Products and Roots of Complex Numbers

Multiplication and Division of Complex Numbers in Trigonometric Form

Complex numbers can be expressed in trigonometric (polar) form as:

• Trigonometric Form: , where is the modulus and is the argument.

Multiplying and dividing complex numbers in this form uses the following properties:

• Product: If and , then:

• Quotient:

Example: Multiply and :

Roots of Complex Numbers in Trigonometric Form

The th roots of a complex number are given by:

, where

• Each value of gives a distinct th root, spaced evenly around the unit circle.

Example: Find the cube roots of :

, for

• For :

• For :

• For :

Additional info: These properties are essential for solving equations involving complex numbers and for understanding their geometric representation in the complex plane.