Drill 81 Solve each equation or inequality Drill

Drill #81: Solve each equation or inequality

Drill #83: Simplify each expression

5 -9 Complex Numbers Objective: To simplify square roots containing negative numbers and to add, subtract, and multiply complex numbers

(1. ) i Definition: i is called the imaginary unit. What is the value of I squared?

(2. ) Pure Imaginary Numbers Definition: For any positive number b, where i is the imaginary unit, and bi is called a pure imaginary number. Example:

Evaluating the Square Root of Negative Numbers* To find the square root of negative numbers: 1. First separate the negative 2. Evaluate each root separately and multiply

More Examples Ex 1. Ex 2.

Powers of I *

Finding Powers of i* Powers of i are cyclical. They repeat after To find : 1. Divide n by 4 and keep only the remainder r 2. , where r is the remainder of n/4 Note:

Example (Powers of i) Find the following: Ex 1. Ex 2. Ex 3.

(3. ) Complex Numbers Definition: A number in the form of a + bi where a and b are real numbers and i is the imaginary unit. a is called the real part. b is called the imaginary part.

Adding Complex Numbers* To add complex numbers: 1. add the real parts together (this is the real part of sum) 2. add imaginary parts together (this is the imaginary part of the solution). (a + bi) + (c + di) = (a + c) + (b + d)i Ex: (5 + 6 i) + (2 + 3 i) = 7 + 9 i

Adding Complex Numbers Examples: Ex 1. (2 + 9 i) + (3 + 4 i) Ex 2. (5 + 6 i) - (2 + 3 i) Ex 3.

Multiplying Complex Numbers* Definition: To multiply imaginary numbers you need to FOIL. (a + bi)(c + di) = = ac (first) + adi (outside) + bci (inside) + bd (last) = (ac - bd) + (ad + bd)i Ex: (2 + 3 i)(4 + 5 i) = 2(4) + 2(5 i) + 3 i(4) + 3 i(5 i) = 8 + 10 i + 12 i – 15 = -7 + 22 i

Multiplying Complex Numbers Ex 1. (2 + 3 i)( 1 + 4 i) Ex 2. (6 + 2 i)( 3 – 2 i) Ex 3. (3 – 5 i)(3 + 5 i)

(4. ) Complex Conjugates Definition: Numbers of the form a + bi and a – bi are called complex conjugates. The product of complex conjugates is: Example: 3 + 2 i and 3 – 2 i are complex conj.

The Product of Complex Conjugates*

Dividing by Complex Numbers* Rationalizing Complex Denominators To rationalize a complex denominator you need to multiply the numerator and denominator by the complex conjugate. Example: Simplify

Solving 2 nd Equations* (no 1 st degree term) To solve equations: 1. Isolate the square term. 2. Take the (+/-) square root of both sides. Example:

(5. ) Equal Complex Numbers Definition: a + bi = c + di if and only if a = c and b = d. The real parts must be equal and the imaginary parts must be equal.

Equal Complex Numbers* Find values of x and y for which each equation is true: