Real Number Systems 15

Electricians use a multi-meter to measure voltage, amps, and resistance in a circuit. Multi-meters are used when troubleshooting a circuit and identifies when no power is traveling to a location.

15.1 The Real Numbers . . . For Realsies! The Numbers of the Real Number System...... 1077

15.2 Getting Real, and Knowing How . . . Real Number Properties...... 1085

15.3 Imagine the Possibilities Imaginary and Complex Numbers...... 1091

15.4 Now It’s Getting Complex . . . But It’s Really Not Difficult! Complex Number Operations...... 1099

15.5 It’s Not Complex—Just Its Solutions Are Complex! Solving Quadratics with Complex Solutions . . . . .1113 © Carnegie Learning

1075 1076

Learning Goals Key Terms

In this lesson, you will: • natural numbers • Define sets of natural numbers, whole numbers, • whole numbers integers, rational numbers, irrational numbers, • closed (closure) and real numbers . • counterexample • Determine under which operations different sets • integers of number are closed . • rational numbers • Create a Venn diagram to show how different • irrational numbers number sets are related . • real numbers • Determine which equations can be solved using • Venn diagram different number sets . • Write repeating decimals as fractions .

athematicians have given some types of numbers special names. A repunit Mnumber is an integer with all 1’s as digits. So, 11, 111, 1111, and so on, are all repunit numbers. A pronic number is a number that is the product of two consecutive counting numbers. The numbers 2, 6, and 12 are the first pronic numbers, because 1 3 2 5 2, 2 3 3 5 6, and 3 3 4 5 12.

Another set of numbers are referred to as the “lucky” numbers by some. To determine the lucky numbers, first start with all the counting numbers (1, 2, 3, 4, 5, and so on). Delete every second number. This will give you 1, 3, 5, 7, 9, and so on. The second number in that list is 3, so cross off every third number remaining. Now you have 1, 3, 7, 9, 13, and so on. The next number that is left is 7, so cross off every seventh number remaining. © Carnegie Learning Can you list all the “lucky” numbers less than 50?

1077 15 1078 Problem

Chapter 15 first set of numbers that you learned whenyouwerefirst setofnumbersthatyoulearned veryyoung,thecountingnumbers,or saw orowned Sometime inthehistoryofhumanity, itbecamenecessaryforpeopletocountobjectsthey numbers thatare addedtogetherresult inasumthatisnaturalnumber For instance,thesetofnaturalnumbersisclosedunder additionbecauseanynatural also inthesameset,setissaidtobe numbers in a set and the result is a number thatis When an operation is performed on any of the numbers numbers thanthoseincludedinthesetofwhole operations on numbers that they discovered more It was probably not until people began performing The setof or zero, theyusedtheset of Once peoplerecognized that theyneededanumbertorepresent thelackof anyquantity, The setof natural numbers 3. 2. 1. 1

List the whole number(s) and 1? How many whole numbers are between How many natural numbers are between Is thesetofnaturalnumbersfiniteorinfinite? Real NumberSystems

Let’s Take aWalk throughNumberHistory . whole numbers natural numbers

. Thisishowtheideaofnumberscameabout . consistsofthesetnaturalnumbersandnumber0 consistsofthenumbersthatyouusetocountobjects whole numbers . closed . 2 2 1 1 and 1? List the natural number(s) (ortohave one more number than the natural numbers (because itincludeszero),thenhowmany and thesetofwholenumbersincludes numbers areinthesetofwhole . natural numbersisinfinite, Theyprobably beganwith the closure If thesetof ) underthatoperation numbers?! . . . . .

example thatshowstheresult isnotpartofthatset To showthatasetis The set of set of numbers called So, the need for people to perform certain operations on numbers is what most likely led to the

5. 4. 7. 6. 8.

closed, provide at least one example to show the set is not closed numbers closed? If the set is closed under an operation, explain why Under which other operations—subtraction, multiplication, division—is the set of natural operation, explain why Under which operations is the set of whole numbers closed? If the set is closed under an What do you notice? Compare the closure properties of whole numbers with the closure properties of integers counterexample for each operation not closed under an operation(s), provide at least one set is closed under an operation(s), explain why Under which operations is the set of integers closed? If the How many integers are between integers consists of the set of whole numbers and their opposites not integers closedunderanoperation,youonlyneedtodetermine . If the set is not closed, provide at least one counterexample . 2 2 and 2? List the integer(s) . 15.1

The NumbersoftheRealNumber System . . If the set is Thisiscalleda . counterexample . . If the set is not are additive inverses if theirsumis0. . Two numbers one

. .

1079 15 15 1080

Chapter 15 The set of more than one person At some point, people were confronted with the problem of having to divide one thing among b 11. 10. are integers, but 9.

numbers Compare the closure properties of integers with the closure properties of rational counterexample for each operation operation(s), provide at least one If the set is not closed under an closed under an operation(s), explain why rational numbers closed? If the set is Under which operations is the set of How many rational numbers are between Real NumberSystems rational numbers . What do you notice? b is not equal to 0 . From this dilemma came the set of consists of all numbers that can be written as . . . 2

1 and 1? That is, it does not result in an answer. So, it is not to be determining closure by 0 is not defined. Remember, division considered when rational numbers properties. . __ b a

, where a and

© Carnegie Learning © Carnegie Learning such as The set of irrational numbers are related A of irrational numbers The set of examples are Eventually, people realized there are some numbers that are not rational numbers  and Irrational numberscanberepresented byasymbol,suchas 13. 12.

Venn diagram <

3 b Write eachnumbersetfrom theword boxinitsappropriate placeintheVenn diagram operation, explainwhy Under whichoperationsisthesetofreal numbersclosed?Ifthesetisclosedunderan are integers . 14 or √ __ 2 rational numbers irrational numbers real numbers natural numbers . .

Irrationalnumbersare often approximated byadecimalorfraction,suchas  

< , √

usescircles toshowhow elementsamongsetsofnumbersorobjects ___ 22 . __ 2 7 . . , and

However, they have no exact numerical representation . consists of the set of rational numbers and the set √ __ 3 .

. These numbers cannot be written as fractions Ifthesetisnotclosed,provide atleastonecounterexample consists of all numbers that cannot be written as irrational numbers whole numbers Real Numbers 15.1

The NumbersoftheRealNumber System  integers , orbyusingothernotation, . They are called . . Some __ b a

, where

a .

. 1081 15 15 1082 Problem

Chapter 15 Consider theequationsshown

5. 4. 3. 2. 1. 2

• numbers Suppose thattheonlynumbersyoucanusetoevaluate thegivenequationsare real rational numbers Suppose thattheonlynumbersyoucanusetoevaluategivenequationsare the integers Suppose that the only numbers you may use to evaluate the given equations are the whole numbers Suppose thattheonlynumbersyoumayusetoevaluategivenequationsare the natural numbers Suppose thattheonlynumbersyoumayusetoevaluategivenequationsare the • • • Real NumberSystems

Equation A:3 Equation G: Equation E: Equation C:8 Real Numbersand . Which equations could you solve? . Whichequationscouldyousolve? x x .

Whichequationscouldyousolve? x

. x 1 . 1 Whichequationscouldyousolve?

