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Algebra 2 Complex Numbers Lesson 4-8 Part 1. Goals Goal To identify, graph, and perform operations with complex numbers. Rubric Level 1 – Know the goals.

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Presentation on theme: "Algebra 2 Complex Numbers Lesson 4-8 Part 1. Goals Goal To identify, graph, and perform operations with complex numbers. Rubric Level 1 – Know the goals."— Presentation transcript:

1 Algebra 2 Complex Numbers Lesson 4-8 Part 1

2 Goals Goal To identify, graph, and perform operations with complex numbers. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3 Vocabulary Imaginary Unit Imaginary Number Complex Number Pure Imaginary Number Complex Number Plane Absolute Value of a Complex Number Complex Conjugate

4 Essential Question Big Idea: Solving Equations 1.What are complex numbers? 2.How do you perform basic operations on complex numbers?

5 What is a imaginary number? It is a tool to solve an equation. –Invented to solve quadratic equations of the form. It has been used to solve equations for the last 200 years or so. “Imaginary” is just a name, imaginary do indeed exist, they are numbers.

6 You can see in the graph of f(x) = x 2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x 2 + 1, you find that x =,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as. You can use the imaginary unit to write the square root of any negative number. Imaginary Numbers

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8 Powers of i We could continue but notice that they repeat every group of 4. For every i 4 it will = 1 To simplify higher powers of i then, we'll group all the i 4ths and see what is left. 4 will go into 33 8 times with 1 left. 4 will go into 83 20 times with 3 left.

9 Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, –1, –i, or 1. Helpful Hint Powers of i

10 Always reduce powers of i to; i, -1, 1, or –i. To reduce, divide the exponent by 4 and the remainder determines the reduced value. –remainder 1 ⇒ i –remainder 2 ⇒ -1 –remainder 3 ⇒ -i –remainder 0 (no remainder) ⇒ 1 Negative exponent, the term goes in the denominator and the exponent becomes positive.

11 For exponents larger than 4, divide the exponent by 4, then use the remainder as your exponent instead. Example: Powers of i

12 Your Turn: Simplify each of the following powers. 1) i 22 2) i 36 1 3) i 13 i 4) i 47 -i

13 Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i. Example: Simplifying Square Roots of Negative Numbers

14 Express the number in terms of i. Factor out –1. Product Property. Simplify. Express in terms of i. Example: Simplifying Square Roots of Negative Numbers

15 Express the number in terms of i. Factor out –1. Product Property. Simplify. Express in terms of i. Product Property. Your Turn:

16 Express the number in terms of i. Factor out –1. Product Property. Simplify. Express in terms of i. Multiply. Your Turn:

17 Express the number in terms of i. Factor out –1. Product Property. Simplify. Express in terms of i. Multiply. Your Turn:

18 The Complex-Number System The complex-number system is used to find zeros of functions that are not real numbers. When looking at a graph of a function, if the graph does not cross the x-axis, it has no real-number zeros.

19 Every complex number has a real part a and an imaginary part b. Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i =. The set of real numbers is a subset of the set of complex numbers C.

20 The Complex-Number System R Real Numbers R Rational Numbers Q Integers Z Whole numbers W Natural Numbers N Irrational Numbers Q -bar Complex Numbers C Imaginary Numbers i

21 Standard Form of Complex Numbers Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers. a is a real number and bi would be an imaginary number. If b = 0, a + bi is a pure real number (2, -56, 34 etc.). If a = 0, a + bi is an pure imaginary number (2i, -56i, 34i etc.).

22 Standard Form of Complex Numbers Since all numbers belong to the Complex number system, C, all numbers can be classified as complex. Real numbers, R, and Imaginary numbers, i, are subsets of Complex Numbers. Real Numbers a + 0i Imaginary Numbers 0 + bi Complex Numbers a + bi

23 Standard Form of Complex Numbers Zero is the only number that is both real and imaginary.

24 Write each of the following in the form of a complex number in standard form a + bi. 6 =6 + 0i 8i =0 + 8i i 6 + 5i i Standard Form of Complex Numbers Example:

25 Your turn: Write each of the following in the form of a complex number in standard form a + bi.

26 Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Equity of Complex Numbers

27 Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true. Equate the real parts. 4x + 10i = 2 – (4y)i Real parts Imaginary parts 4x = 2 Solve for y. Solve for x. Equate the imaginary parts. 10 = –4y Example:

