Warm Up Simplify Solve SAT Review question Dana walks from home to school at a rate of 5 mph. It takes her 2 hours longer to walk home from school than it did to walk to school. If her total walking time to and from school was 8 hours, what was Dana’s rate of speed walking home from school? a) 3 b) 4 c) 5 d) 8 e) 15

Section 5.4 Complex Numbers Essential Question What is an imaginary number?

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

Adding and Subtracting complex numbers 1 Adding and Subtracting complex numbers 1. add or subtract the real parts 2. add or subtract the imaginary parts Ex: Ex:

You try!

Multiplying 1. Treat the i’s like a variable 2 Multiplying 1. Treat the i’s like a variable 2. Change any that are not to the first power Ex: Ex:

You try!

Conjugates The conjugate of a complex number has the same real part and the opposite imaginary part Ex. Find the conjugate of 5 + 3i 5 – 3i Ex. Find the conjugate of 3 – 2i 3 + 2i

Imaginary numbers in the denominator i’s cannot be in the denominator (like radicals) To get rid of the i’s, multiply numerator and denominator by the conjugate If there is only an imaginary part in the denominator, multiply by the same imaginary number

Example

You try!

Assignment Pg. 277: #17-21(odd), 29-33(odd), 37-41(odd), 47-51(odd), 57-61(odd), 65-69(odd), 92

Assessment Concept circles