1. Imaginary and Complex Numbers 18 October 2010

2. Question: If I can take the, can I take the ? Not quite…. 

3. Answer??? But I can get close!

4. = Imaginary Number or

5. Simplifying with Imaginary Numbers Step 1: Factor out -1 from the radicand (the number or expression underneath the radical sign) Step 2: Substitute for Step 3: If possible, simplify the radicand

6. Example:

7. Your Turn: 1. 2. 3.

8. *Powers of

9. *Powers of, your turn: Observations?

10. Simplifying Powers of i To simplify a power of i, divide the exponent by 4, and the remainder will tell you the appropriate power of i. Example: i 54 54 ÷ 4 = 13 remainder 2 i 54 = i 2 = -1

11. Complex Numbers Real PartImaginary Part a + bi where a and b are both real numbers, including 0.

12. Complex Number System Complex Numbers: -5, -√3, 0, √5, ⅝, ⅓, 9, i, -5 + -4i Imaginary Numbers: -4i 2i√2 √-1 i Real Numbers: -5, -√3, 0, √5, ⅝, ⅓, 9 Irrational Numbers -√3 √5 Rational Numbers: -5, 0, ⅓, 9 Integers: -5, 0, 9 Whole Numbers: 0, 9 Natural Numbers: 9 The complex numbers are an algebraically closed set!

13. Writing Complex Numbers Step 1: Simplify the radical expression Step 2: Rewrite in the form a + bi.

14. Examples:

15. Examples, cont.

16. Your Turn: 1.2. 3.4.

17. Operations on Complex Numbers Operations (adding, subtracting, multiplying, and dividing) on complex numbers are the same as operations on radicals!!! Remember: the imaginary number is really just a radical with a negative radicand.

18. Addition and Subtraction of Complex Numbers You can only add or subtract like terms. Translation: You must add or subtract the real parts and the imaginary parts of a complex number separately. Step 1: Distribute any negative/subtraction signs. Step 2: Group together like terms. Step 3: Add or subtract the like terms.

19. Addition Example

20. Subtraction Example

21. Your Turn: 1. 2. 3. 4.

22. Multiplication of Complex Numbers Doesn’t require like terms! Translation: You can multiply real parts by imaginary parts and imaginary parts by real parts! Multiply complex numbers like you would multiply expression with radicals. Monomials: Group together like terms, then multiply. Binomials: FOIL!

23. Multiplication Example 1

24. Multiplication Example 2

25. Your Turn: 1. 2. 3. 4.

26. Division of Complex Numbers When dividing two complex numbers, use the same rules for rationalizing the denominator. Monomial: Multiply by the denominator over the denominator. Binomial: Multiply by the conjugate of the denominator. You must FOIL! The conjugate of a complex number is called a complex conjugate.

27. Division Example 1

28. Division Example 2

29. Your Turn: 1.2. 3.4.

30. *Your Turn: 5.6. 7.8.