1. Imaginary Numbers Today’s Warm-up: Solve using the quadratic formula 15x 2 – 2x – 1 = 0

2. What uses imaginary numbers? Electricity AC (Alternating Current) Electricity changes between positive and negative in a sine wave.sine wave When we combine two AC currents they may not match properly, and it can be very hard to figure out the new current. But using imaginary numbers and real numbers together makes it a lot easier to do the calculations.

3. Whenever we have a negative number under a radical sign, we need to factor out a negative one. Example:

4. Let’s Try Some Examples

5. Powers of i Let us begin with i 0, which is 1. (Any number with exponent 0 is 1.) Each power of i can be obtained from the previous power by multiplying it by i. exponent 0 i 1 = i i 2 = -1 i 3 = -1 ∙ i = - i i4 = -i ∙ i = -i 2 = -(-1) = 1

6. Let’s simplify some powers of i i 9 i 15 i 32

7. Warm-up 10.2.14

8. The Real and Imaginary components Here is the standard form of a complex number: a + bi. Both a and b are real. For example, 3 + 2i. a -- that is, 3 in the example -- is called the real component (or the real part). b (2 in the example) is called the imaginary component (or the imaginary part). Again, the components are real.

10. Operations with Imaginary Numbers Addition/Subtraction – Treat anything with an i as a like term – For example, I can add two terms with i’s together – Example: (4 – 3i) + (-4 + 3i) Rearrange: (4 + (-4)) + (-3i + 3i) Simplify: 0 + 0 = 0

11. Adding/Subtracting Examples 1.) (5 – 3i) – (-2 + 4i) 2.) (7 – 2i) + (-3 + i) 3.) (1 + 5i) – (3 – 2i) 4.) (8 + 6i) – (8 – 6i)

12. Multiplying Imaginary Numbers (3i)(-5 + 2i) Distribute: 3i * (-5) + 3i * 2i Simplify: 15i + 6i 2 Substitute -1 for i 2 Simplify: 15i + 6 (-1) = 15i + (-6)

13. Let’s try a couple (4 + 3i)(-1 – 2i) (-6 + i)(-6 – i) (2 – 3i)(4 + 5i)

14. Warm-up 10.7.14 1.(2 + 4i) + (4 – i) 2.(12 + 5i) – (2 – i) 3.(-6 – 7i) – (1 + 3i)

15. Graphing Imaginary numbers -5 + 3i Notice that we are NOT dealing with an x/y grid

16. Some Examples Graph: 5 – i -3 – 2i 1 + 4i

17. Visualizing Imaginary Numbers Remember: Distance is always Positive In order to find distance, we need to use absolute value.

18. |3 - 2i| Absolute value of a complex number = a = 3 b = -2i 3 2 = 9 -2i 2 = 4 9 + 4 = 13 We are technically squaring two things: -2 and i 2. (-2) 2 = 4 i 2 = -1 Can I have the absolute value of a negative? Let’s Think About This Step

19. Complex Conjugate a + bi and a – bi are complex conjugates Complex conjugates are used to simplify denominators The product of complex conjugates is a real number.

20. What is the complex conjugate? If I have 3 + 4i, what is the complex conjugate? If I have 4 – 2i, what is the complex conjugate?

21. Dividing Imaginary Numbers Multiply both sides by the complex conjugate of the denominator Substitute -1 for i 2 Simplify

22. Your turn