2. Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts

3. Definition The lowercase letter “i” is equal to the square root of negative one.

4. Examples

8. Powers of i i 2 = -1 i 3 = -i i 4 = 1 http://supermanjaviolivares.iespana.es/poster1.jpg

9. Powers of i i 2 = -1 i 3 = -i i 4 = 1 Higher powers of i? http://supermanjaviolivares.iespana.es/poster1.jpg

10. Let’s exploit

16. Complex Numbers  What is a complex number?  A number with a real and an imaginary component  Standard form:  a + bi

17. Adding and Subtracting

22. FOIL-ing Complex Numbers  Remember: i 2 =-1  Example:

23. FOIL-ing Complex Numbers  Remember: i 2 =-1  Example:

24. FOIL-ing Complex Numbers  Remember: i 2 =-1  Example:

25. FOIL-ing Complex Numbers  Remember: i 2 =-1  Example:

26. FOIL-ing Complex Numbers  Remember: i 2 =-1  Example:

27. Conjugates of Complex Numbers  The conjugate of a+bi is a-bi.  The conjugate of a-bi is a+bi.

28. Dividing Complex Numbers  We cannot leave i in the denominator of a fraction. (Why not?)  Multiply the numerator and denominator of the fraction by the conjugate of the denominator.

29. Sample Problem

35. Graphing Calculator  Press MODE.  Select a+bi mode.  The i key is the decimal key.

36. Sample Problem

37.  Type into the calculator with parenthesis around the numerator and the denominator.

38. Sample Problem  Type into the calculator with parenthesis around the numerator and the denominator.  (4-5i)/(3-i)

39. Sample Problem  Type into the calculator with parenthesis around the numerator and the denominator.  (4-5i)/(3-i)  Press ENTER.

40. Sample Problem  Type into the calculator with parenthesis around the numerator and the denominator.  (4-5i)/(3-i)  Press ENTER.  1.7-1.1i

41. Sample Problem  Type into the calculator with parenthesis around the numerator and the denominator.  (4-5i)/(3-i)  Press ENTER.  1.7-1.1i  Prefer fractions? Press Math  Frac.

42. Sample Problem  Type into the calculator with parenthesis around the numerator and the denominator.  (4-5i)/(3-i)  Press ENTER.  1.7-1.1i  Prefer fractions? Press Math  Frac.  17/10-11/10i

43. Sample Problem  Type into the calculator with parenthesis around the numerator and the denominator.  (4-5i)/(3-i)  Press ENTER.  1.7-1.1i  Prefer fractions? Press Math  Frac.  17/10-11/10i IMPORTANT NOTE: You must WRITE the answer as

44. Practice Questions 1. (3 + 2i) + (4 – 5i) 2. (5 – 6i) – (3 – 2i) 3. (2 + 4i)(3 – 5i) 4. (4 + i) 2 5.