Whichequationscouldyousolve? 5 5 5 6 9 4 5 5

1 6

. E quations • • •

Equation F: Equation D: Equation B: x x x 2

5 5 1 4 2 5 5

© Carnegie Learning © Carnegie Learning Problem it isarationalnumber decimal thatrepeats single digitsorblocksofcanbewrittenasafraction;therefore, All irrationalnumbershaveaninfinitenumberofnon-repeating decimalplaces

1. 3

a. Represent eachdecimalasafraction

0 It’s Repeating,It’s Repeating,It’s Repeating. So, the repeating decimal 0 Step 4: Step 3: Step 2: Step 1: repeating decimal0 You canusealgebratowritearepeating decimalasafraction . 2222 . Solvefor Subtractthefirstequationfrom thesecondequation Setthedecimalequaltoavariable

equation by100 the decimalhas2repeating digits,somultiplybothsidesofthe same numberofzeros as repeating digitsinthedecimal Multiply bothsidesoftheequationbyamultiple10thathas .

. . v . . 313131 . . 31 ___ . 100 100

2 isequaltothefraction . ____ 99 99

99 . v (

v v v v v 15.1 v 5 .

Showyourwork

5 5 5 5 5 5 0 31 31 0 31

. ___ ___ 31 99 31 99 313131 b. . The NumbersoftheRealNumber System 313131 . .

313131 313131 0 . . 512512 bar notationmeans . that thenumbers

. . Remember, the

below thebar . . . .

. . .

) repeat. . . . .

___ 99 31 . .

. . Considerthe . Inthiscase, . Anyinfinite

1083 15 15 1084

Chapter 15

Be prepared toshare yoursolutionsandmethods 2.

Consider eachstudent in theirreasoning? Doyouthinkthat0 Are anyofthestudentscorrect intheirreasoning? Are anyofthestudentsincorrect Real NumberSystems Donnie So, 0.999 . . . is equal to 1. -(x =0.999 . . .) Suppose thatx=0.999.

10x =9.999 . . . __ 9x 9x =9 x = 1 9

= __ 9 9

__ that know I Nathaniel So, 1 equals 0.999 ...... 0.999 equals 1 So, 3 1

+ + 1 = 0.999 . . . . . 0.999 = 1

__ __ 2 3 3 3

= 0.333 . . . + 0.666 . . . . . 0.666 + . . . 0.333 = = 0.999 . . . . . 0.999 = ’ s workshown __ 3 1

= 0.333 . . . and and . . . 0.333 = . . 9 __ isequalto1?Explainyourreasoning . __ 2 3

= 0.666 ...... 0.666 =

to 0.999. The number1isequal Kris

9·

_ 9 1

1=0.999. _

9 =0.111. 1

=9·0.111. .

© Carnegie Learning © Carnegie Learning mathematics uses a type of shorthand? aswell? mathematics usesatypeofshorthand? Doyouthink Wheredoyouthinkmostpeopleuseatypeofshorthand? shorthand. If youuseacellphoneorchatonyourcomputer, youprobablyknowsomethingabout press conferencecanbecapturedandthenadded toastory. toensurethataquotetheyhearduring also useshorthand Newspaper reporters maybeableto writemorekeywordsorideasduringapresentation. shorthand speed ofwritingincomparisontofullwords.Therefore,aperson isanabbreviatedsymbolicwritingmethodthatincreasesthe Shorthand shorthand. mayuse reporters, People whotakenotesoften,suchassecretariesorcourt written ortypedkeywords. likely, youdidn’twriteword-for-word whatyourteachersaid,but youmighthave At somepointinschool,youprobablyhavehadtotakenotesduringclass.Morethan Y In thislesson,youwill: Le Properties Number Real . . . How Knowing and Real, Getting • • • arning additive identity, multiplicativeidentity, additiveinverse,andmultiplicativeinverse Identify the properties of the real number systemMake statementsaboutreal numberproperties usingsetnotation including: commutative, associative, distributive, setnotation Learn ikes! Whatdidourteachersay? Go als . . . 15.2 1085 15 1086 Problem

Chapter 15 The AssociativeProperty ofAdditioncanbewrittenthisway: are addedormultipliedcanbegrouped inanywayandthesumorproduct isthesame Addition andmultiplicationare alsoassociative To showtheCommutativeProperty ofAddition,youcanusethesetnotation shown: be addedormultipliedinanyorder andthesumorproduct willbethesame Addition andmultiplicationare commutative set ofreal numbers, For allnumbers So, theentire statementis read as: is read as “ The symbol

is anelementof, 1. 2. 3. 1

b. using similarsetnotation Write theCommutativeProperty ofMultiplication If so,writetheproperties insimilarsetnotation Are subtractionanddivisioncommutative? a. Write thisstatementinwords Real NumberSystems

notation Are divisionandsubtractionassociative?Ifso,writetheproperties insimilarset Write theAssociativeProperty ofMultiplicationusingsimilarnotation O “ perations onRealNumbers the setofreal numbers ; isread as a . Ifnot,givecounterexamples and ” or a plus “ b are elementsof thatare elementsofthe “ for all b ; equals a . , . b ” , Thesymbol ; . c . ”

a e b

R 1 b . , ( ”

e a Finally, thesymbol a

R .

1 , . Recall,commutativemeansthatnumberscan

a b .

e 1 ) . isread as Recall,associativemeansthatnumbers 1

c 5

. 5 Ifnot,givecounterexamples

a ( b R

c ) real numberfor b andthisstatement substitute any will betrue. You can . . . a and .

© Carnegie Learning © Carnegie Learning 4. over Additionthisway: Multiplication is distributive over addition. You can write the Distributive Property of Multiplication 5. 9. multiplicative identity. When anumberismultipliedbyitsmultiplicativeinverse, theproduct isthe 8. When anumberisaddedtoitsadditiveinverse,thesumidentity. 7. real number Similarly, there isanumbercalledthemultiplicativeidentitythatwhenmultipliedbyany 6. number iscalledtheadditiveidentity. Ther e isanumberthatwhenaddedtoanyreal number Is multiplicationdistributiveoversubtraction?Ifso,writetheproperty usingsimilarset not true. a counterexample forthe property thatis in similarsetnotation.Ifnot,provide subtraction? Ifso,writetheproperty thatistrue Is divisiondistributiveoveradditionand notation. Ifnot,giveacounterexample. For anyreal number For anyreal number What numberisthemultiplicativeidentity?Explainyourreasoning. What numberistheadditiveidentity?Explainyourreasoning. a , theproduct isequalto a a , whatisitsmultiplicativeinverse?Explainyourreasoning. , whatisitsadditiveinverse?Explainyourreasoning. ; a , b , c a

e .

R , a ( b

c ) 5

ab a , thesumisequalto

1 15.2 denominator ever

ac be 0inyour expressions? Real Can the

Number

Properties a . Recall,this

1087 15 15 1088 Problem

Chapter 15

1. 2

transformation, orsimplificationusedinthestep at atime Each expression hasbeen simplifiedorsolvedonestep b. a. c. Real NumberSystems

7 7 5 5 5 4 4 2 4 4 4( 4( x (7 7 Simplifying

x x x x x x x x 1 x x x x 1 1

1 1 1 1 1 1 2 1 1 1 2 25 1 (5 5( 4 4 39 (4 4 12 6 . 3) 3) 6 22 2 Nexttoeachstep,identifytheproperty, x x x

x 5 2 2 1

1 1 1

) 1 2 2 x 2 6 3(2 35 1

35) 35) 7) 4 1 5 5( 5 x (4 12

x x

1 1 x x

1 1

2 1

2 21 21

10 10

1 2) 21) 7) 10

xpressions

. you canusefor Combining like justification terms isa a step.

