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Section 3.2 Complex Numbers

• Complex Numbers: Represented as ( z = x + yi ), where ( i = \sqrt{-1} ).
• Graphing Complex Numbers: On a plane; horizontal axis is the real axis, and vertical axis is the imaginary axis.

Absolute Value and Argument

• Absolute Value: ( |z| = r = \sqrt{x^2 + y^2} ) (distance from origin).
• Argument: ( arg(z) ) is the angle ( \theta ) from the positive real axis to ( z ).
  • ( x = r \cos(\theta), y = r \sin(\theta) )

Trigonometric Form of Complex Numbers

• Trigonometric Form: ( z = r(\cos(\theta) + i \sin(\theta)) = r \text{cis} \theta )
• Conversion Relations:
  • Back to standard form: ( r = \sqrt{x^2 + y^2}, \cos(\theta) = \frac{x}{r}, \sin(\theta) = \frac{y}{r} )

Product and Quotient Theorems

• Product Theorem: ( (r1 \text{ cis } \theta1)(r2 \text{ cis } \theta2) = r1r2 \text{ cis } (\theta1 + \theta2) )
• Quotient Theorem: ( \frac{r1 \text{ cis } \theta1}{r2 \text{ cis } \theta2} = \frac{r1}{r2} \text{ cis } (\theta1 - \theta2) )

De Moivre's Theorem

• De Moivre's Theorem: ( [r(\cos(\theta) + i \sin(\theta))]^n = r^n (\cos(n\theta) + i \sin(n\theta)) )

Roots of Complex Numbers

• Root Theorem: For a positive integer ( n ), the complex number ( a + bi ) has ( n ) distinct ( n^{th} ) roots:
  [ \sqrt[n]{r} \text{ cis } \left( \frac{\theta + 360°k}{n} \right), k = 0, 1, 2, …, n-1 ]

Exercises Overview

1. Convert various complex numbers into trigonometric form.
2. Perform operations such as product, quotient, and power on complex numbers.
3. Find roots of complex numbers in both rectangular and trigonometric forms.