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Mathematics B30 Module 1 Lesson 1 Mathematics B30 The Complex Number System 1 Lesson 1 Mathematics B30 2 Lesson 1 The Complex Number System Introduction In this lesson you will learn about a new number called the imaginary number. You will learn how to simplify expressions which involve imaginary numbers. Imaginary numbers then, in turn, will be used to define complex numbers. Complex numbers allows us to be able to solve more mathematical questions. Mathematics B30 3 Lesson 1 Mathematics B30 4 Lesson 1 Objectives After completing this lesson, you will be able to • define and illustrate complex numbers. • express complex numbers in the form a + bi. • add and subtract complex numbers. • multiply and divide complex numbers. • divide complex numbers using conjugates. Mathematics B30 5 Lesson 1 Mathematics B30 6 Lesson 1 1.1 Number Systems Number Systems have been a developmental process over hundreds of years. During the development of Mathematics, number systems were expanded based on the need. Number Systems 1. Natural Numbers 1, 2, 3, ... 2. Whole Numbers 0, 1, 2, ... 3. Integers ..., 1, 0, 1, ... 4. Rational Numbers 5. Irrational Numbers { x x is a real number and x cannot be shown as a quotient of integers.} 2 Examples: 6. Real Numbers { x x is a number that may be shown as a p p and q are integers, and q 0 q 3 1 0 .75 0 .33333 ... Examples: 4 3 decimal.} • The Natural Numbers were the first system of numbers used. That system was used until the need for the zero was addressed. The Whole Numbers were the system that resulted from that need. • From Whole Numbers, Integers were introduced for the use of the negative numbers. Up until the Integers, solutions to 7 9 = ? could not be solved. • Using only Integers, the problem was that division did not always lead to another Integer. Example: 1 4 ? The result was the introduction of the Rational Numbers. • The Rational Numbers were satisfactory until the problem arose that not all numbers could be expressed as the quotient of integers. The Irrational Numbers were then developed and this meant that equations such as x 2 2 0 could have a solution. • The system of Irrational Numbers together with the rational numbers is called the Real Number system. Mathematics B30 7 Lesson 1 Here, in diagram form, are the number systems you have seen and worked with so far on your mathematical journey. Irrational Numbers Zero Real Numbers Integers Natural Numbers Rational Numbers Non-Integers You are in a similar situation where you have solved problems that have given you an impossible solution, or an undefined solution. Here are two such cases. 