Name: Date: IMAGINARY NUMBERS COMMON CORE ALGEBRA II Recall that in the Real Number System, it is not possible to take the square root of a negative quantity because whenever a real number is squared it is non-negative. This fact has a ramification for finding the x-intercepts of a parabola, as Exercise #1 will illustrate. Exercise #1: On the axes below, a sketch of given in function notation as f x x 1. (a) How is the graph of the graph of f x? y x shifted to produce y x is shown. Now, consider the parabola whose equation is (b) Create a quick sketch of below. y f x on the axes (c) What can be said about the x-intercepts of the function y f x? (d) Algebraically, show that these intercepts do not exist, in the Real Number System, by solving the incomplete quadratic x 1 0. x Since we cannot solve this equation using Real Numbers, we introduce a new number, called i, the basis of imaginary numbers. Its definition allows us to now have a result when finding the square root of a negative real number. Its definition is given below. THE DEFINITION OF THE IMAGINARY NUMBER i 1 Exercise #: Simplify each of the following square roots in terms of i. (a) 9 (b) 100 (c) 3 (d) 18 COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #1 emathinstruction, RED HOOK, NY 1571, 015
Exercise #: Solve each of the following incomplete quadratics. Place your answers in simplest radical form. (a) 5x 8 1 (b) 1 0 x (c) x 10 36 Exercise #3: Which of the following is equivalent to 5i 6i? (1) 30i (3) 30 () 11i (4) 11 Powers of i display a remarkable pattern that allow us to simplify large powers of i into one of 4 cases. This pattern is discovered in Exercise #4. Exercise #4: Simplify each of the following powers of i. 1 i i 3 4 5 6 7 8 We see, then, from this pattern that every power of i is either 1, 1, i, or i. And the pattern will repeat. Exercise #5: From the pattern of Exercise #4, simplify each of the following powers of i. (a) i 38 (b) i 1 (c) i 83 (d) i 40 Exercise #6: Which of the following is equivalent to 16 3 6 5 3 i? (1) 8 i (3) 5 4i () 4 3i (4) 7i COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #1 emathinstruction, RED HOOK, NY 1571, 015
Name: Date: FLUENCY IMAGINARY NUMBERS COMMON CORE ALGEBRA II HOMEWORK 1. The imaginary number i is defined as (1) 1 (3) 4 () 1 (4) 1. Which of the following is equivalent to 18? (1) 8 (3) 8 () 8i (4) 8i 3. The sum 9 16 is equal to (1) 5 (3) 7i () 5i (4) 7 4. Which of the following powers of i is not equal to one? (1) i 16 (3) i 3 () i 6 (4) i 48 5. Which of the following represents all solutions to the equation 1 10 7 3 x? (1) x 3i (3) x i () x 5i (4) x i 6. Solve each of the following incomplete quadratics. Express your answers in simplest radical form. (a) x 100 6 (b) 0 3 x COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #1 emathinstruction, RED HOOK, NY 1571, 015
7. Which of the following represents the solution set of 1 1 37 x? (1) 7i (3) 5i () 7i (4) 3i 8. Simplify each of the following powers of i into either 1, 1, i, or i. (a) i (b) i 3 (c) i 4 (d) i 11 (e) i 41 (f) i 30 (g) i 5 (h) i 36 (i) i 51 (j) i 45 (k) i 80 (l) i 70 9. Which of the following is equivalent to i 7 i 8 i 9 i 10? (1) 1 (3) 1 i () i (4) 0 10. When simplified the sum 18 5 8 43 57 6i is equal to (1) 4i (3) 5 7i () 3 i (4) 8 i 11. The product 6 i 4 3i can be written as (1) 4 6i (3) 5i () 18 10i (4) 30 10i COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #1 emathinstruction, RED HOOK, NY 1571, 015
Name: Date: COMPLEX NUMBERS COMMON CORE ALGEBRA II All numbers fall into a very broad category known as complex numbers. Complex numbers can always be thought of as a combination of a real number with an imaginary number and will have the form: a bi where a and b are real numbers We say that a is the real part of the number and bi is the imaginary part of the number. These two parts, the real and imaginary, cannot be combined. Like real numbers, complex numbers may be added and subtracted. The key to these operations is that real components can combine with real components and imaginary with imaginary. Exercise #1: Find each of the following sums and differences. (a) 7i 6 i (b) 8 4i 1 i (c) 5 3i 7i (d) 3 5i 8 i Exercise #: Which of the following represents the sum of 6 i and 8 5i? (1) 5i (3) 3i () 3i (4) 5i Adding and subtracting complex numbers is straightforward because the process is similar to combining algebraic expressions that have like terms. The complex numbers are closed under addition and subtraction, i.e. when you add or subtract two complex numbers the results is a complex number as well. But, is multiplication closed? Exercise #3 Find the following products. Write each of your answers as a complex number in the form a bi. (a) 3 5i 7 i (b) 6i 3 i (c) 4 i 5 3i COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON # emathinstruction, RED HOOK, NY 1571, 015
