checks (for very high n's), remove both if statements.

import java.util.Locale;
 
public class Test {
 
public static void main(String[] a) {
for (int n = 2; n < 6; n++)
unity(n);
}
 
public static void unity(int n) {
System.out.printf("%n%d: ", n);
 
//all the way around the circle at even intervals
for (double angle = 0; angle < 2 * Math.PI; angle += (2 * Math.PI) / n) {
 
double real = Math.cos(angle); //real axis is the x axis
 
if (Math.abs(real) < 1.0E-3)
real = 0.0; //get rid of annoying sci notation
 
double imag = Math.sin(angle); //imaginary axis is the y axis
 
if (Math.abs(imag) < 1.0E-3)
imag = 0.0;
 
System.out.printf(Locale.US, "(%9f,%9f) ", real, imag);
}
}
}
2: ( 1.000000, 0.000000) (-1.000000, 0.000000) 
3: ( 1.000000, 0.000000) (-0.500000, 0.866025) (-0.500000,-0.866025) 
4: ( 1.000000, 0.000000) ( 0.000000, 1.000000) (-1.000000, 0.000000) ( 0.000000,-1.000000) 
5: ( 1.000000, 0.000000) ( 0.309017, 0.951057) (-0.809017, 0.587785) (-0.809017,-0.587785) ( 0.309017,-0.951057)

JavaScript[edit]

function Root(angle) {
with (Math) { this.r = cos(angle); this.i = sin(angle) }
}
 
Root.prototype.toFixed = function(p) {
return this.r.toFixed(p) + (this.i >= 0 ? '+' : '') + this.i.toFixed(p) + 'i'
}
 
function roots(n) {
var rs = [], teta = 2*Math.PI/n
for (var angle=0, i=0; i<n; angle+=teta, i+=1) rs.push( new Root(angle) )
return rs
}
 
for (var n=2; n<8; n+=1) {
document.write(n, ': ')
var rs=roots(n); for (var i=0; i<rs.length; i+=1) document.write( i ? ', ' : '', rs[i].toFixed(5) )
document.write('<br>')
}
 
Output:
2: 1.00000+0.00000i, -1.00000+0.00000i
3: 1.00000+0.00000i, -0.50000+0.86603i, -0.50000-0.86603i
4: 1.00000+0.00000i, 0.00000+1.00000i, -1.00000+0.00000i, -0.00000-1.00000i
5: 1.00000+0.00000i, 0.30902+0.95106i, -0.80902+0.58779i, -0.80902-0.58779i, 0.30902-0.95106i
6: 1.00000+0.00000i, 0.50000+0.86603i, -0.50000+0.86603i, -1.00000+0.00000i, -0.50000-0.86603i, 0.50000-0.86603i
7: 1.00000+0.00000i, 0.62349+0.78183i, -0.22252+0.97493i, -0.90097+0.43388i, -0.90097-0.43388i, -0.22252-0.97493i, 0.62349-0.78183i

jq[edit]

Using the same example as in the Julia section, and representing x + i*y as [x,y]:

def nthroots(n):
(8 * (1|atan)) as $twopi
| range(0;n) | (($twopi * .) / n) as $angle | [ ($angle | cos), ($angle | sin) ];
 
nthroots(10)
$ uname -a
Darwin Mac-mini 13.3.0 Darwin Kernel Version 13.3.0: Tue Jun 3 21:27:35 PDT 2014; root:xnu-2422.110.17~1/RELEASE_X86_64 x86_64
 
$ time jq -c -n -f Roots_of_unity.jq
[1,0]
[0.8090169943749475,0.5877852522924731]
[0.30901699437494745,0.9510565162951535]
[-0.30901699437494734,0.9510565162951536]
[-0.8090169943749473,0.5877852522924732]
[-1,1.2246467991473532e-16]
[-0.8090169943749475,-0.587785252292473]
[-0.30901699437494756,-0.9510565162951535]
[0.30901699437494723,-0.9510565162951536]
[0.8090169943749473,-0.5877852522924732]
 
real 0m0.015s
user 0m0.004s
sys 0m0.004s
 

Julia[edit]

nthroots(n::Integer) = [ cospi(2k/n)+sinpi(2k/n)im for k = 0:n-1 ]

(One could also use complex exponentials or other formulations.) For example, `nthroots(10)` gives:

10-element Array{Complex{Float64},1}:
            1.0+0.0im
  0.809017+0.587785im
  0.309017+0.951057im
 -0.309017+0.951057im
 -0.809017+0.587785im
           -1.0+0.0im
 -0.809017-0.587785im
 -0.309017-0.951057im
  0.309017-0.951057im
  0.809017-0.587785im

Kotlin[edit]

import java.lang.Math.*
 
data class Complex(val r: Double, val i: Double) {
override fun toString() = when {
i == 0.0 -> r.toString()
r == 0.0 -> i.toString() + 'i'
else -> "$r + ${i}i"
}
}
 
fun unity_roots(n: Number) = (1..n.toInt() - 1).map {
val a = it * 2 * PI / n.toDouble()
var r = cos(a); if (abs(r) < 1e-6) r = 0.0
var i = sin(a); if (abs(i) < 1e-6) i = 0.0
Complex(r, i)
}
 
fun main(args: Array<String>) {
(1..4).forEach { println(listOf(1) + unity_roots(it)) }
println(listOf(1) + unity_roots(5.0))
}
Output:
[1]
[1, -1.0]
[1, -0.4999999999999998 + 0.8660254037844387i, -0.5000000000000004 + -0.8660254037844385i]
[1, 1.0i, -1.0, -1.0i]
[1, 0.30901699437494745 + 0.9510565162951535i, -0.8090169943749473 + 0.5877852522924732i, -0.8090169943749475 + -0.587785252292473i, 0.30901699437494723 + -0.9510565162951536i]

