9231 · 2.5
Complex numbers
We'll use De Moivre's theorem to find all the solutions to equations like $z^n=w$, which appear as beautifully symmetric points on a circle. This technique also provides a powerful shortcut for summing certain trigonometric series.
Need to know
What you need to know
- There are always $n$ distinct $n$-th roots.
- The root corresponding to $k=0$ is called the principal root.
- All roots have the same modulus, $r^{1/n}$.
- Geometrically, the roots form the vertices of a regular $n$-gon inscribed in a circle of radius $r^{1/n}$ centred at the origin. The angle between successive roots is $2\pi/n$.
Explanation
Unlocking Hidden Symmetries
- Convert the complex number into its exponential or polar form, $r e^{i\theta}$.
- Generalise the argument by adding $2k\pi$, giving $r e^{i(\theta + 2k\pi)}$.
- Apply the $n$-th root to find the general form of the roots: $r^{1/n} e^{i(\theta + 2k\pi)/n}$.
- Find the distinct roots by substituting integer values for $k$ (e.g., $k=0, 1,..., n-1$) and adjust arguments to the principal range if required.