1. x 2 6 x 6 • 2. Solutions with a negative sign under the square root were not allowed by early mathematicians. Imaginary is what situations like this were called. If the graph of a parabola (quadratic equation) has no intersection with the x-axis, we say that the equation has no real solution. Even though real roots (x-intercepts) do not exist, the equation and its graph do exist. y x2 1 Graph: Table of Values • y x y 2 1 0 1 2 5 2 1 2 5 x The desire by mathematicians to be able to describe every quadratic equation by some sort of roots was not possible with the real number system. Mathematics B30 8 Lesson 1 Gradually, uses were found that needed the application of the imaginary numbers, making it necessary to enlarge the set of real numbers to form the set of complex numbers. By doing so, the set of complex numbers provides a solution for just about any equation that can be written. The definition of a complex number is motivated by the equation x 2 1 0 . x2 1 0 x 2 1 x 1 No real solution. To provide a solution for the equation x 2 1 0 , a new number i is defined. Definition: 1 i The square root of any negative number can now be defined in terms of i. r 1 r Definition: 1 r i r This definition emphasizes that roots of negative numbers must be dealt with in only one way. Convert to i times the root of a positive number. For any real number x, ix is called an imaginary number. Example 1 Convert Mathematics B30 9 into the imaginary number form, ix . 9 Lesson 1 Solution: 9 19 1 9 i3 A common error is mistaking reason we write 3 i for 3 i , with the i under the radical. For this 3 i as i 3 . Using the equation x 2 1 0 along with the definition original equation to obtain another useful equation. 1 i we can substitute into the x2 1 0 x 2 1 x 1 Definition: 1 i x i Substitute. x2 1 0 i 2 1 0 i 2 1 i 2 1 When finding the product of 3 1 , the product rule for radicals can’t be used. The rule applies only when both radicals represent real numbers. Therefore, always change r (where r 0 ) to the form i r before multiplying or dividing. Mathematics B30 10 Lesson 1 Right Multiply: 3 1 Wrong Multiply: 3 1 Solution: 3 1 Solution: 3 1 3 1 i 3 i 1 ii 3 1 3 i 3 2 1 3 * Wrong answer due to incorrect method. 3 Example 2 Simplify: a. b. c. 1 1 2 6 4 9 16 Solution: a. 1 1 ii i2 1 2 6 i 2 i 6 b. i2 2 6 1 12 2 3 c. 4 9 16 i 4 i 9 i 16 i2i 2 3 4 1 i 24 i 24 Mathematics B30 11 Lesson 1 Use the following keystroke pattern to find the product of MODE (six times) 1 1 . (once) to have a + bi highlighted. ENTER 2nd QUIT (2nd x2 ) – 1 ) negative sign (not the subtraction key) – 1 ) ENTER negative sign Answer displayed: 1 Find the product of Mathematics B30 2 6 and the product of 12 4 9 16 using the calculator. Lesson 1 Activity 1.1 A pattern develops when reviewing the various powers of i. Complete the chart. Powers of i Working Room Value i1 i i2 1 i3 By definition i 3 i i 2 1 i4 i5 i6 i7 i8 i9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 Conclusion The pattern of the powers of i is _____, _____, _____, _____. Using the fact that i 4 1 , larger powers of i can be simplified. Mathematics B30 13 Lesson 1 Example 3 Find each power of i. a) b) i 32 i 103 Solution: a) i 32 8 4 32 i 4 8 1 8 1 b) i 103 i 100 i 3 25 i4 25 R3 4 103 i3 1 i 3 25 1 i2 i i There are many combinations that can be used to solve questions like Example 3. Alternate Solution: a) i 32 i 2 16 b) i2 16 i i 2 51 1 i i2 16 51 i 1 i 1 i 1 Mathematics B30 i 103 i i 102 51 14 Lesson 1 Example 4 2 3 4 . Simplify Solution: 2 3 4 i 2 i 3 i 4 i 2 3 2 Exercise 1.1 1. 