Roots of unity
![Task](http://static.miraheze.org/rosettacodewiki/thumb/b/ba/Rcode-button-task-crushed.png/64px-Rcode-button-task-crushed.png)
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 n-th roots of unity.
ALGOL 68
FOR root FROM 2 TO 10 DO printf(($g(4)$,root)); FOR n TO root -1 DO printf(($xg(5,3)g(5,3)"i"$,complex exp( 0 I 2*pi*n/root))) OD; printf($l$) OD
Output:
+2 -1.00+.000i +3 -.500+.866i -.500-.866i +4 +.000+1.00i -1.00+.000i -.000-1.00i +5 +.309+.951i -.809+.588i -.809-.588i +.309-.951i +6 +.500+.866i -.500+.866i -1.00+.000i -.500-.866i +.500-.866i +7 +.623+.782i -.223+.975i -.901+.434i -.901-.434i -.223-.975i +.623-.782i +8 +.707+.707i +.000+1.00i -.707+.707i -1.00+.000i -.707-.707i -.000-1.00i +.707-.707i +9 +.766+.643i +.174+.985i -.500+.866i -.940+.342i -.940-.342i -.500-.866i +.174-.985i +.766-.643i +10 +.809+.588i +.309+.951i -.309+.951i -.809+.588i -1.00+.000i -.809-.588i -.309-.951i +.309-.951i +.809-.588i
J
rou=: [: ^ i. * (o. 0j2) % ] 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
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-15). Instead, they are simply represented as 0. To remove those checks (for very high n's), remove both if statements.
public static void unity(int 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-15) 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-15) imag = 0.0; //get rid of annoying sci notation System.out.print(real + " + " + imag + "i\t"); //tab-separated answers } }
Python
Interpreter: Python 2.5
Function nthroots()
returns all n-th roots of unity.
from cmath import exp, pi def nthroots(n): return [exp(k * 2j * pi / n) for k in range(1, n + 1)]
Example:
>>> f = nthroots >>> for n in range(1, 4): ... print map(lambda c: ("%.3f " + ("+" if c.imag > 0 else "-") + " %.3gj") % (c.real, abs(c.imag)), f(n)) ['1.000 - 2.45e-016j'] ['-1.000 + 1.22e-016j', '1.000 - 2.45e-016j'] ['-0.500 + 0.866j', '-0.500 - 0.866j', '1.000 - 2.45e-016j']