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Complex Numbers, Polar Equations, and Parametric Equations

This page is meant to serve as a quick overview of complex numbers, polar equations, and parametric equations.

**The Imaginary Unit i**

*i* = √(-1) or *i²* = -1

For positive real numbers *a*, √(-a) = *i*√a.

The conjugate of a + bi is a - bi.

Addition of Complex Numbers

(*a* + *bi*) + (*c* + *di*) = (*a* + *c*) + (*b* + *d*)*i*

Subtraction of Complex Numbers

(*a* + *bi*) - (*c* + *di*) = (*a* - *c*) + (*b* - *d*)*i*

Multiplication of Complex Numbers

For (*a* + *bi*)(*c* + *di*), simply foil it out as you normally would if *i* were a variable and then replace *i²* with -1.

Division of Complex Numbers

For (*a* + *bi*)/(*c* + *di*), multiply the numberator and denominator by the conjugate of the denominator and then simplify.

**Trigonometric (Polar) Form of Complex Numbers**

Trigonometric or polar form is writen as *r* (cos Θ + *i*sin Θ) or *r* cis Θ. If the complex number *x* + *yi* corresponds to the vector with direction angle Θ and magnitude *r*, then the following are the translations.

x = r cos Θ |
y = r sin Θ |

r = √(x² + y²) |
tan Θ = y / x, if x ≠ 0. |

**The Product and Quotient Theorems**

Product Theorem: [*r _{1}* (cos Θ

Quotient Theorem: [*r _{1}* (cos Θ

**De Moivre's Theorem**

[*r* (cos Θ + *i*sin Θ)]^{n} = *r*^{n}(cos *n*Θ + *sin nΘ)*

*n*th Root Theorem

So long that n is any positive integer and r is a positive real number, then the nonzero complex number has exactly n distinct *n*th roots, given by

**Polar Graphs**

The polar coordinates dtermine a point by locating it Θ degrees from the polar axis (the positive *x*-axis) and *r* units from the origin. Polar equations are graphed in teh same way as Cartesian equations, by point plotting or with a graphing calculator.

**PPlane Curve**

A plane curve is a set of points (*x*,*y*) such that *x* = *f*(*t*), *y* = *g*(*t*), and *f* and *g* are both defined on an interval *I*. The equations *x* = *f*(*t*) and *y* = *g*(*t*) are paramentric equations with parameter *t*.