Chapter 4 Section 8 Complex Numbers Objective: I will be able to identify, graph, and perform operations with complex numbers I will be able to find complex number solutions of quadratic equations.

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 ¡. ¡= It is used to write the square root of a negative number. Remember: i 2 = -1

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

SOLVING QUADRATIC EQUATIONS

SOLVING QUADRATIC EQUATIONS Solve 2x = -10

SOLVING QUADRATIC EQUATIONS Solve -1/2(x + 1) 2 = 5

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

Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals (no fractions) pi, e Imaginary i, 2i, -3-7i, etc.

The Complex plane Imaginary Axis Real Axis

Graphing in the complex plane imaginary real

Adding and Subtracting Imaginary Numbers Combine like terms! Treat i like a variable and simplify: (6 + 7i) + (4 – 2i) – 2i? i Be sure to write it in standard form a+bi!

Adding and Subtracting Ex: First, add or subtract the real parts. Then, add or subtract the imaginary parts.

Multiplying Ex: Ex: Treat the i’s like variables, But remember to change any that are not to the first power Ex: (6 + 3i)(6 – 3i)

Complex Conjugate a + bi and a – biExample 6 + 3i and 6 – 3i

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 1. 2.

Exit Ticket Simplify (4 + 6i) + (11 – 2i) 4. (-9 – i) – (-2 – 3i) 5. (8 – 2i)(5 + 2i) 6.Find absolute value of 5 + 2i 7.Solve x = 3