3.2 Complex Numbers - Weebly
3.2COMMONCORELearning bComplex NumbersEssential QuestionWhat are the subsets of the set ofcomplex numbers?In your study of mathematics, you have probably worked with only real numbers,which can be represented graphically on the real number line. In this lesson, thesystem of numbers is expanded to include imaginary numbers. The real numbersand imaginary numbers compose the set of complex numbers.Complex NumbersReal NumbersRational NumbersImaginary NumbersIrrational NumbersThe imaginary unit iis defined asIntegers—i 1 .Whole NumbersNatural NumbersClassifying NumbersWork with a partner. Determine which subsets of the set of complex numberscontain each number.—ATTENDINGTO PRECISIONTo be proficient in math,you need to use cleardefinitions in yourreasoning and discussionswith others.—a. 9b. 0 49e. 2—d.—c. 4———f. 1Complex Solutions of Quadratic EquationsWork with a partner. Use the definition of the imaginary unit i to match eachquadratic equation with its complex solution. Justify your answers.a. x2 4 0b. x2 1 0c. x2 1 0d. x2 4 0e. x2 9 0f. x2 9 0A. iB. 3iC. 3D. 2iE. 1F. 2Communicate Your Answer3. What are the subsets of the set of complex numbers? Give an example of anumber in each subset.4. Is it possible for a number to be both whole and natural? natural and rational?rational and irrational? real and imaginary? Explain your reasoning.Section 3.2Complex Numbers103
3.2 LessonWhat You Will LearnDefine and use the imaginary unit i.Add, subtract, and multiply complex numbers.Core VocabulVocabularylarryFind complex solutions and zeros.imaginary unit i, p. 104complex number, p. 104imaginary number, p. 104pure imaginary number, p. 104The Imaginary Unit iNot all quadratic equations have real-number solutions. For example, x2 3has no real-number solutions because the square of any real number is never anegative number.To overcome this problem, mathematicians createdan expanded system of numbers—using the imaginary unit i, defined as i 1 . Note that i 2 1. The imaginaryunit i can be used to write the square root of any negative number.Core ConceptThe Square Root of a Negative NumberPropertyExample———— 3 i 3( i —3 )2 i 2 3 31. If r is a positive real number, then r i r .—2. By the first property, it follows that ( i r )2 r. Finding Square Roots of Negative NumbersFind the square root of each number.——a. 25—b. 72c. 5 9SOLUTION b. 72 72 1 36 2 i 6 2 i 6i 2c. 5 9 5 9 1 5 3 i 15i——————a. 25 25 1 5i———————Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the square root of the number.—1. 4—2. 12——4. 2 543. 36A complex number written in standard formis a number a bi where a and b are real numbers.The number a is the real part, and the number biis the imaginary part.Complex Numbers (a bi )RealNumbers(a 0i)a biIf b 0, then a bi is an imaginarynumber. If a 0 and b 0, then a biis a pure imaginary number. The diagramshows how different types of complex numbersare related.104Chapter 3Quadratic Equations and Complex Numbers 153π2ImaginaryNumbers(a bi, b 0)2 3i 9 5iPureImaginaryNumbers(0 bi, b 0) 4i6i
Two complex numbers a bi and c di are equal if and only if a c and b d.Equality of Two Complex NumbersFind the values of x and y that satisfy the equation 2x 7i 10 yi.SOLUTIONSet the real parts equal to each other and the imaginary parts equal to each other.2x 10x 5Equate the real parts. 7i yiEquate the imaginary parts.Solve for x. 7 ySolve for y.So, x 5 and y 7.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the values of x and y that satisfy the equation.5. x 3i 9 yi6. 9 4yi 2x 3iOperations with Complex NumbersCore ConceptSums and Differences of Complex NumbersTo add (or subtract) two complex numbers, add (or subtract) their real parts andtheir imaginary parts separately.Sum of complex numbers:(a bi) (c di) (a c) (b d)iDifference of complex numbers:(a bi) (c di) (a c) (b d)iAdding and Subtracting Complex NumbersAdd or subtract. Write the answer in standard form.a. (8 i ) (5 4i )b. (7 6i) (3 6i)c. 13 (2 7i) 5iSOLUTIONa. (8 i ) (5 4i ) (8 5) ( 1 4)i 13 3iDefinition of complex additionWrite in standard form.b. (7 6i ) (3 6i ) (7 3) ( 6 6)iDefinition of complex subtraction 4 0iSimplify. 4Write in standard form.c. 13 (2 7i ) 5i [(13 2) 7i] 5iDefinition of complex subtraction (11 7i ) 5iSimplify. 11 ( 7 5)iDefinition of complex addition 11 2iWrite in standard form.Section 3.2Complex Numbers105
