ALGEBRA TWO CHAPTER FIVE QUADRATIC FUNCTIONS 5.4 Complex Numbers

LEARNING GOALS Solve Quadratic Equations with Complex Solutions and perform operations with Complex Numbers. Apply Complex Numbers to Fractal Geometry.

What is the solution of?:

Imaginary Unit Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can! The imaginary unit is i. i = It is used to write the square root of a negative number.

Property of the square root of negative numbers If r is a positive real number, then Examples:

*For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:

Examples

Complex Numbers A complex number has a real part & an imaginary part. Standard form is: Real part Imaginary part Example: 5 + 4i

The Complex plane Real Axis Imaginary Axis

Graphing in the complex plane

Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex: Ex: Ex:

Multiplying Treat the i ’s like variables, then change any that are not to the first power Ex: Ex:

Dividing Complex Numbers 1) multiply the numerator and denominator by the complex conjugate of the denominator (conjugate: change sign between the terms ex: conjugate of x-1 is x+1)

Dividing Complex Numbers

Absolute Value of a Complex Number The distance the complex number is from the origin on the complex plane. If you have a complex number the absolute value can be found using:

Examples Which of these 2 complex numbers is closest to the origin? -2 + 5i

ASSIGNMENT WRITE pg. 277-279. #19, 21, 29, 31, 39, 41, 49, 51, 57, 59, & 65