All the n-th roots of a complex number

A complex number is two real numbers travelling together, so it cannot sit on a number line; it needs a plane. Rectangular form $a + bi$ says "go $a$ across and $b$ up". Polar form describes the SAME point as "face this direction, walk this far". The reason to bother: when complex numbers are multiplied, their lengths multiply and their directions add. Rectangular form hides that completely; polar form makes it visible.

The arrow to $z$ is the long side of a right-angled triangle whose other sides are $\left| a \right|$ across and $\left| b \right|$ up. Pythagoras gives its length. We square $a$ and $b$ first, which also throws away their signs: a distance cannot care whether we went left or right.

The direction is an angle $\theta$ measured from the positive real axis, anticlockwise counted as positive. One reading is not enough to fix it: the cosine alone cannot tell up from down, and the sine alone cannot tell left from right. Two readings together leave exactly one direction.

From the triangle, $a = r \cos \theta$ and $b = r \sin \theta$: cosine is "across over long side" and sine is "up over long side". Putting those into $a + bi$ and taking $r$ out as a common factor gives the polar form. We then multiply it back out, because a conversion that cannot be reversed has not been checked.

A root of $z$ is a number $w$ with $w^{3} = z$. We do not know $w$, but we know what raising to a power does in polar form: it raises the length to that power and multiplies the angle. So we give the unknown $w$ an unknown length and an unknown angle, and let De Moivre's theorem turn one hard complex equation into two easy real ones.

$\rho$ is a length, so it is a positive real number, and a positive real number has exactly ONE positive real cube root. There is no choice to make here. All the variety among the roots will come from the angle.

Divide $3\varphi = \theta + 2k\pi$ by $3$. Each whole number $k$ gives an angle, but they are not all new: raising $k$ by $3$ adds exactly one full turn to $\varphi$ and returns to a direction already found. So $k = 0, 1, 2$ gives every root once, and neighbours are $\frac{2\pi}{3}$ apart.

Each root is the common length in one of the directions just found. Where a direction is one of the special angles the root comes out exactly; otherwise its polar form IS the exact answer and the decimals are only a description of it.

The method could in principle produce numbers that merely look plausible. The definition of a root is the test that cannot be fooled: raise each one to the power $3$ and see whether $z$ comes back. All $3$ must pass, including the ones that look nothing like each other.