28 Find the values of x and y that make each equation true. Equate the real parts. 2x – 6i = –8 + (20y)i Real parts Imaginary parts 2x = –8 Solve for y. Solve for x. –6 = 20y 2x – 6i = –8 + (20y)i Equate the imaginary parts. x = –4 Your Turn:

29 Find the values of x and y that make each equation true. Equate the real parts. Real parts Imaginary parts –8 = 5x Solve for y. Solve for x. Equate the imaginary parts. –8 + (6y)i = 5x – i Your Turn:

30 Just as you can represent real numbers graphically as points on a number line, you can represent complex numbers in a special coordinate plane. The complex plane is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Complex Plane

31 The real axis corresponds to the x-axis, and the imaginary axis corresponds to the y-axis. Think of a + bi as x + yi. Helpful Hint Complex Plane

32 Graph each complex number. A. 2 – 3i B. –1 + 4i C. 4 + i D. –i 2 – 3i –i–i 4 + i –1+ 4i Example:

33 Graph each complex number. a. 3 + 0i b. 2i c. –2 – i d. 3 + 2i 3 + 0i 2i –2 – i 3 + 2i Your Turn:

34 Recall that absolute value of a real number is its distance from 0 on the number line. Similarly, the absolute value of a complex number is its distance from the origin in the complex plane. Absolute Value of a Complex Number

35 Find each absolute value. A. |3 + 5i| |–13 + 0i| 13 B. |–13| C. |–7i| |0 +(–7)i| 7 Example:

36 Find each absolute value. a. |1 – 2i| b.c. |23i| |0 + 23i| 23 Your Turn:

37 Adding and Subtracting Complex Numbers Sum or Difference of Complex Numbers If a + bi and c + di are complex numbers, then their sum is (a + bi) + (c + di) = (a + c) + (b + d)i Their difference is (a + bi) – (c + di) = (a – c) + (b – d)i Combine like Terms: add/sub. real parts and add/sub. Imaginary parts.

38 Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts. The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions. Adding and Subtracting Complex Numbers

39 Complex numbers also have additive inverses. The additive inverse of a + bi is –(a + bi), or –a – bi. Helpful Hint

40 Procedure: 1.Pretend that “i” is a variable and that a complex number is a binomial 2.Add and subtract as you would binomials Example: (2 + i) – (-5 + 7i) + (4 – 3i) 2 + i + 5 – 7i + 4 – 3i 11 – 9i

41 Add or subtract. Write the result in the form a + bi. (4 + 2i) + (–6 – 7i) Add real parts and imaginary parts. (4 – 6) + (2i – 7i) –2 – 5i Example:

42 Add or subtract. Write the result in the form a + bi. (5 –2i) – (–2 –3i) Distribute. Add real parts and imaginary parts. (5 – 2i) + 2 + 3i 7 + i (5 + 2) + (–2i + 3i) Example:

43 Add or subtract. Write the result in the form a + bi. (1 – 3i) + (–1 + 3i) Add real parts and imaginary parts. 0 (1 – 1) + (–3i + 3i) Example:

44 Add or subtract. Write the result in the form a + bi. (–3 + 5i) + (–6i) Add real parts and imaginary parts. (–3) + (5i – 6i) –3 – i Your Turn:

45 Add or subtract. Write the result in the form a + bi. 2i – (3 + 5i) Distribute. Add real parts and imaginary parts. (2i) – 3 – 5i –3 – 3i (–3) + (2i – 5i) Your Turn:

46 Add or subtract. Write the result in the form a + bi. (4 + 3i) + (4 – 3i) Add real parts and imaginary parts. 8 (4 + 4) + (3i – 3i) Your Turn:

47 Multiplying Complex Numbers The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms. You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i 2 = –1.

48 Multiplying Complex Numbers Note that the product rule for radicals does NOT apply for imaginary numbers. Before performing operations with imaginary numbers, be sure to rewrite the terms or factors in i-form first and then proceed with the operations.