2. Convert each of the following to the imaginary number form ix. a. 1 6 d. 2 3 6 b. 3 e. 2 c. 100 f. 6 1 7 Simplify: a. b. 4 2 4 2 8 1 2 h. i. i7 j. i8 1 3 1 2i c. 3. 3 4 d. 2i i i 2 k. i 39 e. i4 l. i 93 f. i 17 m. i 46 g. i4 i9 n. i 84 Simplify: a. 1 2 1 4 b. 4 6i 7i 3 c. 5 7 9 Mathematics B30 15 Lesson 1 1.2 Complex Numbers The complex number system entails the use of the imaginary numbers, ix. A complex number is an expression of the form a bi , where a and b are real numbers. 2 3i Example: real part • imaginary part Every real number may be considered as a complex number with imaginary part equal to zero. Example: Solution: • Express 2 as a complex number. 2 2 0i If the real part is zero, the remaining part bi is called a pure imaginary number. Example: 0 7i Our diagram on the number system we know now looks like the following. Complex Numbers a + bi a and b are reals Imaginary numbers a + bi, b0 Irrational Numbers Real Numbers a + bi b0 Zero Integers Natural Numbers Rational Numbers Non-Integers Mathematics B30 16 Lesson 1 Complex numbers can be represented in a coordinate plane by letting x yi correspond to the point x, y in the following graph. y 4i –3 + 3i 2 + 2i 3 x 1 – 3i –4 – 3i When the coordinate plane is used for complex numbers • the plane is called the complex plane. • the horizontal axis (x-axis) is called the real axis. • the vertical axis is called the imaginary axis. The absolute value of a complex number can also be found. y c = x + yi The absolute value of a complex number, c , is the distance from the point c to the origin 0. x Mathematics B30 17 Lesson 1 The distance can be determined using the Pythagorean Theorem. hypotenuse 2 side 2 side 2 c x2 y2 Absolute Value of a complex number. a bi a 2 b 2 Example 1 Plot 3 4 i on a complex plane and find 3 4 i . Solution: y 3 + 4i 3 4i x 2 y2 3 2 4 2 5 3 x Use the following keystroke pattern to find the absolute value of 3 4 i MATH twice (to highlight CPX) 5 ( to select abs( ) 3 + 4 2nd . ) ENTER Answer displayed: 5 Mathematics B30 18 Lesson 1 Example 2 Simplify 4 2 5i . Solution: 4 ( 2 + 5 i ) = 8 + 2 0 i * Only the real numbers are multiplied. Exercise 1.2 1. 2. 3. Write as a complex number. a. 4 2 b. 6 5 c. 7 Plot each of the following on a complex plane. Find a bi for each as well. a. 5 i b. 3 2i c. 7 4i d. 6 8i Simplify. a. 65 3i b. 6 4 3 c. 8 i 3 3i 5 Mathematics B30 19 Lesson 1 1.3 Addition and Subtraction of Complex Numbers Addition of complex numbers is performed by adding real parts to real parts and imaginary parts to imaginary parts. Subtraction is similar. a ib c id a c i b d a ib c id a c i b d Example 1 Complete the indicated operation. 