Liberty BASIC[edit]

WindowWidth  =400
WindowHeight =400
 
'nomainwin
 
open "N'th Roots of One" for graphics_nsb_nf as #w
 
#w "trapclose [quit]"
 
for n =1 To 10
angle =0
#w "font arial 16 bold"
print n; "th roots."
#w "cls"
#w "size 1 ; goto 200 200 ; down ; color lightgray ; circle 150 ; size 10 ; set 200 200 ; size 2"
#w "up ; goto 200 0 ; down ; goto 200 400 ; up ; goto 0 200 ; down ; goto 400 200"
#w "up ; goto 40 20 ; down ; color black"
#w "font arial 6"
#w "\"; n; " roots of 1."
 
for i = 1 To n
x = cos( Radian( angle))
y = sin( Radian( angle))
 
print using( "##", i); ": ( " + using( "##.######", x);_
" +i *" +using( "##.######", y); ") or e^( i *"; i -1; " *2 *Pi/ "; n; ")"
 
#w "color "; 255 *i /n; " 0 "; 256 -255 *i /n
#w "up ; goto 200 200"
#w "down ; goto "; 200 +150 *x; " "; 200 -150 *y
#w "up  ; goto "; 200 +165 *x; " "; 200 -165 *y
#w "\"; str$( i)
#w "up"
 
angle =angle +360 /n
 
next i
 
timer 500, [on]
wait
[on]
timer 0
next n
 
wait
 
[quit]
close #w
 
end
 
function Radian( theta)
Radian =theta *3.1415926535 /180
end function

Lua[edit]

Complex numbers from the Lua implementation on the complex numbers page.

--defines addition, subtraction, negation, multiplication, division, conjugation, norms, and a conversion to strgs.
complex = setmetatable({
__add = function(u, v) return complex(u.real + v.real, u.imag + v.imag) end,
__sub = function(u, v) return complex(u.real - v.real, u.imag - v.imag) end,
__mul = function(u, v) return complex(u.real * v.real - u.imag * v.imag, u.real * v.imag + u.imag * v.real) end,
__div = function(u, v) return u * complex(v.real / v.norm, -v.imag / v.norm) end,
__unm = function(u) return complex(-u.real, -u.imag) end,
__concat = function(u, v)
if type(u) == "table" then return u.real .. " + " .. u.imag .. "i" .. v
elseif type(u) == "string" or type(u) == "number" then return u .. v.real .. " + " .. v.imag .. "i"
end end,
__index = function(u, index)
local operations = {
norm = function(u) return u.real ^ 2 + u.imag ^ 2 end,
conj = function(u) return complex(u.real, -u.imag) end,
}
return operations[index] and operations[index](u)
end,
__newindex = function() error() end
}, {
__call = function(z, realpart, imagpart) return setmetatable({real = realpart, imag = imagpart}, complex) end
} )
n = io.read() + 0
val = complex(math.cos(2*math.pi / n), math.sin(2*math.pi / n))
root = complex(1, 0)
for i = 1, n do
root = root * val
print(root .. "")
end

Maple[edit]

RootsOfUnity := proc( n )
solve(z^n = 1, z);
end proc:
for i from 2 to 6 do
printf( "%d: %a\n", i, [ RootsOfUnity(i) ] );
end do;

Output:

2: [1, -1]
3: [1, -1/2-1/2*I*3^(1/2), -1/2+1/2*I*3^(1/2)]
4: [1, -1, I, -I]
5: [1, 1/4*5^(1/2)-1/4+1/4*I*2^(1/2)*(5+5^(1/2))^(1/2), -1/4*5^(1/2)-1/4+1/4*I*2^(1/2)*(5-5^(1/2))^(1/2), -1/4*5^(1/2)-1/4-1/4*I*2^(1/2)*(5-5^(1/2))^(1/2), 1/4*5^(1/2)-1/4-1/4*I*2^(1/2)*(5+5^(1/2))^(1/2)]
6: [1, -1, 1/2*(-2-2*I*3^(1/2))^(1/2), -1/2*(-2-2*I*3^(1/2))^(1/2), 1/2*(-2+2*I*3^(1/2))^(1/2), -1/2*(-2+2*I*3^(1/2))^(1/2)]

Mathematica[edit]

Setting this up in Mathematica is easy, because it already handles complex numbers:

RootsUnity[nthroot_Integer?Positive] := Table[Exp[2 Pi I i/nthroot], {i, 0, nthroot - 1}]

Note that Mathematica will keep the expression as exact as possible. Simplifications can be made to more known (trigonometric) functions by using the function ExpToTrig. If only a numerical approximation is necessary the function N will transform the exact result to a numerical approximation. Examples (exact not simplified, exact simplified, approximated):

RootsUnity[2]
RootsUnity[3]
RootsUnity[4]
RootsUnity[5]

RootsUnity[2]//ExpToTrig
RootsUnity[3]//ExpToTrig
RootsUnity[4]//ExpToTrig
RootsUnity[5]//ExpToTrig

RootsUnity[2]//N
RootsUnity[3]//N
RootsUnity[4]//N
RootsUnity[5]//N

gives back:



MATLAB[edit]

function z = rootsOfUnity(n)
 
assert(n >= 1,'n >= 1');
z = roots([1 zeros(1,n-1) -1]);
 
end

Sample Output:

