# Roots of unity

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.

Given   n,   find the   nth   roots of unity.

## 11l

F polar(r, theta)
R r * (cos(theta) + sin(theta) * 1i)

F croots(n)
R (0 .< n).map(k -> polar(1, 2 * k * math:pi / @n))

L(nr) 2..10
print(nr‘ ’croots(nr))
Output:
2 [1, -1]
3 [1, -0.5+0.866025404i, -0.5-0.866025404i]
4 [1, 1i, -1, -1i]
5 [1, 0.309016994+0.951056516i, -0.809016994+0.587785252i, -0.809016994-0.587785252i, 0.309016994-0.951056516i]
6 [1, 0.5+0.866025404i, -0.5+0.866025404i, -1, -0.5-0.866025404i, 0.5-0.866025404i]
7 [1, 0.623489802+0.781831482i, -0.222520934+0.974927912i, -0.900968868+0.433883739i, -0.900968868-0.433883739i, -0.222520934-0.974927912i, 0.623489802-0.781831482i]
8 [1, 0.707106781+0.707106781i, 1i, -0.707106781+0.707106781i, -1, -0.707106781-0.707106781i, -1i, 0.707106781-0.707106781i]
9 [1, 0.766044443+0.64278761i, 0.173648178+0.984807753i, -0.5+0.866025404i, -0.939692621+0.342020143i, -0.939692621-0.342020143i, -0.5-0.866025404i, 0.173648178-0.984807753i, 0.766044443-0.64278761i]
10 [1, 0.809016994+0.587785252i, 0.309016994+0.951056516i, -0.309016994+0.951056516i, -0.809016994+0.587785252i, -1, -0.809016994-0.587785252i, -0.309016994-0.951056516i, 0.309016994-0.951056516i, 0.809016994-0.587785252i]


with Ada.Text_IO;                 use Ada.Text_IO;

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

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


## Arturo

rect: function [r,phi][
to :complex @[ r * cos phi, r * sin phi ]
]
roots: function [n][
map 0..dec n 'k -> rect 1.0 2 * k * pi / n
]

loop 2..10 'nr ->
print [pad to :string nr 3 "=>" join.with:", " to [:string] .format:".3f" roots nr]

Output:
  2 => 1.000+0.000i, -1.000+0.000i
3 => 1.000+0.000i, -0.500+0.866i, -0.500-0.866i
4 => 1.000+0.000i, 0.000+1.000i, -1.000+0.000i, -0.000-1.000i
5 => 1.000+0.000i, 0.309+0.951i, -0.809+0.588i, -0.809-0.588i, 0.309-0.951i
6 => 1.000+0.000i, 0.500+0.866i, -0.500+0.866i, -1.000+0.000i, -0.500-0.866i, 0.500-0.866i
7 => 1.000+0.000i, 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+0.000i, 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+0.000i, 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+0.000i, 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

## ATS

I compute the roots of unity by the formula exp(-2*pi*k*sqrt(-1)/n), k = 0, 1, ..., n-1. Most of the code is to build part of a general infrastructure for supporting standard C complex types.

(ATS code looks like ML code, and has a very unusual and relatively strict type system, but semantically it is essentially C.)

(*

This program has to be compiled without -std=c99, which patscc will
insert unless you override the setting.

##myatsccdef=\
patscc \
--gline \
-atsccomp gcc \
-I"${PATSHOME}" \ -I"${PATSHOME}/ccomp/runtime" \
-L"${PATSHOME}/ccomp/atslib/lib" \ -DATS_MEMALLOC_LIBC \ -o$fname($1)$1 -lm

*)

(*

I use the C _Complex types, but not the newer _Imaginary types.
Thus I demonstrate how one might add new floating point types
nicely.

(In my opinion, it is good to use m4 or similar tools when writing
such repetitive code. Doing so reduces both work and the frequency
of errors. Also you could then more easily add support for the many
extension types such as "_Float128 complex" (quadruple precision).

One could, of course, define one's own complex types directly in
ATS.

*)

#define ATS_EXTERN_PREFIX "myatspre_"

#define NIL list_nil ()
#define ::  list_cons

(*------------------------------------------------------------------*)

%{^

#include <complex.h>

#define myatspre_inline ATSinline ()

typedef float complex myatstype_fcomplex;
typedef double complex myatstype_dcomplex;
typedef long double complex myatstype_lcomplex;

#define myatspre_CMPLXF CMPLXF
#define myatspre_CMPLX CMPLX
#define myatspre_CMPLXL CMPLXL

myatspre_inline atsvoid_t0ype
myatspre_fprint_fcomplex (atstype_ref r,
myatstype_fcomplex x)
{
double rx = crealf (x);
double ix = cimagf (x);
const char *plus = (ix < 0) ? "" : "+";
fprintf ((FILE *) r, "%f%s%fi", rx, plus, ix);
}

#define myatspre_print_fcomplex(x) myatspre_fprint_fcomplex (stdout, (x))
#define myatspre_prerr_fcomplex(x) myatspre_fprint_fcomplex (stderr, (x))

myatspre_inline atsvoid_t0ype
myatspre_fprint_dcomplex (atstype_ref r,
myatstype_dcomplex x)
{
double rx = creal (x);
double ix = cimag (x);
const char *plus = (ix < 0) ? "" : "+";
fprintf ((FILE *) r, "%f%s%fi", rx, plus, ix);
}

