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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 = -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 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 + isin 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 = √( + ) tan Θ = y / x, if x ≠ 0.

The Product and Quotient Theorems

Product Theorem: [r1 (cos Θ1 + isin Θ1)] * [r2 (cos Θ2 + isin Θ2)] = r1r2[cos(Θ12) + isin(Θ12)]

Quotient Theorem: [r1 (cos Θ1 + isin Θ1)] / [r2 (cos Θ2 + isin Θ2)] = [r1/r2][cos(Θ12) + isin(Θ12)] if r2 cis Θ2 ≠ 0.

De Moivre's Theorem

[r (cos Θ + isin Θ)]n = rn(cos nΘ + sin nΘ)

nth 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 nth 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.