Solving a Real-Life ProblemEElectricalcircuit components, such as resistors, inductors, and capacitors, all opposetthe flow of current. This opposition is called resistance for resistors and reactance foriinductors and capacitors. Each of these quantities is measured in ohms. The symboluused for ohms is Ω, the uppercase Greek letter omega.Resistor Inductor CapacitorComponent andsymbolResistance orreactance (in ohms)RLCImpedance (in ohms)RLi Ci5Ω3Ω4ΩAlternating current sourceThe table shows the relationship between a component’s resistance or reactance andits contribution to impedance. A series circuit is also shown with the resistance orreactance of each component labeled. The impedance for a series circuit is the sumof the impedances for the individual components. Find the impedance of the circuit.SOLUTIONThe resistor has a resistance of 5 ohms, so its impedance is 5 ohms. The inductor hasa reactance of 3 ohms, so its impedance is 3i ohms. The capacitor has a reactance of4 ohms, so its impedance is 4i ohms.Impedance of circuit 5 3i ( 4i ) 5 iThe impedance of the circuit is (5 i ) ohms.To multiply two complex numbers, use the Distributive Property, or the FOIL method,just as you do when multiplying real numbers or algebraic expressions.Multiplying Complex NumbersMultiply. Write the answer in standard form.STUDY TIPWhen simplifying anexpression that involvescomplex numbers, be sureto simplify i 2 as 1.a. 4i( 6 i )b. (9 2i )( 4 7i )SOLUTIONa. 4i( 6 i ) 24i 4i 2Distributive Property 24i 4( 1)Use i 2 1. 4 24iWrite in standard form.b. (9 2i )( 4 7i ) 36 63i 8i 14i 2Multiply using FOIL. 36 71i 14( 1)Simplify and use i 2 1. 36 71i 14Simplify. 22 71iWrite in standard form.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com7. WHAT IF? In Example 4, what is the impedance of the circuit when the capacitoris replaced with one having a reactance of 7 ohms?Perform the operation. Write the answer in standard form.8. (9 i ) ( 6 7i )11. ( 3i )(10i )106Chapter 39. (3 7i ) (8 2i )12. i(8 i )Quadratic Equations and Complex Numbers10. 4 (1 i ) (5 9i )13. (3 i )(5 i )
Complex Solutions and ZerosSolving Quadratic EquationsSolve (a) x2 4 0 and (b) 2x2 11 47.SOLUTIONLOOKING FORSTRUCTUREa. x2 4 0Notice that you can usethe solutions in Example6(a) to factor x2 4 as(x 2i )(x 2i ).x2Write original equation. 4Subtract 4 from each side.—x 4Take square roots of each side.x 2iWrite in terms of i.The solutions are 2i and 2i.b. 2x2 11 47Write original equation.2x2 36Add 11 to each side.x2 18Divide each side by 2.—x 18Take square roots of each side.x i 18Write in terms of i.——x 3i 2Simplify radical.——The solutions are 3i 2 and 3i 2 .Finding Zeros of a Quadratic FunctionFind the zeros of f (x) 4x2 20.FINDING ANENTRY POINTSOLUTIONThe graph of f does notintersect the x-axis, whichmeans f has no real zeros.So, f must have complexzeros, which you can findalgebraically.4x2 20 0Set f (x) equal to 0.4x2 20Subtract 20 from each side.x2 5Divide each side by 4.—x 5Take square roots of each side.—x i 5Write in terms of i.——So, the zeros of f are i 5 and i 5 .y40Check30—— 2 f ( i 5 ) 4( i 5 ) 20 4 5i 2 20 4( 5) 20 0—— 2 f ( i 5 ) 4( i 5 ) 20 4 5i 2 20 4( 5) 20 010 4 224 xMonitoring Progress Help in English and Spanish at BigIdeasMath.comSolve the equation.14. x2 1315. x2 3816. x2 11 317. x2 8 3618. 3x2 7 3119. 5x2 33 3Find the zeros of the function.20. f (x) x2 721. f (x) x2 4Section 3.222. f (x) 9x2 1Complex Numbers107