#define myatspre_print_dcomplex(x) myatspre_fprint_dcomplex (stdout, (x))
#define myatspre_prerr_dcomplex(x) myatspre_fprint_dcomplex (stderr, (x))

myatspre_inline atsvoid_t0ype
myatspre_fprint_lcomplex (atstype_ref r,
myatstype_lcomplex x)
{
long double rx = creall (x);
long double ix = cimagl (x);
const char *plus = (ix < 0) ? "" : "+";
fprintf ((FILE *) r, "%Lf%s%Lfi", rx, plus, ix);
}

#define myatspre_print_lcomplex(x) myatspre_fprint_lcomplex (stdout, (x))
#define myatspre_prerr_lcomplex(x) myatspre_fprint_lcomplex (stderr, (x))

myatspre_inline myatstype_fcomplex
myatspre_g0float_cmplx_float_fcomplex (atstype_float x,
atstype_float y)
{
return myatspre_CMPLXF (x, y);
}

myatspre_inline myatstype_dcomplex
myatspre_g0float_cmplx_double_dcomplex (atstype_double x,
atstype_double y)
{
return myatspre_CMPLX (x, y);
}

myatspre_inline myatstype_lcomplex
myatspre_g0float_cmplx_ldouble_lcomplex (atstype_ldouble x,
atstype_ldouble y)
{
return myatspre_CMPLXL (x, y);
}

myatspre_inline myatstype_fcomplex
myatspre_g0int2float_int_fcomplex (atstype_int x)
{
return x;
}

myatspre_inline myatstype_dcomplex
myatspre_g0int2float_int_dcomplex (atstype_int x)
{
return x;
}

myatspre_inline myatstype_lcomplex
myatspre_g0int2float_int_lcomplex (atstype_int x)
{
return x;
}

myatspre_inline myatstype_fcomplex
myatspre_g0float_mul_fcomplex (myatstype_fcomplex x,
myatstype_fcomplex y)
{
return x * y;
}

myatspre_inline myatstype_dcomplex
myatspre_g0float_mul_dcomplex (myatstype_dcomplex x,
myatstype_dcomplex y)
{
return x * y;
}

myatspre_inline myatstype_lcomplex
myatspre_g0float_mul_lcomplex (myatstype_lcomplex x,
myatstype_lcomplex y)
{
return x * y;
}

myatspre_inline myatstype_fcomplex
myatspre_exp_fcomplex (myatstype_fcomplex x)
{
return cexpf (x);
}

myatspre_inline myatstype_dcomplex
myatspre_exp_dcomplex (myatstype_dcomplex x)
{
return cexp (x);
}

myatspre_inline myatstype_lcomplex
myatspre_exp_lcomplex (myatstype_lcomplex x)
{
return cexpl (x);
}

myatspre_inline myatstype_fcomplex
myatspre_pow_fcomplex (myatstype_fcomplex x,
myatstype_fcomplex y)
{
return cpowf (x, y);
}

myatspre_inline myatstype_dcomplex
myatspre_pow_dcomplex (myatstype_dcomplex x,
myatstype_dcomplex y)
{
return cpow (x, y);
}

myatspre_inline myatstype_lcomplex
myatspre_pow_lcomplex (myatstype_lcomplex x,
myatstype_lcomplex y)
{
return cpowl (x, y);
}

%}

(*------------------------------------------------------------------*)

tkindef fcomplex_kind = "myatstype_fcomplex"

tkindef dcomplex_kind = "myatstype_dcomplex"

tkindef lcomplex_kind = "myatstype_lcomplex"

extern fn print_fcomplex : fcomplex -<1> void = "mac#%"
extern fn prerr_fcomplex : fcomplex -<1> void = "mac#%"
extern fn fprint_fcomplex : fprint_type fcomplex = "mac#%"
implement fprint_val<fcomplex> = fprint_fcomplex

extern fn print_dcomplex : dcomplex -<1> void = "mac#%"
extern fn prerr_dcomplex : dcomplex -<1> void = "mac#%"
extern fn fprint_dcomplex : fprint_type dcomplex = "mac#%"
implement fprint_val<dcomplex> = fprint_dcomplex

extern fn print_lcomplex : lcomplex -<1> void = "mac#%"
extern fn prerr_lcomplex : lcomplex -<1> void = "mac#%"
extern fn fprint_lcomplex : fprint_type lcomplex = "mac#%"
implement fprint_val<lcomplex> = fprint_lcomplex

extern fn g0int2float_int_fcomplex : int -<> fcomplex = "mac#%"
extern fn g0int2float_int_dcomplex : int -<> dcomplex = "mac#%"
extern fn g0int2float_int_lcomplex : int -<> lcomplex = "mac#%"
implement g0int2float<intknd,fcmplxknd> = g0int2float_int_fcomplex
implement g0int2float<intknd,dcmplxknd> = g0int2float_int_dcomplex
implement g0int2float<intknd,lcmplxknd> = g0int2float_int_lcomplex

extern fn g0float_cmplx_float_fcomplex : (float, float) -<> fcomplex = "mac#%"
extern fn g0float_cmplx_double_dcomplex : (double, double) -<> dcomplex = "mac#%"
extern fn g0float_cmplx_ldouble_lcomplex : (ldouble, ldouble) -<> lcomplex = "mac#%"
extern fn {tk2 : tkind} {tk1 : tkind} g0float_cmplx : (g0float tk1, g0float tk1) -<> g0float tk2
implement g0float_cmplx<fcmplxknd><fltknd> = g0float_cmplx_float_fcomplex
implement g0float_cmplx<dcmplxknd><dblknd> = g0float_cmplx_double_dcomplex
implement g0float_cmplx<lcmplxknd><ldblknd> = g0float_cmplx_ldouble_lcomplex

extern fn g0float_mul_fcomplex : g0float_aop_type fcmplxknd = "mac#%"
extern fn g0float_mul_dcomplex : g0float_aop_type dcmplxknd = "mac#%"
extern fn g0float_mul_lcomplex : g0float_aop_type lcmplxknd = "mac#%"
implement g0float_mul<fcmplxknd> = g0float_mul_fcomplex
implement g0float_mul<dcmplxknd> = g0float_mul_dcomplex
implement g0float_mul<lcmplxknd> = g0float_mul_lcomplex