2 3i 1 4 i 7 i Solution: 2 3i 1 4 i 7 i 2 1 7 i 3 4 1 6 0 i 6 Use the following keystroke pattern CLEAR ( 2 + 3 i [ 2nd . ] ( – 1 – 4 i – ) ) + use negative sign button ( 7 – i ) ENTER Addition and subtraction of two complex numbers can be shown graphically by using the origin as a starting point. Example 2 Illustrate 4 3i 3 5i on a graph. Mathematics B30 20 Lesson 1 Solution: y (3 + 5i ) Plot the two complex points. (4 + 3 i) x y (3 + 5 i ) 5i 3 Create a parallelogram. (4 + 3 i ) x y (7 + 8 i ) 4 3 i 3 5 i 7 8 i 3 5 i 4 3 i 7 8 i 3i (3 + 5i ) 4 (4 + 3 i ) x Mathematics B30 21 Lesson 1 Example 3 Illustrate 4 3i 2 4 i on a graph. Solution: 2 4 i 4 3i Plot the two complex points. y (–2 + 4i ) Plot the opposite of 2 4 i 2 4 i . (4 + 3 i ) x (2 – 4 i ) Mathematics B30 22 Lesson 1 y Create a parallelogram with 4 3i and 2 4 i . (4 + 3 i ) 4 3i 2 4 i 6 i x (6 – i ) (2 – 4 i ) Exercise 1.3 1. Complete the indicated operation. a. b. c. d. e. f. 2. 4 7i 3 3i 6 3i 9 2i 8 5i 6 3i 2 5i 3 4i 32 3i 4 6 i 24 3i 35 2i Illustrate the following operations on a graph. State your solution. a. b. c. d. 3 2i 4 3i 3 4 i 2 2i 2 3i 3 2i 23 i Mathematics B30 23 Lesson 1 1.4 Multiplication and Division of Complex Numbers Multiplication of complex numbers is performed by using the distributive law just like multiplication of binomials. (a + bi )(c + di ) = ac + adi + bci + bdi 2 = ac + (ad + bc)i + bd (–1) = (ac – bd ) + (ad + bc)i Example 1 Multiply 2 3i 1 4 i . Solution: 2 3 i 1 4 i 2 1 2 4 i 3 i 1 3 i 4 i 2 2 8 i 3 i 12 i 2 11 i 12 1 10 11 i Use the following keystroke pattern. CLEAR ( 2 + 3 i ) × ( – 1 – 4 negative Mathematics B30 i ) ENTER subtraction 24 Lesson 1 Example 2 Solve 5 2 i . 2 Solution: 5 2 i 2 5 2 i 5 2 i 25 20 i 4 i 2 25 20 i 4 21 20 i Example 3 Multiply 1 6i 1 6i . Solution: 1 6 i 1 6 i 1 6 i 6 i 36 i 2 1 36 1 37 The numbers 1 6 i and 1 6 i are called conjugates. Generally, the conjugate of a bi is a bi . As shown in Example 3, the product of a complex number and its conjugate is a real number a 2 b 2 . The use of conjugates is used to simplify the quotient of two imaginary numbers. Use the following keystroke pattern to determine the conjugate of 1 2i . CLEAR MATH (twice) 1 (to select conj( ) 1 + 2 i ) ENTER negative Answer displayed: 1 2 i Mathematics B30 25 Lesson 1 Division of complex numbers involves multiplying the dividend and divisor by the complex conjugate of the divisor. (2–3 i) ÷ (– 1+2 i) D iv id e n d D iv is o r Example 4 Calculate 2 3i 1 2i . Solution: 2 3 i 1 2 i 2 3i 1 2 i 1 2 i 1 2 i * The complex conjugate of 1 2i is 1 2i . 2 4 i 3 i 6 i 2 1 2 i 2 i 4 i 2 i 4 3 6 1 1 4 1 8 i 8 1 i 5 5 5 2 Different Sign Example 5 Calculate 5 7 i 2i . Solution: 5 7 i 5 7 i 2 i 2i 2 i 2 i * The complex conjugate of 0 2 i is 0 2 i . 10 i 14 i 2 4i 2 10 i 14 4 7 5 i. 2 2 Mathematics B30 26 Lesson 1 Never leave an imaginary number, bi, in the denominator of a fraction! Equality of Complex Numbers Two complex numbers are equal if and only if the real parts are equal and the imaginary parts are equal. a bi c di if and only if, a c and b d . Example 6 For what values of x and y does equality hold? 2 x 5 i 2 5 7 yi Solution: Equate the real parts and solve. 2x 2 5 2x 7 x Equate the imaginary parts and solve. 5 i 7 yi 5 7 y y Check. Mathematics B30 7 2 5 7 2 x 5 i 2 5 7 yi 7 5 2 5 i 2 5 7 i 2 7 5 5 i 5 5 i () 27 Lesson 1 Exercise 1.4 1. Write each in the form a bi . a. b. c. d. e. 2. 