Be prepared toshare yoursolutionsandmethods ultiplicative Inverse 2.

M property set notationfrom thelesson intheappropriate sections Complete thegraphicorganizer ultiplicative Identity . R Properties eal Number . Write eachproperty ofthereal numbersystemusing . Distributive Properties Additive Identity Additive Inverse 15.2 . Thenprovide examplesof each

Real NumberProperties

1089 15 15 1090

Chapter 15

Real NumberSystems

© Carnegie Learning © Carnegie Learning equally imaginary? numbersasrealothernumbers?Orareall Are imaginary people asothernumbersaretoyou. numbersareasrealtothese electronic devicesandoutputs.Imaginary ofvarious numbers toanalyzevoltageandcurrents engineers useimaginary people suchaselectricalengineersandairplanedesigners.Forexample, T In thislesson,youwill: Le Numbers Complex and Imaginary Possibilities the Imagine • • • • arning numbers belong Determine the number sets to which complex numbers Understand properties of thesetof imaginary numbers Simplify expressions involving Determine powersof “imaginary” numbers may seem absurd, but imaginary numbersareusedby numbersmay seemabsurd,butimaginary “imaginary” here arenumbersinmathematicsknownas Go als . . . i .

Key Te imaginary numbers imaginary • • • • • • • imaginary partofacomplexnumber real partofacomplexnumber complex numbers pure imaginarynumber imaginary numbers the number exponentiation r m s i . Theconceptof 15.3 1091 15 1092 Problem

Chapter 15 closed underanotheroperationcalled subtraction, multiplication,anddivision You determinedthatthesetofreal numbersisclosedundertheoperations of addition, exponents number real let’s consider Now, quantity toapower that whenitissquared, itisequaltoanegativenumber way tocalculatethesquare root ofanegativenumber In order forthereal numberstobeclosedforallreal numberexponents,there mustbesome 3. 2. 1. 1

Is the set of real numbers closed for all real number exponents? Explain why or why not e. d. c. b. a. Simplify eachexpression, ifpossible provide acounterexample Are thereal numbersclosedforallintegerexponents?Explainwhyornot Real NumberSystems

( 2 ( 4 4 Imagine, if 2 2 __ 2 2 1 (4)

4 4) 5 5 ) __ 2 2 2 1

5 5 5

. Y ou Will . . exponentiation . Let’s explore whetherthesetofreal numbersis . . 1/2 power is the same thing as taking the . .

Thatis,there mustbeanumbersuch Exponentiation number to the . square root. Raising a

meanstoraisea . Ifnot, .

such that this reason, mathematicians definedwhatiscalled If adefinitionexists,thenitmustbepossibletocalculateany root ofany real number

7. 6. 8. 5. 4.

d. b. c. d. b. d. you notice? Compare youranswersinQuestion5to6 c. a. Part (a)hasbeendoneforyou Use youranswersinQuestion5tocalculateeachpowerof a. Calculate eachpowerof a. of Write thevaluesoffirst fourpowers If b. c. i

2 i .

i i i i i i i i i i i i 5 103 102 101 4 3 2 1 104 8 7 6 5

5 5 5 5 5 5 5 5 i

5 5 5 5 5

i 1, thenwhatisthevalueof 4

? 2

i 1 1

. 5 1

√ ___ 2 1

5 i .

√ ___ 2

1 .

i ?

15.3 the numberi though they are both numbers, each is

special enough that it gets its very is similar to the number Imaginary andComplexNumbers Imaginary

own symbol.

The number .

i . The number

i .

Whatdo  π: even

i isanumber

. For

1093 15 15 1094

Chapter 15

You cansimplifyexpressions involvingnegativeroots byusing 10. 9.

a. d. integer powerof Describe howyoucancalculatethevalueofany c. b. Simplify eachexpression byusing Real NumberSystems

______5 √ √ 6 Factor out So, Rewrite theradicalexpression Rewrite The expression Simplify ______2 2 1 2 12 4 2 √ √

√

5 ______2 2

_____ 2 5

50 8 25

√

√ 5

___ 2

simplifiesto5 ___ 25 5

2 1

as i . . √ i . _____ 2 25

canbesimplified i . . i .

√ _____ 2 25 .

5 5 5 5 5 5

√ √ ______ i √ ( ___ 2 2 when the exponent ___ 2 1 1)(25) the remainder hint: Consider 1

of ?

Here's a i √ is divided by 4. ___ 25 i

11.

d. c. b. a. Simplify eachalgebraicexpression e.

( 3 xi xi (2 x Consider theexpression ( Use thepowersof Combine liketerms Group liketerms Multiply binomials steps shown So, ( Simplify x

x 2 1 1

1 2

xi xi i 5 x )

2 i

)(

i 5 5 2

5 x 2 .

2 i 2 )( 3 x 2

1 i . ) 3 5

i i )

2 . 5 .

xi i x . torewrite 2

x 1

5 . x

i i )( 2 . as Showyourwork x

1 2

i ) 1 . To simplifytheexpression followthe . ( x 15.3

)( x Imaginary andComplexNumbers Imaginary

1 .

variable .eventhoughit's i ) 5 5 5 5 5 like terms, i .

x x x x x When combining 2 2 2 2 2

a constant. 1 2 2 1 1 1 ( (

i xi 2 2 xi

2 1) 2 i actslikea

xi xi

2 ) 2

i 2

1095 15 15 1096 Problem

Chapter 15 The set of notation b represented bythenotation term and The setof are real numbers and 2 Rational Numbers

Natural Numbers Whole Numbers b bi Real NumberSystems

are real numbers iscalledthe Integers It’s GettingaBitComplex I . W A Q imaginary numbers N complex numbers Z a pure imaginary number 1

bi wher Real Numbers imaginary partofacomplexnumber e . b Theterm b is not equal to 0 R 5 0and C Irrational Numbers isthesetofallnumberswritteninform . is the set of all numbers written in the form a a iscalledthe , b is a number of the form J a Complex Numbers e

1 . R The set of imaginary numbers is represented by the

bi wher C real partofacomplexnumber e a , b numbers. Seriously, likeall a e

1 R the entireuniverseof . Thesetofcomplexnumbersis

bi bi Imaginary Numbers Now youknow wher , where of them! e b b I ﬁ is not equal to 0 0and a a

1 1

bi bi , where , where a , andthe , b e

R a a . and

Be prepared toshare your solutionsandmethods 2. 1.

b. List c. e. d. e. c. a. a. Complete eachstatementwith rational number natural number

If a number is a complex number, then it is If anumberisreal number, thenitis If anumberiscomplexnumber, thenitis If anumberisreal number, thenitis __ 3 3 If a number is an imaginary number, then it is 8 7 all i

numbersetsfrom theword boxthatdescribeeachgivennumber whole number imaginary number always , sometimes b. f. d. 15.3 .

√ 6 5 irrational number complex number

. __ 7 45 2 ___

Imaginary andComplexNumbers Imaginary , or

never . animaginarynumber acomplexnumber an imaginary number areal number a complex number real number integer . . .