Exercises3.2Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY What is the imaginary unit i defined as and how can you use i?2. COMPLETE THE SENTENCE For the complex number 5 2i, the imaginary part is and thereal part is .3. WRITING Describe how to add complex numbers.4. WHICH ONE DOESN’T BELONG? Which number does not belong with the other three? Explainyour reasoning.3 0i2 5i— 3 6i0 7iMonitoring Progress and Modeling with Mathematics23. (12 4i ) (3 7i )24. (2 15i ) (4 5i )—25. (12 3i ) (7 3i )26. (16 9i ) (2 9i )—27. 7 (3 4i ) 6i28. 16 (2 3i ) i29. 10 (6 5i ) 9i30. 3 (8 2i ) 7iIn Exercises 5–12, find the square root of the number.(See Example 1.)—5. 366. 64—7. 188. 24——9. 2 1610. 3 49——12. 6 6311. 4 3231. USING STRUCTURE Write each expression as acomplex number in standard form.—13. 4x 2i 8 yi————a. 9 4 16In Exercises 13–20, find the values of x and y thatsatisfy the equation. (See Example 2.)—b. 16 8 3632. REASONING The additive inverse of a complexnumber z is a complex number za such thatz za 0. Find the additive inverse of eachcomplex number.14. 3x 6i 27 yi15. 10x 12i 20 3yia. z 1 i16. 9x 18i 36 6yib. z 3 ic. z 2 8i17. 2x yi 14 12iIn Exercises 33 –36, find the impedance of the seriescircuit. (See Example 4.)18. 12x yi 60 13i33.119. 54 —7 yi 9x 4i34.12Ω9Ω7Ω4Ω6Ω9Ω120. 15 3yi —2 x 2iIn Exercises 21–30, add or subtract. Write the answerin standard form. (See Example 3.)21. (6 i ) (7 3i )108Chapter 3HSCC Alg2 PE 03.02.indd 10835.22. (9 5i ) (11 2i )36.8Ω14Ω3Ω2Ω7Ω8ΩQuadratic Equations and Complex Numbers5/28/14 11:53 AM
In Exercises 37–44, multiply. Write the answer instandard form. (See Example 5.)In Exercises 55–62, find the zeros of the function.(See Example 7.)37. 3i( 5 i )38. 2i(7 i )55. f (x) 3x2 656. g(x) 7x2 2139. (3 2i )(4 i )40. (7 5i )(8 6i )57. h(x) 2x2 7258. k(x) 5x2 12541. (4 2i )(4 2i )42. (9 5i )(9 5i )59. m(x) x2 2760. p(x) x2 9843. (3 6i )244. (8 3i )261. r (x) —2 x2 241162. f (x) —5 x2 10JUSTIFYING STEPS In Exercises 45 and 46, justify eachERROR ANALYSIS In Exercises 63 and 64, describestep in performing the operation.and correct the error in performing the operation andwriting the answer in standard form.45. 11 (4 3i ) 5i63. [(11 4) 3i ] 5i (7 3i ) 5i (3 2i )(5 i ) 15 3i 10i 2i 2 15 7i 2i 2 2i 2 7i 15 7 ( 3 5)i64. 7 2i46. (3 2i )(7 4i ) (4 6i )2 (4)2 (6i )2 16 36i 2 16 (36)( 1) 21 12i 14i 208i 2 21 2i 8( 1)65. NUMBER SENSE Simplify each expression. Then 21 2i 8classify your results in the table below.a. ( 4 7i ) ( 4 7i ) 29 2ib. (2 6i ) ( 10 4i )REASONING In Exercises 47 and 48, place the tiles in thec. (25 15i ) (25 6i )expression to make a true statement.d. (5 i )(8 i )47. ( i ) – ( i ) 2 4ie. (17 3i ) ( 17 6i )7436f. ( 1 2i )(11 i )g. (7 5i ) (7 5i )48. i( i ) 18 10i 592h. ( 3 6i ) ( 3 8i )RealnumbersImaginarynumbersPure imaginarynumbersIn Exercises 49–54, solve the equation. Check yoursolution(s). (See Example 6.)49. x2 9 050. x2 49 051. x2 4 1152. x2 9 1553. 2x2 6 3466. MAKING AN ARGUMENT The Product Property— ——states a b ab . Your friend concludes——— 4 –9 36 6. Is your friend correct?Explain. 54. x2 7 47Section 3.2Complex Numbers109