3 2i 3 2i i. j. 3 i 1 i 5 i 5 i 2 1 2i 2i 2 i 1 3 2i 1 i 1 3 2i 2i 1 i 3i 7 3i 12 4 i f. g. h. i. j. Write the reciprocal of each and then write it in the form a bi . a. b. c. d. e. 4. i 2 3i 1 5i 1 5i i i a bi a bi 2 5 i 2 1 3 i 3 f. g. h. Perform each of the following division and write answer in the form a bi . a. b. c. d. e. 3. 63 2i 5i4 3i 1 2i 2 3i 3i 8 2 7i 5 3i a bi The reciprocal of x is 1 . x Solve for x and y in each of the following equations. a. b. c. d. e. 3 7 i x yi 3 i 2 x yi 2i 6i 3 x 2 yi 2i 1 6i ii 1 2 y xi i(2 i ) 3 x iy 2 xi 4 y Mathematics B30 28 Lesson 1 Self Evaluation X 10 5 –9 5 4 F 7 4 –2 –2 6i –4 –9 –3 –9 –4 8i 10 –i –2 –9 5 3 2 4 8 –6 4 –4 –9 9 –i 4 –2 –2 2 i –9 – i 8 10 –6 10 3 –i 4 –4 –9 6 i 5 F 6i 4 –9 4i 8 10 –9 –6 –2 F 3 2 4 3i 8 i 10 – i –9 3 5i 6i 10 3 2 –9 5 F –4 4 –6 –9 –3 10 3 –9 –3 4 6i 4i 4 10 3 Evaluate the following equations for clues to the sentence above. Once the blanks are filled, see if you can shorten the phrase into a well-known proverb! 1. 3 i 3 i A 11. N 3 4i 2. 5 4 i 2 12. 3 2i 5i 2 0 5i 16 3. 3i2 i 2i3 2i C 13. 4 i 2i P 4. 1 i 3 1 i i D 14. i2 i 3 4 i Q i7 5i 5. 1 i 2 1 i 2 15. 2i 4i R 2 i 1 2i 5 6. 2i1 3i 2i 2 H 16. i 1i S 1i 2i 2 7. 1 2i 2 3i 2 5i 1 3i I 17. 8. 8i1 i 24 3i K x 2 6 x 73 0 x T U 18. z 2 6 z 34 0 z V Y B 40 i E 9. i 63 L 10. 3 2i 5 7i 7i 29 M Mathematics B30 29 Lesson 1 Summary – Lesson 1 • Create a summary of this lesson to assist you come examination time. • Each summary is to be sent in with the assignment to be evaluated. • Items to include in a summary: • definitions • formulas • calculator “shortcuts” Mathematics B30 30 Lesson 1 Answers to Exercises Exercise 1.1 1. a. b. c. d. e. f. 6i i 3 10 i 6i 6i 7i 2. a. b. c. d. 2 i i4 2 h. i. j. k. i 48 4 i 3 i 1 i e. f. g. 1 l. m. n. i 93 i 4 1 1 a. b. c. 5i 15 i i 5 7 3 Exercise 1.2 1. a. b. c. 4 i 2 5i 6 0i 7 2. a. 3. i i 23 i i 5 i x2 y2 5 2 1 2 26 units Mathematics B30 31 Lesson 1 b. 3 2i 13 units c. y 7 + 4i d. x 7 4 i 65 units x 6 8 i 10 units y 6 – 8i 3. Mathematics B30 a. b. 30 18 i 24 6 3 24 6i 3 c. 24 i 3 5 24 i 8 24 i 4 24 1 24 * remember the laws of exponents. 2 32 2 Lesson 1 Exercise 1.3 1. 2. a. 7 10 i b. 3 i c. 2 2i d. 5 9i e. 6 9i 24 4i 30 5i 56 i f. 7 y a. (7 + 5 i ) x 0 b. y x 0 (5 – 2 i ) Mathematics B30 33 Lesson 1 c. d. Exercise 1.4 1. Mathematics B30 a. 18 12 i b. 15 20 i c. 8 i d. 3 2i e. 13 f. 26 g. 1 h. i. a 2 b2 21 20 i j. 26 18 i 34 Lesson 1 2. a. b. c. d. e. f. g. h. i. j. 3. a. b. c. d. e. 4. a. b. c. d. e. Mathematics B30 2i 12 5 i 13 2 4i 5 2 4i 5 3 2i 13 1i 2 2 3i 2 1 i 3 7 0 i 3 0 3i 1 3i 3i 3i 1 1 i 0 i 2 3i 3i 9 i 9 3 3 1 1 i 8 2i 2 i 2 i 2 0 2 4 4 8 i 8 i 8 i 8 1 2 7i 2 7i 2 7i 2 7 i 2 2 7 i 2 7 i 4 49 i 53 53 53 1 5 3i 5 3i 5 3i 5 3 i 2 5 3 i 5 3 i 25 9 i 34 34 34 1 a bi a bi a bi a b 2 2 2 2 i 2 2 2 2 a bi a bi a b i a b a b a b2 3, 7 3 ,5 2 1 ,4 3 1 1, 2 Solve two equations in two unknowns. 7 8 2 y 2 x , 1 3 x 4 y. Solution , . 11 11 35 Lesson 1 Answers to Self Evaluation A N I N I N T O F E T H D E D E I E R I X V H C I D E E U P S A R S L E D I B L E S Q U A L I T V A T I V S P E F O S A R I Y A M O O K P I T I A F F L L O I A M E E D N R S T A H I N T “Too many cooks spoil the broth.” 