. . 1097 15 15 1098

Chapter 15

Real NumberSystems

© Carnegie Learning © Carnegie Learning Why doyouthink“4” is sospecialincreatingmusic? mashups usethissame conceptof4measuresmusictocreatenew musicalpieces? themtogethertocreatea newmusicalpiece.Doyouthinkthat songs) andarranging generally usetwoormorepre-recordedsongs (notjustsamples,butentirelymixed Even morerecently, havebeenusingtechnology tocreatemashups.Mashups artists effects, recordscratches,andlyrics. minutes indurationwiththeDJinfusingother samplesofvinylnoise,ambientsound are repeatedforanentirepiece—andsometimes thesepiecescanlast20to30 measures it tobecomethespineoftheirnewpiece.Sometimes thosefourdrum-beat beats(witheachmeasurehaving4permeasure), andreuse measures ofdrum beats.ManyDJswilltakefour One ofthemostcommonsamplesistakingdrum ora“sample”ofone soundrecordingandrepurposingitintoanothersong. portion ofalmost anyDJ’sThe cornerstone ofsampling.Samplingistaking a musicistheart been mixingitsincethelate1960s. oronalaptop! Ofcourse,thesesolomusiciansarecalledDJswhohave turntables music thatdoesn’thaveanybandmembers—butasinglemusicianmixingitupon with otherbandmembers.Well, forbandmemberstoday, this istrue butthereisalso andorchestrasreliedontempobeatstosyncup ensembles, barbershopquartets, “L In thislesson,youwill: Le Number Complex Difficult! Not Really Complex Getting It’s Now • • • • • • arning Rewriting quotientsofcomplexnumbers conjugates isareal number Understand thattheproduct ofcomplex polynomial expressions Add, subtract,andmultiplycomplex complex numbers Add, subtract,andmultiply of thecomplexnumbersystem Interpret thereal numbersaspart Calculate powersof the band to start a song during a performance. Infactforcenturies,bands, asongduring aperformance. the bandtostart et mehearthedownbeat!”mightbesomethingyouleadsingertell Go als . i .

......

. . But It’s But O . perations Key Te • • • • • • • • • • • monomial complex conjugates imaginary partofacomplexnumber real partofacomplexnumber set ofcomplexnumbers pure imaginarynumber set ofimaginarynumbers principal square root ofanegativenumber the imaginarynumber trinomial binomial r m s i 15.4 1099 15 1100 Problem

Chapter 15 2. number, thenumber no real numberexistssuch thatitssquare isequaltoanegative 4. The 3. irrational numbers. real numbersolutions.Remember, thesetofreal numbersincludesthesetsofrationaland So farwithinthiscourse,youhaveworkedthesetofreal numbersanddetermined 1. 1 Real

imaginary If Describe any patterns yousee inthetable. Describe anypatterns Explain whyornot. Is there areal numbersolutiontothisequation? Enter yourresults inthetableandshowyourwork. Use thevaluesof Consider theequation i i i

5 9

i 5

2 5 5

The Powersof 5

Number

2 1, whatisthevalueof

number

Systems i isnotapartofthereal number system. i i i 6 10 2 i and

5 5

i 5 isanumbersuchthat

x i 2 2

andtheproperties ofexponentstocalculateeachpower 5 i

2 Powers 1. i ? i i i 3 7 11

5 5

5 of

i i 2

2 1. Because i i i 8 4 12

5 5

number, doesthat of mean it’safake calculated powers it’s notareal Use previously next power of i tocalculatethe number? So if i. i .

5 2 √ 2 ___ 2 Libby 1 1

√ i

? 1 ___ 2 ?

√ ? i

1 ___ 2 6. 5. ? √

2 ___ 2

i 1

? b. Explain howtocalculateanyintegerpowerof a. Tristan, Kira,andLibbycalculatedthepower 1 1

? ?

If youhadtocalculate Explain whyeachstudent’ √ i

___ 2 ?

i i 15 1

? ? 5 5 5 5

√ Tristan ? (1)

1( (i ___ 2 2

i 4

f 2 ) ? √ 1 3 3

(

√ i ___ 2 ? 2

? i ___ √ 2

√ 1 3 i 2 ___ 2

1 ___ 2 ?

) 1

1 i

? ?

)

√ i

? ___ 2

i 1 99 ?

? i , whosemethodwouldyouuseandwhy?

√ ? s methodiscorrect

___ 2 i f

√ 2 ___ 2 1 1

√ ___ 2 i 1 15 special about

i multiplying . ? usingdifferent methodsasshown What’s so

√ 15.4 . ___ 2 f by i 1

4 ? ?

Complex NumberOperations √ 2 ___ 2 1 1

√ ___ The exponent of i i remainder of 3. I know 15 by 4, I have a is 15. When I divide i 2 3 15

5 2 1 5

? Kira i

√ 3

___ 2 √ f 5 ___ 2 1

2 ? 1

. So, √ √ 2 ___ 2 ___ 2 1 1 1

. ?

√ ___ 2 .

1101 f 15 15 1102

Chapter 15

number, number involving negativeroots Now thatyouknowabout

7. 8.

a. Calculate eachpowerof Explain whysheisincorrect c. Analyze Georgette’s work Real NumberSystems

i i n 400 93

2 , the

5 5

√ n 5 , isdefinedby ____ 2

principal square root ofanegative 32 Georgette

5 5 5

2 2 2 4 √ √ . √ ____ 16(2) ___ 32 Foranypositivereal __ 2 i

, youcanrewrite expressions √ ___ 2

i . . n

5 . Determine thevalueof

i √ __ n . √ _____ 2 75 d. b.

5 5 5 5 i i 2 206

2 √ i √ √

5 ______ √ ___ ( 2 5 ______25 2 __ 3

1)(75)

? ? 3 √

___ 75 √

_____ 2

. your answertopart(d)writean the propertiesofexponentsand equivalent expressionfor principal squareroot of anegativenumber before performing hint forpart(e):Use definition ofthe So, usethe operations! Here’s a i –1 .

10. 9.

b. Jen andTami eachrewrote theexpression c. a. Rewrite eachexpression using Who’s correct? Explaintheerror intheotherstudent’s reasoning

______√ √ 1 ______2 64 2 13

√ 2 2 _____ 2

1 √ 44 10 _____ 2

5 5

5 5 5 √ Jen ___ 2 4 √ √ 4 ______16 (

2 ?

√ 4)( ___ 2 2 4 i

. 4)

√ ___ 2 4

? 15.4

5 5 5 √ Tami √ ____

___ 2

2 2 2 4

4 i i

4 2 4 ?

Complex NumberOperations 2

? i

√ ____

1103 15 15 1104 Problem

Chapter 15 to 0 the form The called the called the the form The where 4. 3. 2. 1. 2

. set ofcomplexnumbers set ofimaginarynumbers A d. c. b. a. Identify whethereachnumberisacomplex Explain whyornot Can anumberbebothreal andimaginary? Give anexampleofapure imaginarynumber values of Write theimaginarynumber Real NumberSystems

pure imaginarynumber isnotequalto0 p 2 3 i I a a

1 imaginary partofacomplexnumber real partofacomplexnumber M 1 1 . 5216 3

bi bi ust Have . a 2 , where , where and i b ? a a and and . O . b b isthesetofallnumberswrittenin nly ImaginedthisWas Complex are real numbersand are real numbers isthesetofallnumberswrittenin is anumberoftheform i intheform , andtheterm a

electromagnetism, fluid dynamics, and quantum 1 . Theterm .

bi They areusedinthescientificfieldsof actually haveapplicationsinreallife. . Whatare the b isnotequal mechanics, justtonameafew! . Explainyourreasoning bi bi , a is Imaginary numbers is care aboutnumbers that areimaginary? Why shouldI .

