9. You can't take the square root of a negative number, right? When we were young and still in Algebra I, no numbers that, when multiplied by themselves, gave us a negative answer. Squaring a negative number always gives you a positive. (-1)² = 1. (-2)² = 4 (-3)² = 9

10. So here’s what the math people did: They used the letter “i” to represent the square root of (-1). “i” stands for “imaginary.” So, does really exist?

11. Examples of how we use

14. The first four powers of i establish an important pattern and should be memorized. Powers of i

15. Divide the exponent by 4 No remainder: answer is 1. Remainder of 1: answer is i. Remainder of 2: answer is –1. Remainder of 3: answer is –i.

16. Powers of i Find i 23 Find i 2006 Find i 37 Find i 828

17. Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals (no fractions) pi, e Imaginary i, 2i, -3-7i, etc.

18. Express these numbers in terms of i.

19. You try… 4. 5. 6.

20. Multiplying 7. 8. 9.

21. To mult. imaginary numbers or an imaginary number by a real number, it’s important to 1 st express the imaginary numbers in terms of i.

22. a + bi Complex Numbers real imaginary The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

23. Add or Subtract 10. 11. 12.

24. Examples

25. Multiplying Treat the i’s like variables, then change any that are not to the first power Ex:

27. Work Work p. 277 #4 – 10, 17 – 28, 37 – 55