>> rootsOfUnity(3)
 
ans =
 
-0.500000000000000 + 0.866025403784439i
-0.500000000000000 - 0.866025403784439i
1.000000000000000

Maxima[edit]

solve(1 = x^n, x)

Demonstration:

for n:1 thru 5 do display(solve(1 = x^n, x));

Output:

solve(1 = x, x) = [x = 1]
solve(1 = x^2, x) = [x = -1, x = 1]
solve(1 = x^3, x) = [x = (sqrt(3)*%i-1)/2, x = -(sqrt(3)*%i+1)/2, x = 1]
solve(1 = x^4, x) = [x = %i, x = -1, x = -%i, x = 1]
solve(1 = x^5, x) = [x = %e^((2*%i*%pi)/5), x = %e^((4*%i*%pi)/5), x = %e^(-(4*%i*%pi)/5), x = %e^(-(2*%i*%pi)/5), x = 1]

МК-61/52[edit]

П0	0	П1	ИП1	sin	ИП1	cos	С/П	2	пи
* ИП0 / ИП1 + П1 БП 03

Nim[edit]

Translation of: Python
import complex, math
 
proc rect(r, phi: float): Complex = (r * cos(phi), sin(phi))
 
proc croots(n): seq[Complex] =
result = @[]
if n <= 0: return
for k in 0 .. < n:
result.add rect(1, 2 * k.float * Pi / n.float)
 
for nr in 2..10:
echo nr, " ", croots(nr)

Output:

2 @[(1.0, 0.0), (-1.0, 1.224646799147353e-16)]
3 @[(1.0, 0.0), (-0.4999999999999998, 0.8660254037844387), (-0.5000000000000004, -0.8660254037844384)]
4 @[(1.0, 0.0), (6.123233995736766e-17, 1.0), (-1.0, 1.224646799147353e-16), (-1.83697019872103e-16, -1.0)]
5 @[(1.0, 0.0), (0.3090169943749475, 0.9510565162951535), (-0.8090169943749473, 0.5877852522924732), (-0.8090169943749476, -0.587785252292473), (0.3090169943749472, -0.9510565162951536)]
6 @[(1.0, 0.0), (0.5000000000000001, 0.8660254037844386), (-0.4999999999999998, 0.8660254037844387), (-1.0, 1.224646799147353e-16), (-0.5000000000000004, -0.8660254037844384), (0.5000000000000001, -0.8660254037844386)]
7 @[(1.0, 0.0), (0.6234898018587336, 0.7818314824680298), (-0.2225209339563143, 0.9749279121818236), (-0.900968867902419, 0.4338837391175582), (-0.9009688679024191, -0.433883739117558), (-0.2225209339563146, -0.9749279121818236), (0.6234898018587334, -0.7818314824680299)]
8 @[(1.0, 0.0), (0.7071067811865476, 0.7071067811865475), (6.123233995736766e-17, 1.0), (-0.7071067811865475, 0.7071067811865476), (-1.0, 1.224646799147353e-16), (-0.7071067811865477, -0.7071067811865475), (-1.83697019872103e-16, -1.0), (0.7071067811865474, -0.7071067811865477)]
9 @[(1.0, 0.0), (0.766044443118978, 0.6427876096865393), (0.1736481776669304, 0.984807753012208), (-0.4999999999999998, 0.8660254037844387), (-0.9396926207859083, 0.3420201433256689), (-0.9396926207859084, -0.3420201433256687), (-0.5000000000000004, -0.8660254037844384), (0.17364817766693, -0.9848077530122081), (0.7660444431189778, -0.6427876096865396)]
10 @[(1.0, 0.0), (0.8090169943749475, 0.5877852522924731), (0.3090169943749475, 0.9510565162951535), (-0.3090169943749473, 0.9510565162951536), (-0.8090169943749473, 0.5877852522924732), (-1.0, 1.224646799147353e-16), (-0.8090169943749476, -0.587785252292473), (-0.3090169943749476, -0.9510565162951535), (0.3090169943749472, -0.9510565162951536), (0.8090169943749473, -0.5877852522924734)]

OCaml[edit]

open Complex
 
let pi = 4. *. atan 1.
 
let () =
for n = 1 to 10 do
Printf.printf "%2d " n;
for k = 1 to n do
let ret = polar 1. (2. *. pi *. float_of_int k /. float_of_int n) in
Printf.printf "(%f + %f i)" ret.re ret.im
done;
print_newline ()
done

Octave[edit]

for j = 2 : 10
printf("*** %d\n", j);
for n = 1 : j
disp(exp(2i*pi*n/j));
endfor
disp("");
endfor

OoRexx[edit]

Translation of: REXX
/*REXX program computes the  K  roots of unity  (which include complex roots).*/
parse Version v
Say v
parse arg n frac . /*get optional arguments from the C.L. */
if n=='' then n=1 /*Not specified? Then use the default.*/
if frac='' then frac=5 /* " " " " " " */
start=abs(n) /*assume only one K is wanted. */
if n<0 then start=1 /*Negative? Then use a range of K's. */
/*display unity roots for a range, or */
do k=start to abs(n) /* just for one K. */
say right(k 'roots of unity',40,"-") /*display a pretty separator with title*/
do angle=0 by 360/k for k /*compute the angle for each root. */
rp=adjust(rxCalcCos(angle,,'D')) /*compute real part via COS function.*/
if left(rp,1)\=='-' then rp=" "rp /*not negative? Then pad with a blank.*/
ip=adjust(rxCalcSin(angle,,'D')) /*compute imaginary part via SIN funct.*/
if left(ip,1)\=='-' then ip="+"ip /*Not negative? Then pad with + char.*/
if ip=0 then say rp /*Only real part? Ignore imaginary part*/
else say left(rp,frac+4)ip'i' /*show the real & imaginary part*/
end /*angle*/
end /*k*/
exit /*stick a fork in it, we're all done. */
/*----------------------------------------------------------------------------*/
adjust: parse arg x; near0='1e-' || (digits()-digits()%10) /*compute small #*/
if abs(x)<near0 then x=0 /*if near zero, then assume zero.*/
return format(x,,frac)/1 /*fraction digits past dec point.*/
::requires rxMath library
Output:
D:\>rexx nrootoo 5
REXX-ooRexx_4.2.0(MT)_64-bit 6.04 22 Feb 2014
------------------------5 roots of unity
 1
 0.30902 +0.95106i
-0.80902 +0.58779i
-0.80902 -0.58779i
 0.30902 -0.95106i

PARI/GP[edit]

vector(n,k,exp(2*Pi*I*k/n))

sqrtn() can give the first n'th root, from which the others by multiplying or powering.

nth_roots(n) = my(z);sqrtn(1,n,&z); vector(n,i, z^i);

Both the above give floating point complex numbers even when a root could be exact, like -1 or fourth root I.

quadgen() can be used for an exact 6th root. (Quads cannot be mixed with ordinary complex numbers, and they always print as w.)

sixth_root = quadgen(-3);   /* 6th root of unity, exact */
vector(6,n, sixth_root^n) /* all the 6'th roots */

Pascal[edit]

Translation of: Fortran
Program Roots;
 
var
root: record // poor man's complex type.
r: real;
i: real;
end;
i, n: integer;
angle: real;
 
begin
for n := 2 to 7 do
begin
angle := 0.0;
write(n, ': ');
for i := 1 to n do
begin
root.r := cos(angle);
root.i := sin(angle);
write(root.r:8:5, root.i:8:5, 'i ');
angle := angle + (2.0 * pi / n);
end;
writeln;
end;
end.

Output:

2:  1.00000 0.00000i -1.00000 0.00000i 
3:  1.00000 0.00000i -0.50000 0.86603i -0.50000-0.86603i 
4:  1.00000 0.00000i  0.00000 1.00000i -1.00000 0.00000i -0.00000-1.00000i 
5:  1.00000 0.00000i  0.30902 0.95106i -0.80902 0.58779i -0.80902-0.58779i  0.30902-0.95106i 
6:  1.00000 0.00000i  0.50000 0.86603i -0.50000 0.86603i -1.00000-0.00000i -0.50000-0.86603i  0.50000-0.86603i 
7:  1.00000 0.00000i  0.62349 0.78183i -0.22252 0.97493i -0.90097 0.43388i -0.90097-0.43388i -0.22252-0.97493i  0.62349-0.78183i 

Perl[edit]

Works with: Perl version 5.6.0

The root() function returns a list of the N many N'th roots of any complex Z, in this case 1.

use Math::Complex;
 
foreach my $n (2 .. 10) {
printf "%2d", $n;
my @roots = root(1,$n);
foreach my $root (@roots) {
$root->display_format(style => 'cartesian', format => '%.3f');
print " $root";
}
print "\n";
}

Output:

 2 1.000 -1.000+0.000i
 3 1.000 -0.500+0.866i -0.500-0.866i
 4 1.000 0.000+1.000i -1.000+0.000i -0.000-1.000i
 5 1.000 0.309+0.951i -0.809+0.588i -0.809-0.588i 0.309-0.951i
 6 1.000 0.500+0.866i -0.500+0.866i -1.000+0.000i -0.500-0.866i 0.500-0.866i
 7 1.000 0.623+0.782i -0.223+0.975i -0.901+0.434i -0.901-0.434i -0.223-0.975i 0.623-0.782i
 8 1.000 0.707+0.707i 0.000+1.000i -0.707+0.707i -1.000+0.000i -0.707-0.707i -0.000-1.000i 0.707-0.707i
 9 1.000 0.766+0.643i 0.174+0.985i -0.500+0.866i -0.940+0.342i -0.940-0.342i -0.500-0.866i 0.174-0.985i 0.766-0.643i
10 1.000 0.809+0.588i 0.309+0.951i -0.309+0.951i -0.809+0.588i -1.000+0.000i -0.809-0.588i -0.309-0.951i 0.309-0.951i 0.809-0.588i

Perl 6[edit]

Perl 6 has a built-in function cis which returns a unitary complex number given its phase. Perl 6 also defines the tau = 2*pi constant. Thus the k-th n-root of unity can simply be written cis(k*τ/n).

constant n = 10;
for ^n -> \k {
say cis(k*τ/n);
}
Output:
1+0i
0.809016994374947+0.587785252292473i
0.309016994374947+0.951056516295154i
-0.309016994374947+0.951056516295154i
-0.809016994374947+0.587785252292473i
-1+1.22464679914735e-16i
-0.809016994374948-0.587785252292473i
-0.309016994374948-0.951056516295154i
0.309016994374947-0.951056516295154i
0.809016994374947-0.587785252292473i

PL/I[edit]

complex_roots:
procedure (N);
declare N fixed binary nonassignable;
declare x float, c fixed decimal (10,8) complex;
declare twopi float initial ((4*asin(1.0)));
 
do x = 0 to twopi by twopi/N;
c = complex(cos(x), sin(x));
put skip list (c);
end;
end complex_roots;
 