7. 6. 5.

b. c. d. e. What isthedifference betweenacomplexnumberandanimaginarynumber? Use theword boxtocompleteeachstatement • • • • • • • • Create adiagramtoshow therelationship betweeneachsetofnumbersshown a.

whole numbers real numbers rational numbers natural numbers irrational numbers integers imaginary numbers complex numbers If anumberiscomplexnumber, thenitis If anumberisreal number, then itis If anumberisreal number, then itis If anumberiscomplexnumber, thenitis If anumberisanimaginarynumber, thenitis always

sometimes . Explainyourreasoning 15.4

Complex NumberOperations acomplexnumber animaginarynumber areal number animaginarynumber acomplexnumber never ......

1105 15 15 1106 Problem

Chapter 15 complex numbers division onthesetofreal numbers You knowhowtoperformthebasicoperationsofaddition,subtraction,multiplication,and variable (eventhoughitisaconstant) When operatingwithcomplexnumbersinvolving

1. 2. 3

b. a. Simplify eachexpression d. c. c. e. a. Determine eachproduct Real NumberSystems

4 (3 (5 5 (3 What doyounoticeabouteachproduct? (2 Call theDoctor, Stat!It’s Time to i i

(3 1 1 1 1 1 2 3 2 3 2

i )(2 2 i i i ) )(2 )(3 2 2 i ) 2 6 5 2 2 (1 .

i 1 )

3 2 5 2

i i i ) )

6 2 5 5 1 i

) 5 5

. Showyourwork . You canalsoperformtheseoperationsonthesetof .

d. b. i , combineliketermsbytreating (1 (

__ 2 1

2 1 3

) i )(1 (

O __ 2 1

2 1 perate!

3 i

) i 5

) 5

i asa

© Carnegie Learning © Carnegie Learning A polynomialinonevariableisanexpression oftheform extended toincludeimaginarynumbers one ormore variablesmultiplied bycoefficients Remember thatapolynomialismathematicalexpression involvingthesumofpowersin pair ofcomplexconjugatesisalwaysareal numberandequalto Complex conjugates in Question2are called even thoughtheoriginalexpression containedimaginarynumbers You mayhavenoticedthattheproducts inQuestion2didnotcontainanimaginary number, binomial polynomial withonetermiscalled A polynomialcanhaveaspecialname,according tothenumberoftermsitcontains are nonnegativeintegers the coefficients ( ? 4. 3.

c. a. Explain yourreasoning Identify eachexpression asamonomial,binomial,trinomial,orother polynomial equivalent expression (3 Dante saysthattheexpression isabinomialbecauseitcanberewritten asthe Maria saysthattheexpression 3

__ 3 2 3 .

x Apolynomialwiththree terms iscalleda 1 2

5 2 i

__ 4 1 .

x Whoiscorrect? Explainyourreasoning a 2 i 0

, a 1 , are pairsofnumbersthe form a 2 complex conjugates , .

. .

. ) are complexnumbers(real orimaginary)andtheexponents 1

i ) x

a monomial 2 5,whichhastwoterms x

1 .

2 . 5isatrinomialbecauseithasthree terms . . Apolynomialwithtwotermsiscalleda Thedefinitionofapolynomialcannowbe d. b.

trinomial 1 2 . 15.4 4 5 xi x a

1 a 1 0 3 2

Complex NumberOperations

1 bi i . . x

Jermainesaysthatitisnota

and

a 2 1 0 x 5 a

. 1 5 . 2 i Eachpairofexpressions a

x 1 1

3 a 2 i 1 2

b x

bi 2 2

. 1 . . Theproduct ofa

a n x . n A , where

1107 15 15 1108

Chapter 15 used tooperatewithnumericalexpressions involving You cansimplifysomepolynomialexpressions involving 5.

a. Simplify eachpolynomialexpression, ifpossible d. c. e. b. Real NumberSystems

xi ( 2 (2 xi x

i 2 1 1 1

2 . 5

3 xi xy 4 x i

) 1 x

5 2 5 )(

5 3 i

) xi 5

1 . 8 i

1 4 x

1 9 5 . Showyourwork i . i usingmethodssimilartothoseyou rules formultiplyingtwo need torememberthe . binomials. You just

a. methods seemsmore efficient andexplainwhy Analyze eachmethod

(2 (2 2 2 Shania Lindsay i)(1 i)(1 1 1 2i)(2 2i)(2 . Explaineachstudent’s reasoning 1 1 i) i) 5 5 5 5 5 5 5 5 5 5 5 (2 (2 (4 (4 (2 (4 5 5(1 5 8 8 1 1 1 1 1 2 1 2 2 1 1 10i 10i 10i 10i 3i i)(2 3i i ( 3i)(2 2i) 2 2 )(1 2 2 1 1 1))(1 1 2( 1 2i 1 3( 3i i)(1 2i) 2 2 2 i) 2 )(2 1 . 1))(2 1) 15.4 2i) 1 1 2i) 1 i)

i) Complex NumberOperations . Then,identifywhichofthetwo

1109 15 15 1110

Chapter 15

b. a. Simplify eachexpression Real NumberSystems

(2 2

i )(1 ( ( 5 5 5 x x

( 2 ( 1 1 1 x x x E Aiden 2 2

2 2

i i

) ) 1 lijah i 1 )(2 1 1

3 x 6 2 ( ( ) 1 x x

x x 1

1 1

1 1 i ) (3

4 xi 3) 3)

x xi . 1

1 1 1 3 3 12 ( ( i x ) ) x x 1 i

1 1 1 ( ( 3 3 x xi 2 i i

)( )( 1 1 x x 3 3

1 1 xi x )

3) 3) 1 1 b. 3 (3 5 5 5

xi ( 2 ( i (

x 1 1 x x x

1 1 2 1 9 9

i 3)(2 i 3)( ) i ) ) ( 4 x xi

x x 1

1 1 3) 6

4 i

x 1 i 1 )

( x 12 x

1 i 3 3 i ) i )( x

1 3)

1. 4

c. a. For eachcomplexnumber, writeitsconjugate

12 7 Rewriting QuotientsofComplexNumbers number inthedenominator ______You canrewrite divisor intoareal number divisor andthedividendbyconjugateofdivisor, thuschangingthe You canrewrite thedivisionofacomplexnumberbymultiplyingboth ______ 3 4 4 3 1 2 1 1 2 1

2 3 3 2 11 i i i i

5 5 5 5 5 i

______ ______ 25 4 3 12 12 6 6 1 2 2 25

16 2 2 2 3 2 17

9 17

4 i i ___ 25 17

1 . ______ 2 4 3 ? i i Recall,thecomplexconjugateof

1 2 9 i ______ 4 4 i 1 2

2 3 8

2 2 3 2 6 2 i 3 3

i i

withoutacomplex 1

i i

6 . i 2 .

d. b.

2 2 4 5 i 2 3 . Thencalculateeachproduct i 15.4

Complex NumberOperations a

bi is (a multiplying bya a 1

You are just 1 bi)(a formof1. Remember that

bi . 2 bi) . 5 a 2

b 2

. 1111 15 15 1112

Chapter 15 Be prepared toshare yoursolutionsandmethods

d. c. b. a. Rewrite eachquotientwithoutacomplexnumberinthedenominator Real NumberSystems

______20 5 3 2 3 2 1 2 1 2 2 1 2 1 2 2 2 4 3 2 4

5 i i

i i i i

i i

. .

