5.4 Complex Numbers

Square Roots Square root: For any real numbers a and b, if a2 = b, then a is a square root of b.

Examples Simplify Answers

Complex Numbers Imaginary number (unit): denoted by i Pure Imaginary Numbers: 3i, -5i, i√2 Ex.1 Ex.2 a) b)

Multiplying Imaginary #’s Ex.3 Ex.4 **Change to Imaginary numbers 1st Ex.5 Try this one

Powers of i

Powers of i Simplify Ex. 6 Try this one Ex. 7

Equations with Imaginary Solutions Solve the following equation Ex. 8 **Must include + when √ both sides.

Complex Numbers Complex Numbers a + bi (a & b are real #’s) – Standard Form a is the real part, b is the imaginary part Find x and y that make the equation true Ex.9 a b a b

Example Problems Ex. 10 (2 – 5i) – (8 + 6i) FOIL Distribute the negative and combine like terms Ex. 10 (2 – 5i) – (8 + 6i) -6 – 11i FOIL Ex. 11 (3 – 4i)(2 + i) 10 – 5i

Complex Conjugates Two complex numbers of the form a + bi and a – bi Example: 3 – 4i ----- conjugate = 3 + 4i Example: 2 + 3i ----- conjugate = 2 – 3i Ex.12 **Put your answer in the standard form (a + bi).

Homework Assignment #25 p. 264 23-71 odd, 78, 79