1.00000000+0.00000000I
0.80901700+0.58778524I
0.30901697+0.95105654I
-0.30901703+0.95105648I
-0.80901706+0.58778518I
-1.00000000-0.00000008I
-0.80901694-0.58778536I
-0.30901709-0.95105648I
0.30901712-0.95105648I
0.80901724-0.58778494I

PicoLisp[edit]

Translation of: C
(load "@lib/math.l")
 
(for N (range 2 10)
(let Angle 0.0
(prin N ": ")
(for I N
(let Ipart (sin Angle)
(prin
(round (cos Angle) 4)
(if (lt0 Ipart) "-" "+")
"j"
(round (abs Ipart) 4)
" " ) )
(inc 'Angle (*/ 2 pi N)) )
(prinl) ) )

PureBasic[edit]

OpenConsole()
For n = 2 To 10
angle = 0
PrintN(Str(n))
For i = 1 To n
x.f = Cos(Radian(angle))
y.f = Sin(Radian(angle))
PrintN( Str(i) + ": " + StrF(x, 6) + " / " + StrF(y, 6))
angle = angle + (360 / n)
Next
Next
Input()

Python[edit]

Works with: Python version 2.6+
import cmath
class Complex(complex):
def __repr__(self):
rp = '%7.5f'%self.real if not self.pureImag() else ''
ip = '%7.5fj'%self.imag if not self.pureReal() else ''
conj = '' if (self.pureImag() or self.pureReal() or self.imag<0.0) else '+'
return '0.0' if (self.pureImag() and self.pureReal()) else rp+conj+ip
def pureImag(self):
return abs( self.real) < 0.000005
def pureReal(self):
return abs( self.imag) < 0.000005
 
def croots(n):
if n<=0:
return None
return (Complex(cmath.rect(1, 2*k*cmath.pi/n)) for k in range(n))
# in pre-Python 2.6, return (Complex(cmath.exp(2j*k*cmath.pi/n)) for k in range(n))
 
for nr in range(2,11):
print nr, list(croots(nr))

Output:

2 [1.00000, -1.00000]
3 [1.00000, -0.50000+0.86603j, -0.50000-0.86603j]
4 [1.00000, 1.00000j, -1.00000, -1.00000j]
5 [1.00000, 0.30902+0.95106j, -0.80902+0.58779j, -0.80902-0.58779j, 0.30902-0.95106j]
6 [1.00000, 0.50000+0.86603j, -0.50000+0.86603j, -1.00000, -0.50000-0.86603j, 0.50000-0.86603j]
7 [1.00000, 0.62349+0.78183j, -0.22252+0.97493j, -0.90097+0.43388j, -0.90097-0.43388j, -0.22252-0.97493j, 0.62349-0.78183j]
8 [1.00000, 0.70711+0.70711j, 1.00000j, -0.70711+0.70711j, -1.00000, -0.70711-0.70711j, -1.00000j, 0.70711-0.70711j]
9 [1.00000, 0.76604+0.64279j, 0.17365+0.98481j, -0.50000+0.86603j, -0.93969+0.34202j, -0.93969-0.34202j, -0.50000-0.86603j, 0.17365-0.98481j, 0.76604-0.64279j]
10 [1.00000, 0.80902+0.58779j, 0.30902+0.95106j, -0.30902+0.95106j, -0.80902+0.58779j, -1.00000, -0.80902-0.58779j, -0.30902-0.95106j, 0.30902-0.95106j, 0.80902-0.58779j]

R[edit]

for(j in 2:10) {
r <- sprintf("%d: ", j)
for(n in 1:j) {
r <- paste(r, format(exp(2i*pi*n/j), digits=4), ifelse(n<j, ",", ""))
}
print(r)
}

Output:

[1] "2:  -1+0i , 1-0i "
[1] "3:  -0.5+0.866i , -0.5-0.866i , 1-0i "
[1] "4:  0+1i , -1+0i , 0-1i , 1-0i "
[1] "5:  0.309+0.9511i , -0.809+0.5878i , -0.809-0.5878i , 0.309-0.9511i , 1-0i "
[1] "6:  0.5+0.866i , -0.5+0.866i , -1+0i , -0.5-0.866i , 0.5-0.866i , 1-0i "
[1] "7:  0.6235+0.7818i , -0.2225+0.9749i , -0.901+0.4339i , -0.901-0.4339i , -0.2225-0.9749i , 0.6235-0.7818i , 1-0i "
[1] "8:  0.7071+0.7071i , 0+1i , -0.7071+0.7071i , -1+0i , -0.7071-0.7071i , 0-1i , 0.7071-0.7071i , 1-0i "
[1] "9:  0.766+0.6428i , 0.1736+0.9848i , -0.5+0.866i , -0.9397+0.342i , -0.9397-0.342i , -0.5-0.866i , 0.1736-0.9848i , 0.766-0.6428i , 1-0i "
[1] "10:  0.809+0.5878i , 0.309+0.9511i , -0.309+0.9511i , -0.809+0.5878i , -1+0i , -0.809-0.5878i , -0.309-0.9511i , 0.309-0.9511i , 0.809-0.5878i , 1-0i "

Racket[edit]

#lang racket
 
(define (roots-of-unity n)
(for/list ([k n])
(make-polar 1 (* k (/ (* 2 pi) n)))))

Will produce a list of roots, for example:

> (for ([r (roots-of-unity 3)]) (displayln r))
1
-0.4999999999999998+0.8660254037844388i
-0.5000000000000004-0.8660254037844384i

REXX[edit]

REXX doesn't have complex arithmetic, so the (real) values of   cos   and   sin   of multiples of   2 pi   radians (divided by K) are used.