(*------------------------------------------------------------------*)
(* Most "math" functions are not defined in the prelude. Here we will
follow the conventions of libats/libc, which does not use the
floating point typekinds. *)

extern fn exp_fcomplex : fcomplex -<> fcomplex = "mac#%"
extern fn exp_dcomplex : dcomplex -<> dcomplex = "mac#%"
extern fn exp_lcomplex : lcomplex -<> lcomplex = "mac#%"
implement exp<fcomplex> = exp_fcomplex
implement exp<dcomplex> = exp_dcomplex
implement exp<lcomplex> = exp_lcomplex

extern fn pow_fcomplex : (fcomplex, fcomplex) -<> fcomplex = "mac#%"
extern fn pow_dcomplex : (dcomplex, dcomplex) -<> dcomplex = "mac#%"
extern fn pow_lcomplex : (lcomplex, lcomplex) -<> lcomplex = "mac#%"
implement pow<fcomplex> = pow_fcomplex
implement pow<dcomplex> = pow_dcomplex
implement pow<lcomplex> = pow_lcomplex

(*------------------------------------------------------------------*)

fn
nth_roots_of_unity
{n : pos}
(n : int n)
:<> list (dcomplex, n) =
let
val C = cmplx (0.0, ~((2.0 * M_PI) / g0i2f n))

fun
loop {k : nat | k <= n}
.<k>.
(k     : int k,
accum : list (dcomplex, n - k))
:<> list (dcomplex, n) =
if k = 0 then
accum
else
loop (pred k, exp (g0i2f (pred k) * C) :: accum)
in
loop (n, NIL)
end

fn
nth_powers {m   : int}
{n   : pos}
(lst : list (dcomplex, m),
n   : int n)
:<1> list (dcomplex, m) =
let
val nth : dcomplex = g0i2f n
implement list_map$fopr<dcomplex><dcomplex> x = pow (x, nth) in list_vt2t (list_map<dcomplex><dcomplex> lst) end fn show_results {n : pos} (n : int n) :<1> void = let val nth_roots = nth_roots_of_unity n val ones = nth_powers (nth_roots, n) in println! (); println! ("roots of unity = ", nth_roots); println! ("roots raised ", n, " = ", ones) end fun loop_over_args {argc : int} {k : pos | k <= argc} {p_args : addr} .<argc - k>. (pf_args : !array_v (string, p_args, argc) | argc : int argc, p_args : ptr p_args, k : int k) :<1> void = if k <> argc then let macdef args = !p_args val argument = args[k] val n =$extfcall ([n : int] int n, "atoi", argument)
in
if 0 < n then
show_results n;
loop_over_args (pf_args | argc, p_args, succ k)
end

implement
main0 {argc} (argc, argv) =
let
@(pf_args, pf_minus | p_args) =
argv_takeout_strarr {argc} argv

val () = loop_over_args {argc} {1} {p_args}
(pf_args | argc, p_args, 1)

prval () = minus_addback (pf_minus, pf_args | argv)
in
println! ()
end

(*------------------------------------------------------------------*)
Output:
$myatscc roots-of-unity.dats && ./roots-of-unity 1 2 3 4 5 roots of unity = 1.000000+0.000000i roots raised 1 = 1.000000+0.000000i roots of unity = 1.000000+0.000000i, -1.000000-0.000000i roots raised 2 = 1.000000+0.000000i, 1.000000+0.000000i roots of unity = 1.000000+0.000000i, -0.500000-0.866025i, -0.500000+0.866025i roots raised 3 = 1.000000+0.000000i, 1.000000+0.000000i, 1.000000+0.000000i roots of unity = 1.000000+0.000000i, 0.000000-1.000000i, -1.000000-0.000000i, -0.000000+1.000000i roots raised 4 = 1.000000+0.000000i, 1.000000+0.000000i, 1.000000+0.000000i, 1.000000+0.000000i roots of unity = 1.000000+0.000000i, 0.309017-0.951057i, -0.809017-0.587785i, -0.809017+0.587785i, 0.309017+0.951057i roots raised 5 = 1.000000+0.000000i, 1.000000+0.000000i, 1.000000+0.000000i, 1.000000+0.000000i, 1.000000+0.000000i  ## AutoHotkey 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 # 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 Works with: QBasic version 1.1 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  ### BASIC256 twopi = 2 * pi n = 5 for m = 0 to n-1 theta = m*twopi/n real = cos(theta) imag = sin(theta) if imag >= 0 then print ljust(string(real),9,"0"); " + "; ljust(string(imag),13,"0"); "i" else print ljust(string(real),9,"0"); " - "; ljust(string(-imag),13,"0"); "i" end if next m  ### BBC BASIC  @% = &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


### Craft Basic

define theta = 0, real = 0, imag = 0
define pi = 3.14, n = 5

for m = 0 to n - 1

let theta = m * (pi * 2) / n
let real = cos(theta)
let imag = sin(theta)

if imag >= 0 then

print real, comma, " ", imag, "i"

else

print real, comma, " ", imag * -1, "i"

endif

wait

next m

Output:

1, 0i
0.31, 0.95i
-0.81, 0.59i
-0.81, 0.59i
0.30, 0.95i



### FreeBASIC

#define twopi 6.2831853071795864769252867665590057684

dim as uinteger m, n
dim as double real, imag, theta
input "n? ", n

for m = 0 to n-1
theta = m*twopi/n
real = cos(theta)
imag = sin(theta)
if imag >= 0 then
print using "#.##### + #.##### i"; real; imag
else
print using "#.##### - #.##### i"; real; -imag
end if
next m


### FutureBasic

window 1, @"Roots of Unity", (0,0,1050,200)

long n, root
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

HandleEvents

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


### Liberty BASIC

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

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 ### PureBasic 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()  ### Run BASIC 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 ### TI-89 BASIC 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} ### True BASIC Translation of: BASIC LET num_pi = 3.1415926 LET n = 5 !this can be changed for any desired n LET angle = 0 !start at angle 0 DO LET real = COS(angle) !real axis is the x axis IF (ABS(real) < 10^(-5)) THEN !get rid of annoying sci notation LET real = 0 END IF LET imag = SIN(angle) !imaginary axis is the y axis IF (ABS(imag) < 10^(-5)) THEN !get rid of annoying sci notation LET imag = 0 END IF PRINT real; "+"; imag; "i" !answer on every line LET angle = angle+(2*num_pi)/n !all the way around the circle at even intervals LOOP WHILE angle < 2*num_pi END  ### Yabasic twopi = 2 * pi n = 5 for m = 0 to n-1 theta = m*twopi/n real = cos(theta) imag = sin(theta) if imag >= 0 then print real using("##.########"), " + ", imag using("#.########"), "i" else print real using("##.########"), " + ", -imag using("#.########"), "i" end if next m  ## Ursala 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..mul@iiX+ 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>> ## C #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# 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++ #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 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 (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)). ## Crystal Translation of: Ruby require "complex" def roots_of_unity(n) (0...n).map { |k| Math.exp((2 * Math::PI * k / n).i) } end p roots_of_unity(3)  Or alternative def roots_of_unity(n) (0...n).map { |k| Complex.new(Math.cos(2 * Math::PI * k / n), Math.sin(2 * Math::PI * k / n)) } end  Output: [(1+0.0i), (-0.4999999999999998+0.8660254037844387i), (-0.5000000000000004-0.8660254037844384i)]  ## D 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] ## Delphi Translation of: C# program Roots_of_unity; {$APPTYPE CONSOLE}

uses
System.VarCmplx;

function RootOfUnity(degree: integer): Tarray<Variant>;
var
k: Integer;
begin
SetLength(result, degree);
for k := 0 to degree - 1 do
Result[k] := VarComplexFromPolar(1, 2 * pi * k / degree);
end;

const
n = 3;
var
num: Variant;
begin
Writeln('Root of unity from ', n, ':'#10);
for num in RootOfUnity(n) do
Writeln(num);
end.

Output:
Root of unity from 3:

1 + 0i
-0,5 + 0,866025403784438i
-0,5 - 0,866025403784439i


## EasyLang

Translation of: AWK
numfmt 4 0
for n = 2 to 5
write n & ": "
for root = 0 to n - 1
real = cos (360 * root / n)
imag = sin (360 * root / n)
write real & " " & imag & "i"
if root <> n - 1
write ", "
.
.
print ""
.

Output:
2: 1 0i, -1 0.0000i
3: 1 0i, -0.5000 0.8660i, -0.5000 -0.8660i
4: 1 0i, 0.0000 1i, -1 0.0000i, -0.0000 -1i
5: 1 0i, 0.3090 0.9511i, -0.8090 0.5878i, -0.8090 -0.5878i, 0.3090 -0.9511i


## EchoLisp

(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

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. ## Factor USING: math.functions prettyprint ; 1 3 roots .  Output: { 1.0 C{ -0.4999999999999998 0.8660254037844387 } C{ -0.5000000000000003 -0.8660254037844384 } }  ## Forth 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  On the other hand, complex numbers are implemented by the FSL. Works with: gforth version 0.7.9_20170308 Translation of: C++ require fsl-util.fs require fsl/complex.fs : abs= 1E-12 F~ ; : clamp-to-0 FDUP 0E0 abs= IF FDROP 0E0 THEN ; : zclamp-to-0 clamp-to-0 FSWAP clamp-to-0 FSWAP ; : .roots 1+ 2 DO I . ." : " I 0 DO 1E0 2E0 PI F* I S>F F* J S>F F/ polar> zclamp-to-0 z. SPACE LOOP CR LOOP ; 3 SET-PRECISION 5 .roots  ## Fortran ### Sin/Cos + Scalar Loop 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 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  ## Frink Calculates the angles in degrees, since Frink will use rational arithmetic (exact) roots[n] := { a = makeArray[[n], 0] alpha = 360/n degrees theta = 0 degrees for k = 0 to length[a] - 1 { a@k = cos[theta] + i sin[theta] theta = theta + alpha } a } Output: setPrecision[8] roots[3] [1.0, ( -0.5 + 0.86602540498103642 i ), ( -0.5 - 0.86602540139124295 i )]  ## FunL 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}  ## GAP 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 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 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

import Data.Complex (Complex, cis)

rootsOfUnity :: (Enum a, Floating a) => a -> [Complex a]
rootsOfUnity n =
[ cis (2 * pi * k / n)
| k <- [0 .. n - 1] ]

main :: IO ()
for. i.y do.
z=. z,e
e=. e*r
end.
z
)