67. FINDING A PATTERN Make a table that shows the75. OPEN-ENDED Find two imaginary numbers whosepowers of i from i1 to i8 in the first row and thesimplified forms of these powers in the second row.Describe the pattern you observe in the table. Verifythe pattern continues by evaluating the next fourpowers of i.sum and product are real numbers. How are theimaginary numbers related?76. COMPARING METHODS Describe the two differentmethods shown for writing the complex expression instandard form. Which method do you prefer? Explain.68. HOW DO YOU SEE IT? The graphs of three functionsMethod 14i (2 3i ) 4i (1 2i ) 8i 12i 2 4i 8i 2are shown. Which function(s) has real zeros?imaginary zeros? Explain your reasoning. 8i 12( 1) 4i 8( 1)yh4f 20 12ig2 4Method 24i(2 3i ) 4i (1 2i ) 4i [(2 3i ) (1 2i )]4x 4i [3 5i ] 12i 20i 2 4 12i 20( 1) 20 12iIn Exercises 69–74, write the expression as a complexnumber in standard form.77. CRITICAL THINKING Determine whether each69. (3 4i ) (7 5i ) 2i(9 12i )statement is true or false. If it is true, give anexample. If it is false, give a counterexample.70. 3i(2 5i ) (6 7i ) (9 i )71. (3 5i )(2 a. The sum of two imaginary numbers is animaginary number.7i 4)b. The product of two pure imaginary numbers is areal number.72. 2i 3(5 12i )c. A pure imaginary number is an imaginary number.73. (2 4i 5) (1 9i 6) ( 3 i 7 )d. A complex number is a real number.74. (8 2i 4) (3 7i 8) (4 i 9)78. THOUGHT PROVOKING Create a circuit that has animpedance of 14 3i.Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsDetermine whether the given value of x is a solution to the equation.79. 3(x 2) 4x 1 x 1; x 1(Skills Review Handbook)80. x3 6 2x2 9 3x; x 5Write an equation in vertex form of the parabola whose graph is shown.y82.83.( 1, 5)84.y4(0, 3)2(1, 2) 2110(Section 2.4)462y31981. x2 4x —x2; x —43Chapter 32 64x(3, 2) 21x 2( 3, 3)Quadratic Equations and Complex Numbers 46x(2, 1)
system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers. Complex Numbers Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Natural Numbers The imaginary unit i is defi
To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. a bi, bi b a bi. a i2 1. i 1 1. x x2 1 0 Appendix E Complex Numbers E1 E Complex Numbers Definition of a Complex Number For real numbers and the number is a complex number.If then is called an imaginary number.
extend the real numbers to the complex numbers by using the closure property. define i as -1 and plot complex numbers on the complex plane. perform operations on complex numbers, including finding absolute value. Materials: DO NOW; pairwork; homework 4-3 Time Activity 5 min Review Homework Show the answers to hw #4-2 on the overhead.
7.2 Arithmetic with complex numbers 7.3 The Argand Diagram (interesting for maths, and highly useful for dealing with amplitudes and phases in all sorts of oscillations) 7.4 Complex numbers in polar form 7.5 Complex numbers as r[cos isin ] 7.6 Multiplication and division in polar form 7.7 Complex numbers in the exponential form
Oct 07, 2012 · Complex Numbers Misha Lavrov ARML Practice 10/7/2012. A short theorem Theorem (Complex numbers are weird) . the square root, we get p 1 p 1 p 1 p 1: Finally, we can cross-multiply to get p 1 p 1 p 1 p 1, or 1 1. Basic complex number facts I Complex numbers are numbers of the form a b_{, where _ .
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Any complex number is hence expressed as z Re(z) iIm(z). The set of real numbers is denoted by R, and the set of complex numbers by C. From (3) it follows that real numbers are complex numbers with a zero imag-inary part, namely x x 0i. In particular 0 is the complex number having both real and imaginary parts equal to zero. 1
A. Complex numbers 1 Introduction to complex numbers 2 Fundamental operations with complex numbers 3 Elementary functions of complex variable 4 De Moivre’s theorem and applications 5 Curves in the complex plane 6 Roots
APPENDIX I ARCHITECTS AND DESIGNERS’ BIOGRAPHIES Architects at the University of Stirling 1.1 Robert Matthew Johnson-Marshall and Partners Robert Hogg Matthew was born in Edinburgh in 1906 and was educated at the then Edinburgh Institution (now Stewarts Melville College). He trained to be an architect at the Edinburgh College of Art, gaining his diploma in 1930. Upon graduation Matthew .