17. x 2 6 x 73 0 a 1 1 Use the ‘Quadratic Formula’. b 6 c 73 b b 2 4 ac 2a 6 256 x 2 x 3 8i x= x T U Mathematics B30 36 2 Lesson 1 Mathematics B30 Module 1 Assignment 1 Mathematics B30 37 Lesson 1 Mathematics B30 38 Lesson 1 Optional insert: Assignment #1 frontal sheet here. Mathematics B30 39 Lesson 1 Mathematics B30 40 Lesson 1 Assignment 1 Values (20) A. Multiple Choice: Select the correct answer to complete each of the following statements and place a check () beside it. 1. 49 converted to imaginary form, ix, is ***. ____ ____ ____ ____ 2. 4 4 i 2 16 16 a. b. c. d. 1 12 is ***. 1 i i 1 The absolute value of 9 11i is ***. ____ ____ ____ ____ Mathematics B30 a. b. c. d. The simplest form of the expression ____ ____ ____ ____ 4. undefined i7 i 7 7 i49 2 8 simplified is ***. ____ ____ ____ ____ 3. a. b. c. d. a. b. c. d. 202 40 i 202 i 40 41 Lesson 1 5. 2 3i 7 5i in complex number form is ***. ____ ____ ____ ____ 6. Mathematics B30 a. b. c. d. 30 21 i 3i 2 30 9 i 3i 2 27 21 i 27 9 i When simplified, 2 20 3 i 5 2 45 i 3 80 3 i is ***. ____ ____ ____ ____ 8. 9 2i 5 2i 9 8i 5 8i When in complex number form, 5 i 6 3i is ***. ____ ____ ____ ____ 7. a. b. c. d. a. b. c. d. 14 14 38 38 5 20 i 5 2i 5 20 i 5 2i 1 i 2 3i in complex number form is ***. ____ a. ____ b. ____ c. ____ d. 2 2 i 5 5 5 5 i 13 13 2 i 1 1 5 i 13 13 42 Lesson 1 9. 10. Mathematics B30 The reciprocal of 2 3 i in complex number form is ***. ____ a. ____ b. ____ c. ____ d. 1 1 2 3i 2 3 i 13 13 1 2 3i 2 3 i 5 5 The values for x and y, respectively, in the equation 2 3i x 3 yi 4 6i i are ***. ____ a. ____ b. ____ c. ____ d. 7 3 7 8, 3 7 , 8 3 7 ,8 3 8, 43 Lesson 1 Mathematics B30 44 Lesson 1 Answer Part B and Part C in the space provided. Evaluation of your solution to each problem will be based on the following. (15) B. • A correct mathematical method for solving the problem is shown. • The final answer is accurate and a check of the answer is shown where asked for by the question. • The solution is written in a style that is clear, logical, well-organized, uses proper terms, and states a conclusion. Simplify each to the complex form a bi . 1. 2i i 2 2. 1 3 i 2 3. 4. Mathematics B30 2 i 3 1 i 1 1 i 45 Lesson 1 5. 2i 1 i 6. 2 3i 2 3i 7. i 16 8. i 21 9. Mathematics B30 (Note: an imaginary number i can be written as 0 i in complex form) 48 3 2 46 Lesson 1 10. 6 2 i 8 4 i 18 4 3 3 i 50 11. 5i 9i 12. 6 i 9 11 i 42 i 103 2i 121 13. 5 5i 3 7i 2 3i 5 i Mathematics B30 47 Lesson 1 (2) C. 14. 2 x 3 y 5 x 2 yi 13 4 i . 15. 2 3i x yi 1 . 1. Show that i, i , 1 and 1 are each solutions of the equation x 4 1 0 . Mathematics B30 Solve for x and y. Solve for x and y. 48 Lesson 1 2. Illustrate the following operations on a graph. Label all points clearly. (4) a. 4 9i 5 2i (4) b. (12 4 i) (7 4 i) (5) 3. On a separate piece of paper, write a summary of Lesson 1 that would be suitable as a review for an examination. 50 Mathematics B30 49 Lesson 1 Assignment 1 Checklist: Have you attached the requested forms to the back of your assignment? Mathematics B30 Summary for Lesson 1 50 Lesson 1