2. Complex Numbers warm up 4 Solve the following

3. Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

4. Definition of imaginary numbers:

5. Solving for i We know that But what about other exponents?

6. Warm UP 4 (3 + 2i)² 4 (4 + 5i)(2 – 4i) 4 3i(2 + 5i)

7. STEPS for Dividing Complex Numbers 4 Find the conjugate of the denom

8. The Cycle of i 4 And so on…

9. The cycle of i 4

10. The Cycle of i 4

11. You Try – Remember the cycle -1, -i, 1, i (for exponents 2, 3, 4, 5) 4 1

12. Simplify complex numbers Remember 28

14. Try these problems: -15 i

15. When adding or subtracting complex numbers, combine like terms.

16. Try these on your own

17. ANSWERS:

18. Multiplying complex numbers. To multiply complex numbers, you use the same procedure as multiplying polynomials.

19. Examples to do together:

20. Lets do another example. FOIL Next

21. Answer: Now try these: 21-i

22. Next Warm UP

23. Answers:

24. Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

25. 4 The reverse! 4 Take the sign in the middle of the terms and change it 4 Positive becomes negative 4 Negative becomes positive

26. Another Example 3 + 2i3 – 2i 6i-6i -2 + 4i -2 – 4i

27. STEPS TO DIVIDE COMPLEX NUMBERS 4 Find the conjugate of the denominator 4 Multiply the numerator and the denominator by the conjugate 4 Collect like terms 4 Simplify (remember, no i² in the answer) numerator denominator

28. Example 3 x + 1 Take the conjugate of the denominator Multiply the numerator and denominator by conjugate Simplify Collect like terms

29. Another Example 3 2 + i

30. One more example 2 – 2i 1 + 3i

31. Lets do an example: Rationalize using the conjugate Next

32. Reduce the fraction

33. Lets do another example Next

34. Try these problems.