© Carnegie Learning © Carnegie Learning imaginary numbersarenothingtofretabout—theyreallyaren’t imaginary the solutionsyouhaveencounteredsofar beenreal.So,don’tbeworried: roots, intercepts,andzerosofquadratics; withtheexceptionofonequestion,all occur—or if they even occur. You’ve successfully gone through this course determining S In thislesson,youwill: Le Solutions Complex with Quadratics Solving Its Solutions Are Complex! Complex—Just Not It’s • • • • arning an equationinradicalform quadratic equationfrom agraphandfrom Determine thenumberofroots ofa in radicalform solutions from a graph and from an equation Determine whetherafunctionhascomplex quadratic functions equations andcomplexzeros of Interpret complexroots ofquadratic quadratic functions equations andcomplexzeros of Calculate complexroots of quadratic no punintended—but really, it’s wherethe amatterofdetermining numberscanappeartobequite o, allthistalkaboutrealandimaginary Go . als . . . Key Te • • imaginary zeros imaginary roots r m s that complex! x -intercepts 15.5 complex — 1113 15 1114 Problem

Chapter 15

1. 1

f Consider thetwoquadraticfunctionsandtheirgraphsshown c. b. a. Real NumberSystems (

) two functions? Compare zeros, the List allthekeycharacteristicsyouknowabout zeros, the List allthekeycharacteristicsyouknowabout 5 8 Does ItIntersectthe

2 x 2 6

2 2 10 4 f x x x ( 2 x

-intercept(s), the -intercept(s), the 1 ) and 2 2 2 2 2 25 2 4 6 8 8 6 4 2 y

0 g ( x 2 ) . Whatdotheyhaveincommon?isdifferent aboutthe 4 y y 6 -intercept, theaxisofsymmetry, andthevertex -intercept, theaxisofsymmetry, andthevertex x 8 ? x

g 2 ( x 8 ) 5 f g 2 ( x (

x 2 6 ) . ) Besure toincludethenumberof . x Besure toincludethenumberof 2 2 4 2 2 . 2 2 2 2 2 4 6 8 8 6 4 2 y 0 2 4 6 . . 8 x

c Consider thetwoquadraticfunctionsandtheirgraphsshown c. b. a. ( x

) two functions? Compare of zeros, the List allthekeycharacteristicsyouknowabout of zeros, the List allthekeycharacteristicsyouknowabout 8 5

2 2 6 x 2 2

1 4 6 c 2 ( x x

2 ) and x x 2 2 2 2 -intercept(s), the -intercept(s), the 2 4 6 8 8 6 4 2 y 0 d ( x 2 ) . Whatdotheyhaveincommon?isdifferent aboutthe 4 6 y y 15.5 -intercept, theaxisofsymmetry, andthevertex -intercept, theaxisofsymmetry, andthevertex 8

Solving QuadraticswithComplex Solutions x

d 2 ( x 8 ) 5 c d 2 ( (

x x x 6 ) ) 2 . .

Besure toincludethenumber Besure toincludethenumber 1 2 4 12 2 x 2

. 1 2 2 2 2 2 4 6 8 8 6 4 2 32 y 0 2 4 6 8

. .

x 1115 15 15 1116

Chapter 15

p Consider thetwoquadraticfunctionsandtheirgraphsshown c. b. a. Real NumberSystems ( x 2

) two functions? Compare of zeros, the List allthekeycharacteristicsyouknowabout zeros, the List allthekeycharacteristicsyouknowabout 8 5 2 2 6 x 2

2 1 4 2 p x 2

( -intercept(s), the x 2 ) and x 2 2 2 2 -intercept(s), the 2 4 6 8 8 6 4 2 y 0 q ( x 2 ) . Whatdotheyhaveincommon?isdifferent aboutthe 4 y 6 -intercept, theaxisofsymmetry, andthevertex y -intercept, theaxisofsymmetry, andthevertex 8 x

q 2 ( x 8 ) 5 p q 2 ( (

x x 2 6 ) ) . . x Besure toincludethenumber of Besure toincludethenumber 2 2

4 2 8 2 x 2 .

2 2 2 2 2 2 4 6 8 8 6 4 2 18 y 0 2 4 6 . 8 . x

© Carnegie Learning © Carnegie Learning Problem in Problem 1 Question 3 had no real solutions you probably would have said that the quadratic equations Before learning about the set of complex numbers, roots equations that have imaginary solutions have know that they have imaginary solutions 3. 2. 1. 2

a. Recall thefunction examining thegraph?Explainyourreasoning solutions by Do youthinkcandeterminetheimaginary imaginary solutions? equation whetherornotithasreal solutionsor How canyoutellfrom the graphofaquadratic or

Use anymethodtosolve2 I See!No imaginary zeros x p -intercept , which are the solutions ( x ) 5 2 x 2

1 x 2 2

1 .

2 . 15.5 Functions and M 5 0 eans Imaginary!

. imaginary Now you . Solving QuadraticswithComplex Solutions . . the QuadraticFormula. Which the square,factor, anduse I knowhowtoanalyze a graph,complete method works best here? Let's see. includes bothrealandimaginary are real,someimaginary, numbers. So,somesolutions set ofcomplexnumbers but Remember, the all solutionsare

complex.

1117 15 15 1118

Chapter 15

a. b. b. Recall thefunction Real NumberSystems

when thesolutionsofafunctionare real orimaginary Use anymethodtosolve the solutionsare real orimaginary? the equationinpart(a) Suppose youusetheQuadraticFormulatosolve Consider afunctionwrittenintheform a a isnegative ispositive q ( x ) 5

c . 2 Howcanyoutellwhether ispositive x 2 2

2 x 2 8

2 x

8 2 x 18

2 18 . ax c 5 2 isnegative

1 0

. c function? Willthequadratic

5 graphical behaviorofthe and 0 . Completethetabletoshow c the variables pass through the . tellyouaboutthe What do x -axis? hint: Lookatthe discriminant. Here's a a

Consider the function

4. 3. 2. 1. 3

Are thezeros ofthefunctionreal orimaginary?Explainhowyouknow In whatformisthequadraticfunctiongiven? Use anymethodtodeterminethezeros ofthefunction Determine the

Imaginary Imaginary y -intercept ofthefunction f ( x ) 5 ﬁ

x Impossible 2

2 2 x

1 2 . 15.5

. Solving QuadraticswithComplex Solutions . .

1119 15 15 1120

Chapter 15

Recall that a quadratic function in factored form is written in the form

7. 5. 6.

in Question6toverifyyouranswer function instandard form? Simplifythefunctionyouwrote Is thefunctionyouwrote in factored formthesameasoriginal What do Use youranswertoQuestion3writethefunction Real NumberSystems r 1 and r 2 represent forafunctionwritteninthisform? . f ( x ) 5

x 2

2 2 f ( x x

) 1 Distributive Property 5 2infactored form

a with imaginary numbers? ( still usethe x

2 Can I

r 1 )( x

r 2 ) . .

Recall that a quadratic function in vertex form is written in the form

11. 10. 12. 9. 8.

Rewrite thefunction Explain howtodeterminetheaxisofsymmetryusingzeros ofthefunction of Use theaxisofsymmetrytodeterminevertex Determine theaxisofsymmetryfor Question 11 to check form? Simplify the function you wrote in same as the original function in standard Is the function you wrote in vertex form the h f ( , x k ) . ) Showyourwork . . . f ( x . ) 5

x 2 2 2 x

1 15.5 f ( 2invertexform x ) .

