De Moivre's Theorem & Complex Roots in 2026
The Fundamental Theorem of Algebra guarantees that every non-zero complex number $z = r e^{i\theta}$ possesses exactly $n$ distinct $n$-th roots. By De Moivre's Theorem, these roots form the vertices of a regular $n$-sided polygon centered at the origin of the Argand plane.
\text{Principal Root: } w_0 = \sqrt[n]{r} \, e^{i\theta / n}
\text{General Root: } w_k = \sqrt[n]{r} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right] \quad (k = 0, 1, \dots, n-1)