Opening:

1. Opening: The Habitat for humanity project utilizes volunteers to help build house for low – income families who might not be able to afford the purchase of a home. At a recent site, Habitat workers built a small storage shed attached to the house. The electrical blueprint for the shed called for two AC circuits connected in series with a total voltage of 220 volts. One of the circuits must have an impedance of 7-10j ohms, and the other needs to have an impedance of 9+5j ohms. According to the building codes, the impedance cannot exceed 20-5j ohms. Will the circuits, as designed, meet the code?

2. Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number. Definition of pure imaginary numbers:

3. Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

4. Simplify each expression.

7. Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder.

8. Simplify. Divide the exponent by 4 and look at the remainder.

9. Simplify. Divide the exponent by 4 and look at the remainder.

10. Simplify. Divide the exponent by 4 and look at the remainder.

11. Complex Numbers Consider the quadratic equation x2 + 1 = 0. Solving for x , gives x2 = – 1 We make the following definition:

12. Irrational Numbers Rational Numbers Integers Natural Numbers The Real Number System

13. Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers Imaginary Numbers Real Numbers

14. To play video click on it:

15. Definition of a Complex Number If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If a = 0, the number a + biis called an imaginary number. Write the complex number in standard form

16. Addition and Subtraction of Complex Numbers If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows. Sum: Difference:

17. Perform the subtraction and write the answer in standard form. ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i Try another example: 4

18. Now go back and see our question in our opening: • Sol:Total impedance in the circuit is = (7 – 10j) + (9+5j) = (7 + 9) + ( - 10j + 5j) = 16 – 5j Yes, condition is satisfied because total impedance in the circuit is less than 20 – 5j.

19. Simplify.

20. Simplify.

21. Simplify.

22. Work Session: Text Book work Page: 583 Questions: 5 – 10 & 13 - 20

23. Closure/Exit Card: Simplify each expression: • (5 – 3i) + (-2 + 4i) • (10 – 2i) – (14 – 6i)

24. Multiplying Complex NumbersOpening Multiplying complex numbers is similar to multiplying polynomials and combining like terms. Perform the operation and write the result in standard form. a. ( 6 – 2i )( 2 – 3i ) F O I L = 12 – 18i – 4i + 6i2 = 12 – 22i + 6 ( -1 ) = 6 – 22i

25. Consider ( 3 + 2i )( 3 – 2i ) = 9 – 6i + 6i – 4i2 = 9 – 4( -1 ) = 9 + 4 = 13 This is a real number. The product of two complex numbers can be a real number.

26. Multiplying: Ex: Ex:

27. Complex Conjugates and Division Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers. ( a + bi )( a – bi ) a 2 – abi + abi – b 2 i2 a 2 – b 2( -1 ) a 2 + b 2 The product of a complex conjugate pair is a positive real number.

28. To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

29. Perform the operation and write the result in standard form.

30. Perform the operation and write the result in standard form.

32. The Complex plane Real Axis Imaginary Axis

33. Graphing in the complex plane

34. Absolute Value of a Complex Number • The distance the complex number is from the origin on the complex plane. • If you have a complex number the absolute value can be found using

35. Examples 1. 2. Which of these 2 complex numbers is closest to the origin? -2+5i

36. Work Session: Text Book Work: Page: 583 Questions: 21 - 35

37. Closure/Exit card 1. 2. Which of these 2 complex numbers is closest to the origin? -2+7i

38. Date: 04/28/2011Vision: Transforming teaching and learning to improve student achievement.QCC Standard(s) # 21: Define and applies the basic operations and properties of complex numbers22: Uses appropriate theorems and definitions to find powers, roots, and absolute values of complex numbers. 23: Graph and expresses complex numbers In both rectangular polar forms. Elements:Add, subtract, multiply and divide complex numbers In rectangular form. • Convert complex numbers from rectangular to polar and vice versa. • Finds powers and roots of complex numbers in polar form using De Moivre’s theorem. • Essential Question(s): 1. Write a quadratic equation with roots i and –i. • 2. Explain how to use De Moivre’s theorem to find the reciprocal of a complex number in polar form.

39. Opening:Expressing Complex Numbers in Polar Form We can convert a complex number given in rectangular form (x + yi) in to polar form (ie Trigonometric form) r ( cos Ø + i sin Ø).

40. Expressing Complex Numbers in Polar Form Remember these relationships between polar and rectangular form: So any complex number, X + Yi, can be written in polar form: Here is the shorthand way of writing polar form:

41. Expressing Complex Numbersin Polar Form Rewrite the following complex number in polar form: a) 4 - i b) - 3 + 3i c) 5 + 5√3 i d) – 2 – 1i Rewrite the following complex number in polar form in to standard (rectangular) form:

42. Expressing Complex Numbers GIVEN in Polar Form TO STANDARD FORM Express the following complex number in rectangular(standard)form:

43. Closure/Exit pass a) Express the following complex number in polar form: 1 + √3i b) Express the following complex number in standard form 2(cos 3Π + i sin 3Π) 2 2

44. Date: 05/4/2011Vision: Transforming teaching and learning to improve student achievement.QCC Standard(s) # 21: Define and applies the basic operations and properties of complex numbers22: Uses appropriate theorems and definitions to find powers, roots, and absolute values of complex numbers. 23: Graph and expresses complex numbers In both rectangular polar forms. Elements:Add, subtract, multiply and divide complex numbers In rectangular form. • Convert complex numbers from rectangular to polar and vice versa. • Finds powers and roots of complex numbers in polar form using De Moivre’s theorem. • Essential Question(s): 1. Write a quadratic equation with roots i and –i. • 2. Explain how to use De Moivre’s theorem to find the reciprocal of a complex number in polar form.

45. Products and Quotients of Complex Numbers in Polar Form The product of two complex numbers, and Can be obtained by using the following formula:

46. Products and Quotients of Complex Numbers in Polar Form The quotient of two complex numbers, can be obtained by using the following formula:

47. Products and Quotients of Complex Numbers in Polar Form Find the product of 5cis30 and –2cis120 Next, write that product in rectangular form

48. Products and Quotients of Complex Numbers in Polar Form Find the quotient of 36cis300 divided by 4cis120 Next, write that quotient in rectangular form

49. Products and Quotients of Complex Numbers in Polar Form Find your answer in polar form. Based on how you answered this problem, what generalization can we make about raising a complex number in polar form to a given power?

50. De Moivre’s Theorem De Moivre's Theorem is the theorem which shows us how to take complex numbers to any power easily. De Moivre's Theorem – Let r(cos F+isin F) be a complex number and n be any real number. Then [r(cos F+isin F]n = rn(cos nF+isin nF) What is this saying? The resulting r value will be r to the nth power and the resulting angle will be n times the original angle.