49 Procedure: 1.Pretend that “i” is a variable and that a complex number is a binomial 2.Multiply as you would binomials 3.Simplify by changing “i 2” to “-1” and combining like terms Example: (-4 + 3i)(5 – i) -20 + 4i + 15i - 3i 2 -20 + 4i + 15i + 3 -17 + 19i

50 Multiply. Write the result in the form a + bi. –2i(2 – 4i) Distribute. Write in a + bi form. Use i 2 = –1. –4i + 8i 2 –4i + 8(–1) –8 – 4i Example:

51 Multiply. Write the result in the form a + bi. (3 + 6i)(4 – i) Multiply. Write in a + bi form. Use i 2 = –1. 12 + 24i – 3i – 6i 2 12 + 21i – 6(–1) 18 + 21i Example:

52 Multiply. Write the result in the form a + bi. (2 + 9i)(2 – 9i) Multiply. Write in a + bi form. Use i 2 = –1. 4 – 18i + 18i – 81i 2 4 – 81(–1) 85 Example:

53 Multiply. Write the result in the form a + bi. (–5i)(6i) Multiply. Write in a + bi form. Use i 2 = –1 –30i 2 –30(–1) 30 Example:

54 Multiply. Write the result in the form a + bi. 2i(3 – 5i) Distribute. Write in a + bi form. Use i 2 = –1. 6i – 10i 2 6i – 10(–1) 10 + 6i Your Turn:

55 Multiply. Write the result in the form a + bi. (4 – 4i)(6 – i) Distribute. Write in a + bi form. Use i 2 = –1. 24 – 4i – 24i + 4i 2 24 – 28i + 4(–1 ) 20 – 28i Your Turn:

56 Multiply. Write the result in the form a + bi. (3 + 2i)(3 – 2i) Distribute. Write in a + bi form. Use i 2 = –1. 9 + 6i – 6i – 4i 2 9 – 4(–1) 13 Your Turn:

57 If a quadratic equation with real coefficients has nonreal solutions, those solutions are complex conjugates. When given one complex solution, you can always find the other by finding its conjugate. Helpful Hint The complex numbers and are related. These complex numbers are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi. Complex Conjugates

58 Find each complex conjugate. A. 8 + 5i B. 6i 0 + 6i 8 + 5i 8 – 5i Write as a + bi. Find a – bi. Write as a + bi. Find a – bi. Simplify. 0 – 6i –6i Example:

59 Find each complex conjugate. A. 9 – i 9 + (–i) 9 – (–i) Write as a + bi. Find a – bi. B. C. –8i 0 + (–8)i Write as a + bi. Find a – bi. Simplify. 0 – (–8)i 8i8i Write as a + bi. Find a – bi. 9 + i Simplify. Your Turn:

60 Product of Complex Conjugates Complex Conjugates The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and (a + bi)(a – bi) = a 2 + b 2

61 Complex Conjugate Examples:  6 + 7i and  6  7i 8  3i and 8 + 3i 14i and  14i The product of a complex number and its conjugate is a real number. Example: (4 + 9i)(4  9i) = 4 2  (9i) 2 = 16  81i 2 = 16  81(  1) = 97 Example: (7i)(  7i) =  49i 2 =  49(  1) = 49

62 Your Turn: Multiply each complex number by its complex conjugate. 1)3 + i 10 2)5 – 4i 41

63 Recall that expressions in simplest form cannot have square roots in the denominator. Because the imaginary unit represents a square root, you must rationalize any denominator that contains an imaginary unit. To do this, multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a – bi. Helpful Hint Dividing Complex Numbers

64 1.Write the division problem in fraction form. 2.The trick is to make the denominator real, this is done using the complex conjugate. 3.Multiply the fraction by a special “1” where “1” is the conjugate of the denominator over itself. 4.Simplify and write answer in standard form: a + bi. Procedure:

65 Simplify. Multiply by the conjugate. Distribute. Simplify. Use i 2 = –1. Example:

66 Simplify. Multiply by the conjugate. Distribute. Simplify. Use i 2 = –1. Example:

67 Simplify. Multiply by the conjugate. Distribute. Simplify. Use i 2 = –1. Your Turn:

68 Simplify. Multiply by the conjugate. Distribute. Simplify. Use i 2 = –1. Your Turn:

69 1) 2)

70 Essential Question Big Idea: Solving Equations 1.What are complex numbers? –The set of complex numbers includes real numbers and imaginary numbers. The basis for the complex numbers is the imaginary unit i, whose square is -1. Use a complex number of the form a + bi to simplify the square root of a negative number. 2.How do perform basic operations on complex numbers? –To perform basic operations on complex numbers, treat the imaginary parts like variable terms of binomials. Add, subtract, or multiply as you would with algebraic expressions. To divide complex numbers, multiply numerator and denominator by the complex conjugate of the denominator and simplify.

71 Assignment Section 4-8 Part 1, Pg 273;# 1 – 6 all, 8 – 30 even.


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