Showyourwork Solving QuadraticswithComplex Solutions functions isstilltrue even ifthesolutions Hey! Everything we havelearned about quadratic are imaginary! . the vertexislocatedon . the axisof Remember that symmetry. f ( x ) 5

a ( x

h ) 2

. k with

1121 15 15 1122 Talk theTalk

Chapter 15 ? Be prepared toshare yoursolutionsandmethods 2. 1.

solutions (doubleimaginaryroot) Explain whyitisnotpossibleforaquadraticequationtohave2equalimaginary Who’s correct? Explainyourreasoning • • • • Karl saysthatanyquadraticequationhasonlyoneofthese4typessolutions: • • • Brandon saysthatanyquadraticequationhasonlyoneofthese3typessolutions: • • • Casey saysthatanyquadraticequationhasonlyoneofthese3typessolutions: Real NumberSystems

1 real and1imaginarysolution 2 imaginarysolutions 2 equalreal numbersolutions (adoubleroot) 2 uniquereal numbersolutions 2 imaginarysolutions 2 equalreal numbersolutions (adoubleroot) 2 uniquereal numbersolutions 1 real and1imaginarysolution 2 equalreal numbersolutions (adoubleroot) 2 uniquereal numbersolutions . . .

© Carnegie Learning © Carnegie Learning Chapter Chapter Key Te 15.1 • • • • • • • • • • • • imaginary numbers(15 the number exponentiation (15 Venn diagram(15 real numbers(15 irrational numbers(15 rational numbers(15 integers (15 counterexample (15 closed (closure) (15 whole numbers(15 natural numbers(15 r Examples Numbers canbeclassifiedintosetsbasedontheircharacteristics Defining SetsofNumbersintheRealNumberSystem m √ 2 2 14 isanaturalnumber, wholenumber, integer, rationalnumber, andreal number Rational numbers Integers Whole numbers Natural numbers Irrational numbers Real numbers __ __ 2 1 3 s 1000 isaninteger, rationalnumber, and real number isarationalnumberandreal number isanirrationalnumberand areal number . i 1) (15 Set Name 15 . . 3) 1) . 1) . . 3) . 1) . . 1) 1) . 1) 1) Summary . 1) . 3) • • • • • • • • b b written as All numbersthatcanbe additive inverses Whole numbersandtheir number zero Natural numbersandthe Counting numbers be writtenas All numbersthatcannot numbers All rationalandirrational are integers are integers pure imaginary number ( ( set of negative number principal square root ofa the imaginary number number (15 imaginary partofacomplex number (15 real partofacomplex complex numbers(15 pure imaginary number (15 15 . 4 Description ) imaginary number __ b a

, where __ b a . .

, where 3) 3) . a and . a and . i 3) ( s 15 15 . . . 4 4 3) . ) ) • • • • • • • • • 2 0, 1,2,3, 1, 2,3, 2 √ __ 1 2 __

, 2

3, __ 2 3 imaginary zeros (15 imaginary roots (15 binomial (15 monomial (15 number (15 real partofacomplex trinomial (15 complex conjugates(15 number (15 imaginary partofacomplex ( set of 2 , 15

, 0,4, . p 2 __ 7 4 . , 0

, 0 4 2, Examples ) . .

. complex number 342359 2 . 75, 0

√ . 1, 0,2,3, . __ 5

. , 3

. . . . 01, 3,6 . 4) 4) . . 222, 4) 4) . . 4)

. 2 . . 10 . . 51 ___ 5) 5) .

s . . 4) 1123 15 1124

15.1 15.1

Chapter 15 The repeating decimal0 under an operation, a counterexample can be used to show a result thatsame is set,not partthe setof theis saidset to be closed, or have closure, under that operationWhen an operation is performed on any of the numbers in a set and the result is a numberReal NumberSystem in that Closure forSetsofNumbersinthe Determining x 99 100 Let Write thedecimal0 Example variable willresult intheequivalentfraction multiple often by settingthedecimalequaltoavariableandmultiplyingbothsidesofequation therefore, arationalnumber decimal thatrepeats singledigitsorblocksofcanbewrittenasafractionandis All irrationalnumbershaveaninfinitenumberofnon-repeating decimalplaces Writing RepeatingDecimals asFractions For example, Multiply anytwoirrationalnumbersandtheproduct maynotbeanirrationalnumber not haveclosure undersubtraction For example,2 Subtract anytwonaturalnumbersandthedifference maynotbeanaturalnumber Add anytwointegersandthesumisaninteger Examples under division 3 Divide anytwowholenumbersandthequotientmaynotbeanumber not haveclosure undermultiplication 2 4 ( x

x 4 x x

Real NumberSystems 5 5 5 5 5 5 5 0 41 41 ___ 99 41 0 0 . . 414141 . . 414141 75, and0

4141 √ . . Subtractingthefirstequationfrom thesecondequationandsolvingfor __ 3 2

. ?

5 . .

√ .

. . .

. 5 414141 __ 3

. 75 isnotawholenumber .

)

5 2 3,and3isnotanirrationalnumber . 3, and 41 ___ isequaltothefraction . . Algebracanbeusedtowritearepeating decimalasafraction

. asafraction 2 3 isnotanaturalnumber . . . . . Thesetofwholenumbersare notclosed . Thesetofintegersisclosedunderaddition ___ 99 41

. Thesetofnaturalnumbersdoes . Thesetofirrationalnumbersdo . If a set is not closed . Forexample, . Anyinfinite .

. .

© Carnegie Learning © Carnegie Learning 15.2 15.2 An exampleoftheCommutativeProperty ofMultiplicationis4 An exampleoftheadditiveinverseis An exampleoftheAssociativeProperty ofAdditionis14 • • The properties ofthereal numbersinclude: ofRealNumbers Understanding theProperties • • • • • • • The properties ofreal numberscanbeusedtosimplifyalgebraicexpressions Simplifying An exampleofthemultiplicativeidentityis 4( An exampleoftheDistributiveProperty ofMultiplicationoverSubtractionis Examples • • • 23 15 15 Simplify 5(3 Example

x x x x Commutative Property of Multiplication: Commutative Property of Addition: Distributive Property ofMultiplication overAddition: Associative Property ofMultiplication: Associative Property ofAddition: Distributive Property ofMultiplication overSubtraction: the multiplicativeidentity Multiplicative Inverse:anumberthatwhenmultipliedbyanyreal number identity Additive Inverse:anumberthatwhenaddedtoanyreal number Distributive Property ofDivision overSubtraction: equal to Multiplicative Identity:anumberthatwhenmultipliedbyanyreal number Additive Identity:anumberthatwhenaddedtoanyreal number Distributive Property ofDivision overAddition:

1 1 2 3) 5 8 35 x 5

2 1 4 a 35 8 x x

2 x 2

7) 1 1 12 40 40 E 1 . xpressions Using the Properties ofRealNumbers xpressions UsingtheProperties 8(

1 5) Combine liketerms Associative Property ofAddition Distributive Property of Multiplication over Subtraction and Addition . ; 2 ; a 8 , a . b, c 5 ; , b 2 1 ; a P ,

35 P a 8 b, c R , . R 5 , b ? , ( a 1 P 5 P ;

a 1 R 0 R 5 a

, , ; , ( . b

a 2 b, c

b a a

5 ; ? 35

, ) ?

