1. Pre Clac Chapter 2 Section 4

2. Imaginary #’s Let’s pretend that x 2 + 1 = 0 had a solution That would mean x 2 = -1 That can’t be… if you square a number it’s always positive…..

3. i So let’s make a number up that’s if I did square it, it would be -1 lets call that # i i * i = -1

4. Complex numbers you can add real numbers to imaginary numbers and call them complex numbers a + bi

5. Complex numbers Add em’ Add the real parts, add the imaginary parts (3+2i) + (5-7i) = 8 – 5i

6. Complex numbers Multiply em’ FOIL it (3+2i) (5-7i) = 15 – 21i + 10i – 14i 2 isn’t i 2 = -1 15 – 11i + 14 29 - 11i

7. Complex Conjugates 2 complex numbers can multiply to make a real number (a + bi) (a – bi) = a 2 + b 2 so (5 + 3i) has a conjugate of (5 – 3i) Because if I multiply them I get a real # 34

8. I hate imaginary numbers on the bottom of fractions If you have one… multiply by the conjugate over the conjugate It gives me the creeps! 3 + 5i 4 + 2i

9. Principal square root of a number if you have a negative number and you take the square root of it then the principal square root is a i

10. Sometimes you get complex numbers as the solution in the quadratic formula

11. Simplify