AHL 1.14 (HL)—Complex roots and De Moivre's theorem

Use De Moivre to find powers and roots.

De Moivre gives (r cisθ)ⁿ=rⁿ cis(nθ); roots share magnitude r^(1/n) and differ by 2π/n in argument.

Worked example
The cube roots of 8 include arguments 0, 120° and 240°, all with modulus 2.

Worked example
Why are there n roots? rotating by 2π/n gives distinct solutions.

Common boundary
Taking only the principal root misses the other roots.

Two required extensions: if a polynomial $P$ has real coefficients and $P(c)=0$, then $P(\overline c)=\overline{P(c)}=0$, so non-real roots occur in conjugate pairs. For positive integers, prove De Moivre by induction: the $n=1$ case is immediate; assuming $(\mathrm{cis}\,\theta)^k=\mathrm{cis}(k\theta)$, multiply by $\mathrm{cis}\,\theta$ and use angle addition to obtain $\mathrm{cis}((k+1)\theta)$. This completes the inductive step.