MAT 3730 Complex Variables Section 1.3 Vectors and Polar Forms

Preview More on Vector Representation of complex numbers Triangle Inequalities Polar form of complex numbers (Need to begin 1.4,may be?)

Recall We can identify z as the position vector

Recall We can identify z as the position vector

Triangle Inequality

Geometric Proof of the 1 st Form

(Classwork) Algebraic Proof of the 1 st Form

Geometric Proof of the 2 st Form

2nd Form from the 1 st Form

Polar Form of Complex Numbers

Recall We can identify z as the ordered pair (x,y).

Polar Form of Complex Numbers We can also use the polar coordinate

Polar Form of Complex Numbers We can also use the polar coordinate Note that is undefined if z=0.

Polar Form of Complex Numbers We can also use the polar coordinate

Example 1

Problems 1. 2.

The argument of a complex number z is not unique.  is called the Principal Argument if Notation: Property of Arguments

Example 1 (Remedy)

Example 1

Polar Form of Complex Numbers We can also use the polar coordinate

Product of Complex Numbers in Polar Form

Next Class Read Section 1.4 We will introduce the Complex Exponential and Euler Formula Review Maclaurin Series (Stewart section 12.10?)