Complex Numbers include Real numbers and Imaginary Numbers Define i = Practice Complex Numbers include Real numbers and Imaginary Numbers Powers of i: i1 = i2 = i3 = i4 = i5 = i6 = i7 = i8 = What about i12? What about i15? A short cut to find any power of i: Divide the exponent by 4 and find the remainder i5 → i15 → i12 → i43 → Simplify i2349 Simplify i234 1

e) Challenge: Simplify Simplify the following by getting all negatives outside the square root a) b) c) d) e) Challenge: Simplify Operations with Complex Numbers: Add/subtract complex numbers by combining real and imaginary separately Multiply complex numbers by FOIL (use i properties to simplify) Simplify the following: (3 + 5i) + (8 – i) b) (9 – 3i) – (2 – 8i) c) -4 – (1 + i) – (5 + 9i) d) 10 – (6 + 7i) + 4i 2

Simplify the following (8 – 7i)(3 + 4i) b) c) d) ( 6 + 2i)(6 – 2i) (a + bi) and (a – bi) are called complex conjugates (look at b and d above) What is the difference between the two? What do you notice when complex conjugates are multiplied? Find the following for f(x) = 2x2 – 6x + 5 Vertex: y-intercept zeros 3