## Java

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

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

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

$time jq -c -n -f Roots_of_unity.jq [1,0] [0.8090169943749475,0.5877852522924731] [0.30901699437494745,0.9510565162951535] [-0.30901699437494734,0.9510565162951536] [-0.8090169943749473,0.5877852522924732] [-1,1.2246467991473532e-16] [-0.8090169943749475,-0.587785252292473] [-0.30901699437494756,-0.9510565162951535] [0.30901699437494723,-0.9510565162951536] [0.8090169943749473,-0.5877852522924732] real 0m0.015s user 0m0.004s sys 0m0.004s ## Julia nthroots(n::Integer) = [ cospi(2k/n)+sinpi(2k/n)im for k = 0:n-1 ]  (One could also use complex exponentials or other formulations.) For example, nthroots(10) gives: 10-element Array{Complex{Float64},1}: 1.0+0.0im 0.809017+0.587785im 0.309017+0.951057im -0.309017+0.951057im -0.809017+0.587785im -1.0+0.0im -0.809017-0.587785im -0.309017-0.951057im 0.309017-0.951057im 0.809017-0.587785im  ## Kotlin import java.lang.Math.* data class Complex(val r: Double, val i: Double) { override fun toString() = when { i == 0.0 -> r.toString() r == 0.0 -> i.toString() + 'i' else -> "$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] ## Lambdatalk // cleandisp just to display 0 when n < 10^-10 {def cleandisp {lambda {:n} {if {<= {abs :n} 1.e-10} then 0 else :n}}} -> cleandisp {def uroots {lambda {:n} {S.map {{lambda {:n :i} {let { {:theta {/ {* 2 {PI} :i} :n}} } {cons {cleandisp {cos :theta}} {cleandisp {sin :theta}}}}} :n} {S.serie 0 {- :n 1}}} }} -> uroots {S.map {lambda {:i} {hr}i = :i -> {uroots :i}} {S.serie 2 10}} -> i = 2 -> (1 0) (-1 0) i = 3 -> (1 0) (-0.4999999999999998 0.8660254037844388) (-0.5000000000000004 -0.8660254037844384) i = 4 -> (1 0) (0 1) (-1 0) (0 -1) i = 5 -> (1 0) (0.30901699437494745 0.9510565162951535) (-0.8090169943749473 0.5877852522924732) (-0.8090169943749475 -0.587785252292473) (0.30901699437494723 -0.9510565162951536) i = 6 -> (1 0) (0.5000000000000001 0.8660254037844386) (-0.4999999999999998 0.8660254037844388) (-1 0) (-0.5000000000000004 -0.8660254037844384) (0.5 -0.8660254037844386) i = 7 -> (1 0) (0.6234898018587336 0.7818314824680297) (-0.22252093395631434 0.9749279121818236) (-0.900968867902419 0.43388373911755823) (-0.9009688679024191 -0.433883739117558) (-0.2225209339563146 -0.9749279121818235) (0.6234898018587334 -0.7818314824680299) i = 8 -> (1 0) (0.7071067811865476 0.7071067811865475) (0 1) (-0.7071067811865475 0.7071067811865476) (-1 0) (-0.7071067811865477 -0.7071067811865475) (0 -1) (0.7071067811865475 -0.7071067811865477) i = 9 -> (1 0) (0.7660444431189781 0.6427876096865393) (0.17364817766693041 0.984807753012208) (-0.4999999999999998 0.8660254037844388) (-0.9396926207859083 0.3420201433256689) (-0.9396926207859084 -0.34202014332566866) (-0.5000000000000004 -0.8660254037844384) (0.17364817766692997 -0.9848077530122081) (0.7660444431189779 -0.6427876096865396) i = 10 -> (1 0) (0.8090169943749475 0.5877852522924731) (0.30901699437494745 0.9510565162951535) (-0.30901699437494734 0.9510565162951536) (-0.8090169943749473 0.5877852522924732) (-1 0) (-0.8090169943749475 -0.587785252292473) (-0.30901699437494756 -0.9510565162951535) (0.30901699437494723 -0.9510565162951536) (0.8090169943749473 -0.5877852522924732)  ## Lua 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 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 /Wolfram Language 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: ${\displaystyle \{1,-1\}}$ ${\displaystyle \left\{1,e^{\frac {2i\pi }{3}},e^{-{\frac {2i\pi }{3}}}\right\}}$ ${\displaystyle \{1,i,-1,-i\}}$ ${\displaystyle \left\{1,e^{\frac {2i\pi }{5}},e^{\frac {4i\pi }{5}},e^{-{\frac {4i\pi }{5}}},e^{-{\frac {2i\pi }{5}}}\right\}}$ ${\displaystyle \{1,-1\}}$ ${\displaystyle \left\{1,-{\frac {1}{2}}+{\frac {i{\sqrt {3}}}{2}},-{\frac {1}{2}}-{\frac {i{\sqrt {3}}}{2}}\right\}}$ ${\displaystyle \{1,i,-1,-i\}}$ ${\displaystyle \left\{1,-{\frac {1}{4}}+{\frac {\sqrt {5}}{4}}+i{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}},-{\frac {1}{4}}-{\frac {\sqrt {5}}{4}}+i{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},-{\frac {1}{4}}-{\frac {\sqrt {5}}{4}}-i{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},-{\frac {1}{4}}+{\frac {\sqrt {5}}{4}}-i{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right\}}$ ${\displaystyle \{1.,-1.\}}$ ${\displaystyle \{1.,-0.5+0.866025i,-0.5-0.866025i\}}$ ${\displaystyle \{1.,0.+1.i,-1.,0.