Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit defined by . Operations with complex numbers include addition, subtraction, multiplication, and division, and results are typically written in standard a + bi form.

• Addition/Subtraction: Combine real parts and imaginary parts separately.

• Multiplication: Use distributive property and the fact that .

• Division: Multiply numerator and denominator by the conjugate of the denominator to rationalize.

Example: Perform the indicated operation and write the result in standard a + bi form.

• Example 1:

• Example 2: Recall for . (cannot combine further; this is in a + bi form with )

Standard Form: Always express complex numbers as .

Quadratic Equations

Solving Quadratic Equations with Complex Solutions

A quadratic equation is an equation of the form . When the discriminant is negative, the solutions are complex numbers.

• Quadratic Formula:

• Complex Solutions: If , then is imaginary, and solutions are complex conjugates.

Example: Solve .

• Discriminant:

• Solutions:

Summary Table: Complex Number Operations

Key Concepts

• Imaginary Unit: is defined such that .

• Complex Conjugate: For , the conjugate is .

• Standard Form: Always write complex numbers as .

• Quadratic Equations: Use the quadratic formula to find complex solutions when the discriminant is negative.

Additional info: Some context and examples have been inferred based on standard Precalculus curriculum and the fragmentary nature of the provided material.