DeMoivre's Theorem (powers and roots of complex numbers)
Theory
Raise the modulus to the power and multiply the argument by \(n\).
Reversing it gives the \(n\)th roots: a nonzero complex number has exactly \(n\) of them, equally spaced by \(\dfrac{360^\circ}{n}\) around a circle of radius \(r^{1/n}\).
Powers and roots in polar form:
How to use DeMoivre's Theorem
- Convert the number to polar form.
- Power: raise \(r\) to the power, multiply \(\theta\) by \(n\).
- Roots: take \(r^{1/n}\); use arguments \(\dfrac{\theta+360^\circ k}{n}\) for \(k=0,\dots,n-1\).
- Convert back to rectangular form if required.
Raise the modulus to the power and multiply the argument by it.
| \(=\) | \(2^4\,\text{cis}\,(4\cdot 30^\circ)\) | |
| \(=\) | \(16\,\text{cis}\,120^\circ\) |
Convert to polar (\(r=\sqrt2,\ \theta=45^\circ\)), then apply DeMoivre.
| \(=\) | \((\sqrt2)^6\,\text{cis}\,(6\cdot 45^\circ)\) | |
| \(=\) | \(8\,\text{cis}\,270^\circ\) | |
| \(=\) | \(-8i\) |
A number has exactly \(n\) distinct \(n\)th roots, equally spaced around a circle.
| \(n\) | \(=\) | \(3\ \text{roots}\) |
| \(\text{spacing}\) | \(=\) | \(\dfrac{360^\circ}{3}=120^\circ\) |
Take \(\sqrt9=3\) for the modulus; the arguments are \(\dfrac{60^\circ+360^\circ k}{2}\) for \(k=0,1\).
| \(k=0\) | \(:\) | \(3\,\text{cis}\,30^\circ\) |
| \(k=1\) | \(:\) | \(3\,\text{cis}\,210^\circ\) |
Common pitfalls
Frequently asked questions
What is DeMoivre's Theorem?
\((r\,\text{cis}\,\theta)^n=r^n\,\text{cis}(n\theta)\): to raise a complex number to a power, raise the modulus and multiply the argument by \(n\).
How many nth roots does a complex number have?
Exactly \(n\), equally spaced by \(\dfrac{360^\circ}{n}\) around a circle of radius \(r^{1/n}\).
How do you find the nth roots of a complex number?
Take \(r^{1/n}\) as the modulus and arguments \(\dfrac{\theta+360^\circ k}{n}\) for \(k=0,1,\dots,n-1\).
Why is polar form needed for powers and roots?
Because DeMoivre's Theorem acts on the modulus and argument directly, which only appear in polar form.