מספרים מרוכבים — הכלים החשובים

function fromPolar(radius, angle) {
  return { re: radius*Math.cos(angle), im: radius*Math.sin(angle) };
}

function nthRoots(radius, angle, n) {
  if (radius < 0 || !Number.isInteger(n) || n < 1) {
    throw new RangeError("nonnegative radius and positive integer n required");
  }
  const rootRadius = radius ** (1/n);
  return Array.from({ length: n }, (_, k) =>
    fromPolar(rootRadius, (angle + 2*Math.PI*k)/n)
  );
}

function multiply(z, w) {
  return { re: z.re*w.re-z.im*w.im, im: z.re*w.im+z.im*w.re };
}

function integerPower(z, n) {
  let result = { re: 1, im: 0 };
  for (let k = 0; k < n; k += 1) result = multiply(result, z);
  return result;
}

const roots = nthRoots(8, -Math.PI/2, 3);
console.table(roots.map((root) => ({ ...root, cube: integerPower(root, 3) })));
console.log({
  sumRe: roots.reduce((sum, root) => sum + root.re, 0),
  sumIm: roots.reduce((sum, root) => sum + root.im, 0),
});

from cmath import phase, rect
from math import pi

def nth_roots(z, n):
    if not isinstance(n, int) or n < 1:
        raise ValueError("positive integer n required")
    radius = abs(z)**(1/n)
    angle = phase(z)
    return [rect(radius, (angle + 2*pi*k)/n) for k in range(n)]

roots = nth_roots(-8j, 3)
for root in roots:
    print(root, root**3)
print("sum:", sum(roots))