Also, REXX doesn't have the   pi   constant defined, nor a   sin   or   cos   function, so they are included below within the REXX program.

Note:   this REXX version only shows   5   significant digits past the decimal point,   but this can be overridden by specifying the 2nd argument when invoking the REXX program.   (See the value of the REXX variable   frac,   4th line).

/*REXX program computes the  K  roots of  unity  (which usually includes complex roots).*/
parse arg n frac . /*get optional arguments from the C.L. */
if n=='' | n=="," then n=1 /*Not specified? Then use the default.*/
if frac='' | frac=="," then frac=5 /* " " " " " " */
start=abs(n) /*assume only one K is wanted. */
if n<0 then start=1 /*Negative? Then use a range of K's. */
numeric digits length(pi()) - 1 /*use number of decimal digits in pi. */
pi2=pi*2 /*obtain the value of pi doubled. */
/*display unity roots for a range, or */
do k=start to abs(n) /* just for one K. */
say right(k 'roots of unity', 40, "─") /*display a pretty separator with title*/
do angle=0 by pi2/k for k /*compute the angle for each root. */
rp=adjust(cos(angle)) /*compute real part via COS function.*/
if left(rp,1)\=='-' then rp=" "rp /*not negative? Then pad with a blank.*/
ip=adjust(sin(angle)) /*compute imaginary part via SIN funct.*/
if left(ip,1)\=='-' then ip="+"ip /*Not negative? Then pad with + char.*/
if ip=0 then say rp /*Only real part? Ignore imaginary part*/
else say left(rp,frac+4)ip'i' /*display the real and imaginary part. */
end /*angle*/
end /*k*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
adjust: parse arg x; near0='1e-' || (digits()-digits()%10) /*compute a tiny number.*/
if abs(x)<near0 then x=0 /*if it's near zero, then assume zero.*/
return format(x,,frac)/1 /*fraction digits past decimal point. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi: pi=3.141592653589793238462643383279502884197169399375105820974944592307816; return pi
/*──────────────────────────────────────────────────────────────────────────────────────*/
r2r: return arg(1) // (pi()*2) /*reduce #radians: -2pi──� +2pi radians*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure; parse arg x; x=r2r(x); a=abs(x); numeric fuzz min(9, digits()-9)
if a=pi/3 then return .5; if a=pi/2|a=pi*2 then return 0
if a=pi then return -1; if a=pi*2/3 then return -.5; return .sincos(1,1,-1)
/*──────────────────────────────────────────────────────────────────────────────────────*/
sin: procedure; parse arg x; x=r2r(x); numeric fuzz min(5,digits()-3)
if abs(x)=pi then return 0; return .sincos(x,x, 1)
/*──────────────────────────────────────────────────────────────────────────────────────*/
.sincos: parse arg z,_,i; x=x*x; p=z
do k=2 by 2; _=-_*x/(k*(k+i)); z=z+_; if z=p then leave; p=z; end
return z

output   when the input is:   5

────────────────────────5 roots of unity
 1
 0.30902 +0.95106i
-0.80902 +0.58779i
-0.80902 -0.58779i
 0.30902 -0.95106i

output   when the input is:   10   35

───────────────────────10 roots of unity
 1
 0.80901699437494742410229341718281906 +0.58778525229247312916870595463907277i
 0.30901699437494742410229341718281906 +0.95105651629515357211643933337938214i
-0.30901699437494742410229341718281906 +0.95105651629515357211643933337938214i
-0.80901699437494742410229341718281906 +0.58778525229247312916870595463907277i
-1
-0.80901699437494742410229341718281906 -0.58778525229247312916870595463907277i
-0.30901699437494742410229341718281906 -0.95105651629515357211643933337938214i
 0.30901699437494742410229341718281906 -0.95105651629515357211643933337938214i
 0.80901699437494742410229341718281906 -0.58778525229247312916870595463907277i

output   when the input is:   -12

────────────────────────1 roots of unity
 1
────────────────────────2 roots of unity
 1
-1
────────────────────────3 roots of unity
 1
-0.5     +0.86603i
-0.5     -0.86603i
────────────────────────4 roots of unity
 1
 0       +1i
-1
 0       -1i
────────────────────────5 roots of unity
 1
 0.30902 +0.95106i
-0.80902 +0.58779i
-0.80902 -0.58779i
 0.30902 -0.95106i
────────────────────────6 roots of unity
 1
 0.5     +0.86603i
-0.5     +0.86603i
-1
-0.5     -0.86603i
 0.5     -0.86603i
────────────────────────7 roots of unity
 1
 0.62349 +0.78183i
-0.22252 +0.97493i
-0.90097 +0.43388i
-0.90097 -0.43388i
-0.22252 -0.97493i
 0.62349 -0.78183i
────────────────────────8 roots of unity
 1
 0.70711 +0.70711i
 0       +1i
-0.70711 +0.70711i
-1
-0.70711 -0.70711i
 0       -1i
 0.70711 -0.70711i
────────────────────────9 roots of unity
 1
 0.76604 +0.64279i
 0.17365 +0.98481i
-0.5     +0.86603i
-0.93969 +0.34202i
-0.93969 -0.34202i
-0.5     -0.86603i
 0.17365 -0.98481i
 0.76604 -0.64279i
───────────────────────10 roots of unity
 1
 0.80902 +0.58779i
 0.30902 +0.95106i
-0.30902 +0.95106i
-0.80902 +0.58779i
-1
-0.80902 -0.58779i
-0.30902 -0.95106i
 0.30902 -0.95106i
 0.80902 -0.58779i
───────────────────────11 roots of unity
 1
 0.84125 +0.54064i
 0.41542 +0.90963i
-0.14231 +0.98982i
-0.65486 +0.75575i
-0.95949 +0.28173i
-0.95949 -0.28173i
-0.65486 -0.75575i
-0.14231 -0.98982i
 0.41542 -0.90963i
 0.84125 -0.54064i
───────────────────────12 roots of unity
 1
 0.86603 +0.5i
 0.5     +0.86603i
 0       +1i
-0.5     +0.86603i
-0.86603 +0.5i
-1
-0.86603 -0.5i
-0.5     -0.86603i
 0       -1i
 0.5     -0.86603i
 0.86603 -0.5i

