Polar Form of Complex Numbers

Introduction

Complex numbers can be represented in both rectangular (Cartesian) form and polar form. The polar form is especially useful in trigonometry and when multiplying or dividing complex numbers. The polar form expresses a complex number in terms of its magnitude (modulus) and angle (argument).

• Rectangular form:

• Polar form: or

Key Concepts

• Magnitude (r):

• Argument (\theta): (adjust for quadrant)

• Conversion: To convert to polar form, find and , then write

Practice Problems

Problem 1

Express the complex number in polar form.

• Step 1: Find the magnitude:

• Step 2: Find the argument:

• Polar form:

Problem 2

Express the complex number in polar form.

• Step 1:

• Step 2: (or in )

• Polar form:

Problem 3

Express the complex number in polar form.

• Step 1:

• Step 2: , but since the point is in the second quadrant,

• Polar form:

Summary Table: Rectangular to Polar Form

Additional info: The polar form is essential for understanding trigonometric applications of complex numbers, such as De Moivre's Theorem and operations involving multiplication and division of complex numbers.