Complex Numbers and Their Operations

Introduction to Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit i, where i is defined as the square root of -1. Any number of the form a + bi (where a and b are real numbers) is called a complex number.

• Imaginary Unit:

• Standard Form:

• Pure Imaginary Number: A number of the form

Example:

Writing Square Roots of Negative Numbers as Products of a Number and i

To express as a product of a real number and i:

• Use the property

• Example:

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules and the property .

• Addition/Subtraction: Combine like terms:

• Multiplication: Use distributive property and

• Division: Multiply the numerator and denominator by the conjugate of the denominator

Example:

Products and Quotients Involving i

• Remember

• To simplify , multiply the numerator and denominator by the conjugate

Example:

Solving Quadratic Equations

Zero-Factor Property

The zero-factor property states that if , then or . This property is used to solve quadratic equations that can be factored.

• Example: Factor: Solutions: or

Square Root Property

If , then .

• Example: Solutions: or

Quadratic Formula

The quadratic formula solves any quadratic equation of the form :

• Discriminant: determines the nature of the roots:

  • If , two real solutions

  • If , one real solution

  • If , two complex solutions

Example: Solve Discriminant: Solutions:

Factoring and Solving Cubic Equations

Some cubic equations can be solved by factoring or by using the quadratic formula after factoring out a common term.

• Example: Factor: Further factor: Solutions:

Summary Table: Properties of Complex Numbers

Additional info:

• These notes cover key concepts from College Algebra, specifically from chapters on complex numbers and quadratic equations.

• Practice problems in the original file reinforce understanding of these concepts.