De Moivre's Theorem, Powers, Roots & Loci
De Moivre's Theorem
Use Complex Numbers | Polar Coordinates
De Moivre's theorem is a fundamental result in complex numbers that connects complex numbers and trigonometry. It provides a formula for computing powers of complex numbers expressed in polar form.
The theorem states that for any complex number $z = r(\cos\theta + i\sin\theta)$ (where $r$ is the modulus and $\theta$ is the argument) and any integer $n$:
$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$
Using Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, the complex number can be written in exponential form as $z = re^{i\theta}$. In this form, De Moivre's theorem becomes:
$(re^{i\theta})^n = r^n e^{in\theta}$
This theorem is extremely useful for calculating powers and roots of complex numbers efficiently.
Powers of Complex Numbers ($z^n$)
Calculate $z^n = (a+ib)^n$ or $[r(\cos\theta + i\sin\theta)]^n$
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Roots of Complex Numbers ($z^{1/n}$)
Calculate $n$-th roots of $z = a+ib$ or $r(\cos\theta + i\sin\theta)$
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Exponential Form of Complex Numbers
The exponential form of a complex number $z$ is given by Euler's formula, which states $e^{i\theta} = \cos\theta + i\sin\theta$.
If a complex number has modulus $r = |z|$ and argument $\theta = \arg(z)$ (in radians), its exponential form is:
z = re^{i\theta}
This form is particularly convenient for multiplication, division, powers, and roots of complex numbers, as these operations follow the standard rules of exponents.
Convert Complex Numbers to/from Exponential Form
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Loci in the Complex Plane
A locus (plural: loci) is a set of points whose location is determined by a specific geometric condition. In the complex plane, these conditions often involve the modulus (distance) or argument (angle) of complex numbers.
Understanding loci helps visualize relationships and regions in the complex plane. Common examples include circles, perpendicular bisectors of line segments, and rays.
Describe and Visualize Common Loci
Defines a circle with center $z_0 = a_0 + ib_0$ and radius $R$.
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