Using De Moivre's Theorem to find powers of complex numbers in polar form, z=reiθ, so that zn=rneinθ=rncis(nθ),
Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
And since reiθ=rcisθ:
De Moivre's Theorem can also be used to find the nth roots of complex numbers:
or equivalently
However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have
Note that k stops at n−1 since when k=n we have
Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:
Exercises checked off
Using De Moivre's Theorem to find powers of complex numbers in polar form, z=reiθ, so that zn=rneinθ=rncis(nθ),
Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
And since reiθ=rcisθ:
De Moivre's Theorem can also be used to find the nth roots of complex numbers:
or equivalently
However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have
Note that k stops at n−1 since when k=n we have
Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:
Exercises checked off