Ring[edit]

 
decimals(4)
for n = 2 to 5
see string(n) + " : "
for root = 0 to n-1
real = cos(2*3.14 * root / n)
imag = sin(2*3.14 * root / n)
see "" + real + " " + imag + "i"
if root != n-1 see ", " ok
next
see nl
next
 

RLaB[edit]

RLaB can find the n-roots of unity by solving the polynomial equation

It uses the solver polyroots. Interested user is recommended to check the rlabplus manual for details on the solver and the parameters that tune the solver performance.

// specify polynomial
>> n = 10;
>> a = zeros(1,n+1); a[1] = 1; a[n+1] = -1;
>> polyroots(a)
radius roots success
>> polyroots(a).roots
-0.309016994 + 0.951056516i
-0.809016994 + 0.587785252i
-1 + 5.95570041e-23i
-0.809016994 - 0.587785252i
-0.309016994 - 0.951056516i
0.309016994 - 0.951056516i
0.809016994 - 0.587785252i
1 + 0i
0.809016994 + 0.587785252i
0.309016994 + 0.951056516i

Ruby[edit]

def roots_of_unity(n)
(0...n).map {|k| Complex.polar(1, 2 * Math::PI * k / n)}
end
 
p roots_of_unity(3)
Output:
 [(1+0.0i), (-0.4999999999999998+0.8660254037844387i), (-0.5000000000000004-0.8660254037844384i)]

Run BASIC[edit]

PI = 3.1415926535
FOR n = 2 TO 5
PRINT n;":" ;
FOR root = 0 TO n-1
real = COS(2*PI * root / n)
imag = SIN(2*PI * root / n)
PRINT using("-##.#####",real);using("-##.#####",imag);"i";
IF root <> n-1 then PRINT "," ;
NEXT
PRINT
NEXT
 

Output:

2:   1.00000   0.00000i, -1.00000   0.00000i
3:   1.00000   0.00000i, -0.50000   0.86603i, -0.50000  -0.86603i
4:   1.00000   0.00000i,  0.00000   1.00000i, -1.00000   0.00000i,  0.00000  -1.00000i
5:   1.00000   0.00000i,  0.30902   0.95106i, -0.80902   0.58779i, -0.80902  -0.58779i,  0.30902  -0.95106i

Scala[edit]

Using Complex class from task Arithmetic/Complex.

def rootsOfUnity(n:Int)=for(k <- 0 until n) yield Complex.fromPolar(1.0, 2*math.Pi*k/n)

Usage:

rootsOfUnity(3) foreach println

1.0+0.0i
-0.4999999999999998+0.8660254037844387i
-0.5000000000000004-0.8660254037844385i

Seed7[edit]

$ include "seed7_05.s7i";
include "float.s7i";
include "complex.s7i";
 
const proc: main is func
local
var integer: n is 0;
var integer: k is 0;
begin
for n range 2 to 10 do
write(n lpad 2 <& ": ");
for k range 0 to pred(n) do
write(polar(1.0, 2.0 * PI * flt(k) / flt(n)) digits 4 lpad 15 <& " ");
end for;
writeln;
end for;
end func;

Output:

2:  1.0000+0.0000i -1.0000+0.0000i
3: 1.0000+0.0000i -0.5000+0.8660i -0.5000-0.8660i
4: 1.0000+0.0000i 0.0000+1.0000i -1.0000+0.0000i 0.0000-1.0000i
5: 1.0000+0.0000i 0.3090+0.9511i -0.8090+0.5878i -0.8090-0.5878i 0.3090-0.9511i
6: 1.0000+0.0000i 0.5000+0.8660i -0.5000+0.8660i -1.0000+0.0000i -0.5000-0.8660i 0.5000-0.8660i
7: 1.0000+0.0000i 0.6235+0.7818i -0.2225+0.9749i -0.9010+0.4339i -0.9010-0.4339i -0.2225-0.9749i 0.6235-0.7818i
8: 1.0000+0.0000i 0.7071+0.7071i 0.0000+1.0000i -0.7071+0.7071i -1.0000+0.0000i -0.7071-0.7071i 0.0000-1.0000i 0.7071-0.7071i
9: 1.0000+0.0000i 0.7660+0.6428i 0.1736+0.9848i -0.5000+0.8660i -0.9397+0.3420i -0.9397-0.3420i -0.5000-0.8660i 0.1736-0.9848i 0.7660-0.6428i
10: 1.0000+0.0000i 0.8090+0.5878i 0.3090+0.9511i -0.3090+0.9511i -0.8090+0.5878i -1.0000+0.0000i -0.8090-0.5878i -0.3090-0.9511i 0.3090-0.9511i 0.8090-0.5878i