-1.i\}}$ ${\displaystyle \{1.,0.309017+0.951057i,-0.809017+0.587785i,-0.809017-0.587785i,0.309017-0.951057i\}}$ ## MATLAB 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 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]  ## MiniScript complexRoots = function(n) result = [] for i in range(0, n-1) real = cos(2*pi * i/n) if abs(real) < 1e-6 then real = 0 imag = sin(2*pi * i/n) if abs(imag) < 1e-6 then imag = 0 result.push real + " " + "+" * (imag>=0) + imag + "i" end for return result end function for i in range(2,5) print i + ": " + complexRoots(i).join(", ") end for  Output: 2: 1 +0i, -1 +0i 3: 1 +0i, -0.5 +0.866025i, -0.5 -0.866025i 4: 1 +0i, 0 +1i, -1 +0i, 0 -1i 5: 1 +0i, 0.309017 +0.951057i, -0.809017 +0.587785i, -0.809017 -0.587785i, 0.309017 -0.951057i ## МК-61/52 П0 0 П1 ИП1 sin ИП1 cos С/П 2 пи * ИП0 / ИП1 + П1 БП 03  ## Nim import complex, math, sequtils, strformat, strutils proc roots(n: Positive): seq[Complex64] = for k in 0..<n: result.add rect(1.0, 2 * k.float * Pi / n.float) proc toString(z: Complex64): string = &"{z.re:.3f} + {z.im:.3f}i" for nr in 2..10: let result = roots(nr).map(toString).join(", ") echo &"{nr:2}: {result}"  Output:  2: 1.000 + 0.000i, -1.000 + 0.000i 3: 1.000 + 0.000i, -0.500 + 0.866i, -0.500 + -0.866i 4: 1.000 + 0.000i, 0.000 + 1.000i, -1.000 + 0.000i, -0.000 + -1.000i 5: 1.000 + 0.000i, 0.309 + 0.951i, -0.809 + 0.588i, -0.809 + -0.588i, 0.309 + -0.951i 6: 1.000 + 0.000i, 0.500 + 0.866i, -0.500 + 0.866i, -1.000 + 0.000i, -0.500 + -0.866i, 0.500 + -0.866i 7: 1.000 + 0.000i, 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 + 0.000i, 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 + 0.000i, 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 + 0.000i, 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 ## OCaml 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 for j = 2 : 10 printf("*** %d\n", j); for n = 1 : j disp(exp(2i*pi*n/j)); endfor disp(""); endfor  ## OoRexx 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 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 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 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  ## Phix Translation of: AWK with javascript_semantics for n=2 to 10 do printf(1,"%2d:",n) for root=0 to n-1 do atom real = cos(2*PI*root/n) atom imag = sin(2*PI*root/n) printf(1,"%s %6.3f %6.3fi",{iff(root?",":""),real,imag}) end for printf(1,"\n") end for  Output:  2: 1.000 0.000i, -1.000 0.000i 3: 1.000 0.000i, -0.500 0.866i, -0.500 -0.866i 4: 1.000 0.000i, 0.000 1.000i, -1.000 0.000i, -0.000 -1.000i 5: 1.000 0.000i, 0.309 0.951i, -0.809 0.588i, -0.809 -0.588i, 0.309 -0.951i 6: 1.000 0.000i, 0.500 0.866i, -0.500 0.866i, -1.000 0.000i, -0.500 -0.866i, 0.500 -0.866i 7: 1.000 0.000i, 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 0.000i, 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 0.000i, 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 0.000i, 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  ## PicoLisp 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) ) ) ## PL/I 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 ## Prolog Solves the roots of unity symbolically, as complex powers of e. roots(N, Rs) :- succ(Pn, N), numlist(0, Pn, Ks), maplist(root(N), Ks, Rs). root(N, M, R2) :- R0 is (2*M) rdiv N, % multiple of PI (R0 > 1 -> R1 is R0 - 2; R1 = R0), % adjust for principal values cis(R1, R2). cis(0, 1) :- !. cis(1, -1) :- !. cis(1 rdiv 2, i) :- !. cis(-1 rdiv 2, -i) :- !. cis(-1 rdiv Q, exp(-i*pi/Q)) :- !. cis(1 rdiv Q, exp(i*pi/Q)) :- !. cis(P rdiv Q, exp(P*i*pi/Q)).  Output: ?- roots(2,X). X = [1, -1]. ?- roots(3,X). X = [1, exp(2*i*pi/3), exp(-2*i*pi/3)]. ?- roots(4,X). X = [1, i, -1, -i]. ?- roots(5,X). X = [1, exp(2*i*pi/5), exp(4*i*pi/5), exp(-4*i*pi/5), exp(-2*i*pi/5)]. ?- roots(8,X), forall(member(A,X), format("~w~n", A)). 1 exp(i*pi/4) i exp(3*i*pi/4) -1 exp(-3*i*pi/4) -i exp(-i*pi/4) X = [1, exp(i*pi/4), i, exp(3*i*pi/4), -1, exp(... * ... * pi/4), -i, exp(... / ...)].  ## Python Works with: Python version 3.7 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 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 #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 ## Raku (formerly Perl 6) Raku has a built-in function cis which returns a unitary complex number given its phase. Raku 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 Alternately, you could use the built-in .roots method to find the nth roots of any number. .say for 1.roots(9)  Output: 1+0i 0.766044443118978+0.6427876096865393i 0.17364817766693041+0.984807753012208i -0.4999999999999998+0.8660254037844387i -0.9396926207859083+0.3420201433256689i -0.9396926207859084-0.34202014332566866i -0.5000000000000004-0.8660254037844384i 0.17364817766692997-0.9848077530122081i 0.7660444431189778-0.6427876096865396i ## REXX 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 displays 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, 5th line). /*REXX program computes the K roots of unity (which usually includes complex roots).*/ numeric digits length( pi() ) - length(.) /*use number of decimal digits in pi. */ 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. */ do #=start to abs(n) /*show unity roots (for a range or 1).*/ say right(# 'roots of unity', 40, "─") ' (showing' frac "fractional decimal digits)" do angle=0 by pi*2/# for # /*compute the angle for each root. */ Rp= adj( cos(angle) ) /*the real part via COS function.