Exercise #4: Consider the more general product a bi c di where constants a, b, c and d are real numbers. (a) Show that the real component of this product will always be ac bd. (b) Show that the product of 3i and 4 6i results in a purely real number. (c) Under what conditions will the product of two complex numbers always be a purely imaginary number? Check by generating a pair of complex numbers that have this type of product. Exercise #5: Determine the result of the calculation below in simplest a bi form. 5 i 3 i 4i 3i Exercise #6: Which of the following products would be a purely real number? (1) 4 i 3 i (3) 5 i 5 i () 3 i 4i (4) 6 3i 6 3i COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON # emathinstruction, RED HOOK, NY 1571, 015
Name: Date: FLUENCY COMPLEX NUMBERS COMMON CORE ALGEBRA II HOMEWORK 1. Find each of the following sum or difference. (a) 6 3i 9i (b) 7 i 3 5i (c) 10 3i 6 8i (d) 7i 15 6i (e) 15 i 5 5i (f) 1 i 5 6i. Which of the following is equivalent to 35 i 3 6i? (1) 9 18i (3) 9 6i () 1 8i (4) 1 i 3. Find each of the following products in simplest a bi form. (a) 5 i 1 7i (b) 3 9i 4i (c) 4 i 6i 4. Complex conjugates are two complex numbers that have the form a bi and a bi. Find the following products of complex conjugates: (a) 5 7i 5 7i (b) 10 i 10 i (c) 3 8i 3 8i (d) What's true about the product of two complex conjugates? COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON # emathinstruction, RED HOOK, NY 1571, 015
5. Show that the product of a bi and a bi is the purely real number a b. 6. The product of 8 i and its conjugate is equal to (1) 64 4i (3) 68 () 60 (4) 60 4i 7. The complex computation 6 i 6 i 3 4i 3 4i (1) 15 (3) 10 () 39 (4) 35 can be simplified to 8. Perform the following complex calculation. Express your answer in simples a bi form. 8 5i 3 i 4 i 4 i 9. Perform the following complex calculation. Express your answer in simplest a bi form. i i i 73 5 4 6 7 10. Simplify the following complex expression. Write your answer in simplest a bi form. 5 i i COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON # emathinstruction, RED HOOK, NY 1571, 015
Name: Date: SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS COMMON CORE ALGEBRA II As we saw in the last unit, the roots or zeroes of any quadratic equation can be found using the quadratic formula: x b b 4ac Since this formula contains a square root, it is fair to investigate solutions to quadratic equations now when the quantity b 4ac, known as the discriminant, is negative. Up to this point, we would have concluded that if the discriminant was negative, the quadratic had no (real) solutions. But, now it can have complex solutions. Exercise #1: Use the quadratic formula to find all solutions to the following equation. Express your answers in simplest a bi form. x a 4x 9 0 As long as our solutions can include complex numbers, then any quadratic equation can be solved for two roots. Exercise #: Solve each of the following quadratic equations. Express your answers in simplest a bi form. (a) x x x 5 30 7 10 (b) x x x 16 15 10 4 COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #3 emathinstruction, RED HOOK, NY 1571, 015
There is an interesting connection between the x-intercepts (zeroes) of a parabola and complex roots with nonzero imaginary parts. The next exercise illustrates this important concept. Exercise #3: Consider the parabola whose equation is 6 13. (a) Algebraically find the x-intercepts of this parabola. Express your answers in simplest a bi form. (b) Using your calculator, sketch a graph of the parabola on the axes below. Use the window indicated. y 0 (c) From your answers to (a) and (b), what can be said about parabolas whose zeroes are complex roots with non-zero imaginary parts? x 5 5 10 Exercise #4: Use the discriminant of each of the following quadratics to determine whether it has x-intercepts. (a) 3 10 (b) 6 10 (c) y x x 3 5 Exercise #5: Which of the following quadratic functions, when graphed, would not cross the x-axis? (1) 5 3 (3) y x x 4 4 5 () 6 (4) y x x 3 13 4 COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #3 emathinstruction, RED HOOK, NY 1571, 015
Name: Date: FLUENCY SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS COMMON CORE ALGEBRA II HOMEWORK 1. Solve each of the following quadratic equations. Express your solutions in simplest a bi form. Check. (a) x x x 4 0 1 5 (b) x x 1 (c) x 5x 7 15x 10 (d) 8x 36x 4 1x 5 (e) x x x 6 15 8 (f) 4x 38x 50 10x 35 COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #3 emathinstruction, RED HOOK, NY 1571, 015
. Which of the following represents the solution set to the equation x x 0? (1) x 1 or (3) x i () x 1 i (4) x 1 i 3. The solutions to the equation x 6x 11 0 are (1) x 3 i (3) x 6 i 11 () x 3 i (4) x 6 i 11 4. Using the determinant, zeroes. b 4ac, determine whether each of the following quadratics has real or imaginary (a) 7 6 (b) 3 1 (c) 8 14 (d) 1 6 (e) 6 5 (f) y x x 4 4 1 5. Which of the following quadratics, if graphed, would lie entirely above the x-axis. Try to use the discriminant to solve this problem and then graph to check. (1) 1 (3) 4 7 () 6 (4) 10 16 REASONING 6. For what values of c will the quadratic for this problem. y x 6x c have no real zeros? Set up and solve an inequality COMMON CORE ALGEBRA II, UNIT #9 COMPLEX NUMBERS LESSON #3 emathinstruction, RED HOOK, NY 1571, 015