Scheme[edit]

(define pi (* 4 (atan 1)))
 
(do ((n 2 (+ n 1)))
((> n 10))
(display n)
(do ((k 0 (+ k 1)))
((>= k n))
(display " ")
(display (make-polar 1 (* 2 pi (/ k n)))))
(newline))

Sidef[edit]

Translation of: Perl 6
func roots_of_unity (n) {
n.of { |j|
exp(2.i * Complex.pi / n * (j-1))
}
}
 
roots_of_unity(5).each { |c|
printf("%+.5f%+.5fi\n", c.reals);
}
Output:
+1.00000+0.00000i
+0.30902+0.95106i
-0.80902+0.58779i
-0.80902-0.58779i
+0.30902-0.95106i
Translation of: Perl
5.times { |n|
var roots = Complex(1).roots(n);
printf ("%2d. ", n);
say roots.map{ "%+.5f%+.5fi" % .reals }.join(' ');
}
Output:
 1. +1.00000+0.00000i
 2. +1.00000+0.00000i -1.00000+0.00000i
 3. +1.00000+0.00000i -0.50000+0.86603i -0.50000-0.86603i
 4. +1.00000+0.00000i +0.00000+1.00000i -1.00000+0.00000i -0.00000-1.00000i
 5. +1.00000+0.00000i +0.30902+0.95106i -0.80902+0.58779i -0.80902-0.58779i +0.30902-0.95106i

Sparkling[edit]

function unity_roots(n) {
// nth-root(1) = cos(2 * k * pi / n) + i * sin(2 * k * pi / n)
return map(range(n), function(idx, k) {
return {
"re": cos(2 * k * M_PI / n),
"im": sin(2 * k * M_PI / n)
};
});
}
 
// pirnt 6th roots of unity
foreach(unity_roots(6), function(k, v) {
printf("%.3f%+.3fi\n", v.re, v.im);
});

Tcl[edit]

package require Tcl 8.5
namespace import tcl::mathfunc::*
 
set pi 3.14159265
for {set n 2} {$n <= 10} {incr n} {
set angle 0.0
set row $n:
for {set i 1} {$i <= $n} {incr i} {
lappend row [format %5.4f%+5.4fi [cos $angle] [sin $angle]]
set angle [expr {$angle + 2*$pi/$n}]
}
puts $row
}

TI-89 BASIC[edit]

cZeros(x^n - 1, x)

For n=3 in exact mode, the results are

{-1/2+√(3)/2*i, -1/2-√(3)/2*i, 1}

Ursala[edit]

The roots function takes a number n to the nth root of -1, squares it, and iteratively makes a list of its first n powers (oblivious to roundoff error). Complex functions cpow and mul are used, which are called from the host system's standard C library.

#import std
#import nat
#import flo
 
roots = ~&htxPC+ c..mul:-0^*DlSiiDlStK9\iota c..[email protected]+ c..cpow/-1.+ div/1.+ float
 
#cast %jLL
 
tests = roots* <1,2,3,4,5,6>

The output is a list of lists of complex numbers.

<
   <1.000e+00-2.449e-16j>,
   <
      1.000e+00-2.449e-16j,
      -1.000e+00+1.225e-16j>,
   <
      1.000e+00-8.327e-16j,
      -5.000e-01+8.660e-01j,
      -5.000e-01-8.660e-01j>,
   <
      1.000e+00-8.882e-16j,
      2.220e-16+1.000e+00j,
      -1.000e+00+4.441e-16j,
      -6.661e-16-1.000e+00j>,
   <
      1.000e+00-5.551e-17j,
      3.090e-01+9.511e-01j,
      -8.090e-01+5.878e-01j,
      -8.090e-01-5.878e-01j,
      3.090e-01-9.511e-01j>,
   <
      1.000e+00-1.221e-15j,
      5.000e-01+8.660e-01j,
      -5.000e-01+8.660e-01j,
      -1.000e+00+6.106e-16j,
      -5.000e-01-8.660e-01j,
      5.000e-01-8.660e-01j>>

zkl[edit]

Translation of: C
PI2:=(0.0).pi*2;
foreach n,i in ([1..9],n){
c:=s:=0;
if(not i) c = 1;
else if(n==4*i) s = 1;
else if(n==2*i) c = -1;
else if(3*n==4*i) s = -1;
else a,c,s:=PI2*i/n,a.cos(),a.sin();
 
if(c) print("%.2g".fmt(c));
print( (s==1 and "i") or (s==-1 and "-i" or (s and "%+.2gi" or"")).fmt(s));
print( (i==n-1) and "\n" or ", ");
}
Output:
1
1,  -1
1,  -0.5+0.87i,  -0.5-0.87i
1,  i,  -1,  -i
1,  0.31+0.95i,  -0.81+0.59i,  -0.81-0.59i,  0.31-0.95i
1,  0.5+0.87i,  -0.5+0.87i,  -1,  -0.5-0.87i,  0.5-0.87i
1,  0.62+0.78i,  -0.22+0.97i,  -0.9+0.43i,  -0.9-0.43i,  -0.22-0.97i,  0.62-0.78i
1,  0.71+0.71i,  i,  -0.71+0.71i,  -1,  -0.71-0.71i,  -i,  0.71-0.71i
1,  0.77+0.64i,  0.17+0.98i,  -0.5+0.87i,  -0.94+0.34i,  -0.94-0.34i,  -0.5-0.87i,  0.17-0.98i,  0.77-0.64i