*/ Ip= adj( sin(angle) ) /* " imaginary " " SIN " */ if Rp>=0 then Rp= ' 'Rp /*Not neg? Then pad with a blank char.*/ if Ip>=0 then Ip= '+'Ip /* " " " " " " plus " */ 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 /*#*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ adj: parse arg x; if abs(x) < ('1e-')(digits()*9%10) then x= 0; return format(x,,frac)/1 pi: pi=3.141592653589793238462643383279502884197169399375105820974944592307816; return pi r2r: pi2= pi() + pi; return arg(1) // pi2 /*reduce #radians: -2pi ─► +2pi radians*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; x= r2r(x); a= abs(x); numeric fuzz min(9, digits() - 9) pi1_3=pi/3; if a=pi1_3 then return .5; if a=pi*.5 | a=pi2 then return 0 if a=pi then return -1; if a=pi1_3*2 then return -.5; z= 1; _= 1;$x= x * x
do k=2  by 2  until p=z;  p=z;  _= -_ * $x / (k*(k-1)); z= z + _; end; return z /*──────────────────────────────────────────────────────────────────────────────────────*/ sin: procedure; parse arg x; x= r2r(x); numeric fuzz min(5, digits() - 3) if abs(x)=pi then return 0;$x= x * x;       z= x;  _= x
do k=2  by 2  until p=z;  p=z;  _= -_ * $x / (k*(k+1)); z= z + _; end; return z  output when using the input of: 5 ────────────────────────5 roots of unity (showing 5 fractional decimal digits) 1 0.30902 +0.95106i -0.80902 +0.58779i -0.80902 -0.58779i 0.30902 -0.95106i  output when using the input of: 10 36 ───────────────────────10 roots of unity (showing 36 fractional decimal digits) 1 0.809016994374947424102293417182819059 +0.587785252292473129168705954639072769i 0.309016994374947424102293417182819059 +0.951056516295153572116439333379382143i -0.309016994374947424102293417182819059 +0.951056516295153572116439333379382143i -0.809016994374947424102293417182819059 +0.587785252292473129168705954639072769i -1 -0.809016994374947424102293417182819059 -0.587785252292473129168705954639072769i -0.309016994374947424102293417182819059 -0.951056516295153572116439333379382143i 0.309016994374947424102293417182819059 -0.951056516295153572116439333379382143i 0.809016994374947424102293417182819059 -0.587785252292473129168705954639072769i  output when using the input of: -12 (Shown at five-sixths size.) ────────────────────────1 roots of unity (showing 5 fractional decimal digits) 1 ────────────────────────2 roots of unity (showing 5 fractional decimal digits) 1 -1 ────────────────────────3 roots of unity (showing 5 fractional decimal digits) 1 -0.5 +0.86603i -0.5 -0.86603i ────────────────────────4 roots of unity (showing 5 fractional decimal digits) 1 0 +1i -1 0 -1i ────────────────────────5 roots of unity (showing 5 fractional decimal digits) 1 0.30902 +0.95106i -0.80902 +0.58779i -0.80902 -0.58779i 0.30902 -0.95106i ────────────────────────6 roots of unity (showing 5 fractional decimal digits) 1 0.5 +0.86603i -0.5 +0.86603i -1 -0.5 -0.86603i 0.5 -0.86603i ────────────────────────7 roots of unity (showing 5 fractional decimal digits) 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 (showing 5 fractional decimal digits) 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 (showing 5 fractional decimal digits) 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 (showing 5 fractional decimal digits) 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 (showing 5 fractional decimal digits) 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 (showing 5 fractional decimal digits) 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 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 RLaB can find the n-roots of unity by solving the polynomial equation ${\displaystyle x^{n}-1=0.}$ 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 ## RPL ≪ → r n ≪ { } 0 n 1 - FOR q 'r^INV(n)*e^(2*i*π*q/n)' →NUM + NEXT ≫ ≫ 'ROOTS' STO  1 3 ROOTS  Output: 1: { (1,0) (-0.5,0.866025403784) (-0.5,-0.866025403784) }  ## Ruby 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)]  ## Rust Here we demonstrate initialization from polar complex coordinate, radius 1, e^πi/n, and raising the resulting complex number to the power 2k for k in 0..n-1, which generates approximate roots (see the Mathematica answer for a nice display of exact vs approximate). This code will require adding the num crate to one's rust project, typically in Cargo.toml [dependencies] \n num="0.2.0"; use num::Complex; fn main() { let n = 8; let z = Complex::from_polar(&1.0,&(1.0*std::f64::consts::PI/n as f64)); for k in 0..=n-1 { println!("e^{:2}πi/{} ≈ {:>14.3}",2*k,n,z.powf(2.0*k as f64)); } }  e^ 0πi/8 ≈ 1.000+0.000i e^ 2πi/8 ≈ 0.707+0.707i e^ 4πi/8 ≈ 0.000+1.000i e^ 6πi/8 ≈ -0.707+0.707i e^ 8πi/8 ≈ -1.000+0.000i e^10πi/8 ≈ -0.707-0.707i e^12πi/8 ≈ -0.000-1.000i e^14πi/8 ≈ 0.707-0.707i  ## Scala 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  ## Scheme (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))  ## Seed7 $ 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

