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Boundless Algebra
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Chapter 14

Complex Numbers and Polar Coordinates

Book Version 13
By Boundless
Boundless Algebra
Algebra
by Boundless
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Section 1
The Polar Coordinate System
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Introduction to the Polar Coordinate System

The polar coordinate system is an alternate coordinate system where the two variables are $r$ and $\theta$, instead of $x$ and $y$.

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Converting Between Polar and Cartesian Coordinates

Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.

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Conics in Polar Coordinates

Polar coordinates allow conic sections to be expressed in an elegant way.

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Other Curves in Polar Coordinates

Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.

Section 2
Complex Numbers
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Introduction to Complex Numbers

A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Addition and Subtraction of Complex Numbers

Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately. 

Multiplication of Complex Numbers

Complex numbers can be multiplied using the FOIL algorithm.

Complex Numbers and the Binomial Theorem

Powers of complex numbers can be computed with the the help of the binomial theorem.

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Complex Conjugates

The complex conjugate of the number $a+bi$ is $a-bi$. Two complex conjugates of each other multiply to be a real number with geometric significance. 

Division of Complex Numbers

Division of complex numbers is accomplished by multiplying by the multiplicative inverse. The multiplicative inverse of $z$ is $\frac{\overline{z}}{\abs{z}^2}.$

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Complex Numbers in Polar Coordinates

Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$. This leads to a way to visualize multiplying and dividing complex numbers geometrically.

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Boundless Algebra by Boundless
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Complex Numbers and Polar Coordinates
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