Powers of Complex Numbers
Using De Moivre's Theorem to find powers of complex numbers in polar form, âz=reiθ, so that âzn=rneinθ=rncis(nθ), and applying the argument rule âarg(zw)=arg(z)+arg(w).
Powers of complex numbers
Now that we know how to represent complex numbers in the form âreiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
And since âreiθ=rcisθ:
Roots of complex numbers
De Moivre's Theorem can also be used to find the ânthâ roots of complex numbers:
or equivalently
However, since âcisθ=cis(θ+2kĎ)â for any âkâZ, then we actually have
Note that âkâ stops at ânâ1â since when âk=nâ we have
Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:
Skills covered
Exercises checked off