## Sidef

Translation of: Raku
func roots_of_unity(n) {
n.of { |j|
exp(2i * Num.pi / n * j)
}
}

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


## Sparkling

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);
});

## Stata

n=7
exp(2i*pi()/n*(0::n-1))
1
+-----------------------------+
1 |                          1  |
2 |   .623489802 + .781831482i  |
3 |  -.222520934 + .974927912i  |
4 |  -.900968868 + .433883739i  |
5 |  -.900968868 - .433883739i  |
6 |  -.222520934 - .974927912i  |
7 |   .623489802 - .781831482i  |
+-----------------------------+


## Tcl

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
}


## VBA

Public Sub roots_of_unity()
For n = 2 To 9
Debug.Print n; "th roots of 1:"
For r00t = 0 To n - 1
Debug.Print "   Root "; r00t & ": "; WorksheetFunction.Complex(Cos(2 * WorksheetFunction.Pi() * r00t / n), _
Sin(2 * WorksheetFunction.Pi() * r00t / n))
Next r00t
Debug.Print
Next n
End Sub

Output:
 2 th roots of 1:
Root 0: 1
Root 1: -1+1.22460635382238E-16i

3 th roots of 1:
Root 0: 1
Root 1: -0.5+0.866025403784439i
Root 2: -0.5-0.866025403784438i

4 th roots of 1:
Root 0: 1
Root 1: 6.12303176911189E-17+i
Root 2: -1+1.22460635382238E-16i
Root 3: -1.83690953073357E-16-i

5 th roots of 1:
Root 0: 1
Root 1: 0.309016994374947+0.951056516295154i
Root 2: -0.809016994374947+0.587785252292473i
Root 3: -0.809016994374948-0.587785252292473i
Root 4: 0.309016994374947-0.951056516295154i

6 th roots of 1:
Root 0: 1
Root 1: 0.5+0.866025403784439i
Root 2: -0.5+0.866025403784439i
Root 3: -1+1.22460635382238E-16i
Root 4: -0.5-0.866025403784438i
Root 5: 0.5-0.866025403784439i

7 th roots of 1:
Root 0: 1
Root 1: 0.623489801858734+0.78183148246803i
Root 2: -0.222520933956314+0.974927912181824i
Root 3: -0.900968867902419+0.433883739117558i
Root 4: -0.900968867902419-0.433883739117558i
Root 5: -0.222520933956315-0.974927912181824i
Root 6: 0.623489801858733-0.78183148246803i

8 th roots of 1:
Root 0: 1
Root 1: 0.707106781186548+0.707106781186547i
Root 2: 6.12303176911189E-17+i
Root 3: -0.707106781186547+0.707106781186548i
Root 4: -1+1.22460635382238E-16i
Root 5: -0.707106781186548-0.707106781186547i
Root 6: -1.83690953073357E-16-i
Root 7: 0.707106781186547-0.707106781186548i

9 th roots of 1:
Root 0: 1
Root 1: 0.766044443118978+0.642787609686539i
Root 2: 0.17364817766693+0.984807753012208i
Root 3: -0.5+0.866025403784439i
Root 4: -0.939692620785908+0.342020143325669i
Root 5: -0.939692620785908-0.342020143325669i
Root 6: -0.5-0.866025403784438i
Root 7: 0.17364817766693-0.984807753012208i
Root 8: 0.766044443118978-0.64278760968654i

## V (Vlang)

Translation of: Ring
import math

for n in 2..6 {
print("${n}: ") for root in 0..n { real := math.cos(2 * 3.14 * root/n) imag := math.sin(2 * 3.14 * root/n) print("${real:.4} ${imag:.4}i") if root != n - 1 {print(", ")} } println("") }  ## Wren Translation of: Go Library: Wren-complex Library: Wren-fmt import "./complex" for Complex import "./fmt" for Fmt var roots = Fn.new { |n| var r = List.filled(n, null) for (i in 0...n) r[i] = Complex.fromPolar(1, 2 * Num.pi * i / n) return r } for (n in 2..5) { Fmt.print("$d roots of 1:", n)
for (r in roots.call(n)) Fmt.print("  \$ 0.14z", r)
}

Output:
2 roots of 1:
1.00000000000000 + 0.00000000000000i
-1.00000000000000 + 0.00000000000000i
3 roots of 1:
1.00000000000000 + 0.00000000000000i
-0.50000000000000 + 0.86602540378444i
-0.50000000000000 - 0.86602540378444i
4 roots of 1:
1.00000000000000 + 0.00000000000000i
0.00000000000000 + 1.00000000000000i
-1.00000000000000 + 0.00000000000000i
-0.00000000000000 - 1.00000000000000i
5 roots of 1:
1.00000000000000 + 0.00000000000000i
0.30901699437495 + 0.95105651629515i
-0.80901699437495 + 0.58778525229247i
-0.80901699437495 - 0.58778525229247i
0.30901699437495 - 0.95105651629515i


## zkl

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
`