1 a 1 b b, c b

P . , b ;

(11 5 1 b, c ) R c a ?

5 , b

a c ______ ( b, c ? R

a 1 P

( a 5

1 2

5) R a Chapter 15 c 1 ______ (

P , a

15) a a b

( a 2 5 , thesumisadditive , thesumisequalto R ? c ) b (

5 ,

(14

( 5 b 1 b

a 1

)

( __

c a

? b 2 a 5

1 ) 1 c 2 15

) __ ) c a a

a 11)

__ b

, theproduct is

. , theproduct is c 2

Summary ?

) ab 4

__ b c 1 .

1 ab 5

. ac

ac a

1125 15 15 1126

15.4 15.3 15.3

Chapter 15 2 simplified using and 2 The number raise aquantitytopower, unlessthesquare root ofanegativenumbercanbecalculated The setofreal numbersis notclosedunderexponentiation,whichistheoperationusedto Simplifying i Example The imaginary number Calculating Powersof 4 4 √ number 25 isanaturalnumber, wholenumber, integer, rationalnumber, real number, andcomplex Examples comprised ofthesetreal numbersandthesetofimaginary a The setofimaginarynumbers( oftheSetComplexNumbers Understanding theProperties Simplify Examples

2 its square is equal to a negative number, the number

The valuesof 2 5 and . __ 2

6 6 6 1 1 5 5 5 isanirrationalnumber, real number, and complexnumber 66 ___

1 1 7 1

i ( √ ( 4 1 Real NumberSystems i

3 i

isarationalnumber, real number, andcomplex number 4 ____ isanimaginarynumberandacomplex 2 b √ √ 5 ) ) 6 6 √ . ______are real numbersand 9 2 1 ( ( 1 √ 2 i __ 5

45 ? ____ ) 2 . i 6

√ 1

1 __ 5 i

) isdefinedsuchthat i

n ? √ repeat aftereveryfourpowersof

_____ 2 i √ . E ___ 2 45 xpressions Involving Imaginary Numbers xpressions InvolvingImaginary 1 .

i is a number such that b I isnotequaltozero ) isthesetofallnumberswritteninform Simplify (5 5 (5 5 5 5 i x x x x x i 2 2 2 2 2

2 1 1 1 1 5 8 22 22 22 30

2 i )( 1 xi xi xi xi x .

Expressions involvingnegative roots canbe

x 1 1 2 2 2

2 i 6 48 48( 48 8 2

8 5 i xi ) i i

2 )( 2 2 2 i x , where 1) 1

48 . 1 . Thesetofcomplexnumbers( i Because no real number exists such that is not a part of the real number system 6 i . 2 i ) . i

. √ . ___ 2 1

, i 2

. 2 1, i a 3

5 2 1

C , where √ ) is ___ 2 1

, .

© Carnegie Learning © Carnegie Learning 15.4 15.4 15.4 shows one Example Combine liketermstonamethepolynomial with twotermsiscalledabinomial number oftermstheyhave include imaginarynumbers variables multipliedbycoefficients A polynomialisamathematicalexpression involvingthesumofpowersinoneormore Identifying ComplexPolynomials 5 Example principal square root ofanegativenumber, Expressions withnegative roots canberewritten Rewriting 4 4 Example pair ofcomplexconjugatesisalwaysareal numberintheform Complex conjugatesare pairs ofnumberstheform involving called theimaginarypartofacomplexnumber b The setofcomplexnumbersistheallwritteninform of ComplexNumbers Adding, Subtracting,and √ The expression isatrinomialbecauseitcanberewritten as x x x _____ are real numbers 2

2 1 1 63 3 ( (

x x 1 xi

1 1

1 √ i 3 3 , combineliketermsbytreating

_____ 2

x x i i

24 )( )( 2 2

term,one x x 2

2 2 E 5 5 5 3 3 3 xpressions withNegativeRootsUsing 3

i i i

√ √ 1 . i i √

Theterm ) ) _____ (9)(7) ___ 63 __ 9 7 2 2 i

5 5 1 1 x x x

term,andoneconstantterm 1 i

√ 1 1 √ . .

___ 24 Apolynomialwithonetermiscalledamonomial Somepolynomialshavespecialnames,according tothe i __ 7 7 6 √ a

i _____ (6)(4)

5 5 5 iscalledthereal partofa complexnumber, andtheterm

4 x x 2 2 x . .

Thedefinitionofapolynomialcanbeextendedto Apolynomialwiththree termsiscalledatrinomial

2 1 1 M (4

x x

ultiplying ontheSet 2 1 x

2 2 16 i asavariable(eventhoughit’s aconstant) 9 5 2 . i 2 x n

) . , isdefinedby 2 Whenoperatingwithcomplexnumbers 1 5 . Foranypositivereal number (7 x

1 2 7 a . 9(

1 2

bi 1)) x and 2 √

1 ___ 2 a (5 n 2 Chapter 15

a 1

5 2 2

i 3 √ bi 2 . __ n i .

) Theproduct ofa i . x a

1 1 (9

. bi Apolynomial Summary , where n 2 , the 3 i

), which .

and bi

is 1127 15 15 1128

15.4 15.4

Chapter 15 complex conjugatesfirsttogeta real number used tooperatewithnumericalexpressions involving Some polynomialexpressions involving Adding, Subtracting,and ______Example product ofapaircomplex conjugatesisalwaysareal numberintheform by thecomplexconjugateofdivisor, thuschangingthe divisorintoareal number When rewriting thequotient ofcomplexnumbers,multiplyboththedivisoranddividend Rewriting theQuotientofComplexNumbers (3 (3 Example 5 3 x x 1 2

1 1 2

i Real NumberSystems 7 7

5 5 5 5 5 i i

)(2 )(2

______ ______ 5 15 29 15 13 13 3

x x 1

1 1 29 25 2 2 2 2 2

5 5 i 6 11 11

i ___ 29 11

1 i i 2 ? i

) )

5 5 1 2

______ i i 5 5 i 5

2 6 6 2 5

2 2 6 2 x x 2 i 2 2 x

2 2 1

i i

1 1

1 2 29 29 15 i 2

xi xi xi

2 1 1 35 35( 14 2 xi M

1) 1 ultiplying ComplexPolynomials i 35 canbesimplifiedusingmethodssimilartothose i 2 . i . Wheneverpossible,multiplythe a 2

b 2 . . The

© Carnegie Learning © Carnegie Learning 15.5 x The zeros ofthefunctionare discriminant oftheQuadraticFormulaispositive,equationhastwouniquereal number solutions, twoequalreal number solutions(adoubleroot), ortwoimaginarysolutions All quadraticequationshaveonlyoneof3typessolutions:twouniquereal number Complex ZerosofQuadraticFunctions Complex RootsofQuadratic Determining x x a The quadraticequationcanbeusedtodeterminetheimaginaryzeros The graphdoesnotintersectthe Consider thefunction Example solutions x x x

5 5 5 5 5 5 5 2

1, ______ ______ 4 2 4 4 2 6 b ( 6 6 6 2 b

2 6

4 4) √ 5 2 . √

Ifthediscriminantisnegative,equationhastwoimaginarysolutions √ √ _____

______16 √

2 __ 6 2 __ 2 2 2 32 ______b

i a i

2 √ 2

2 ______( 2

2(1) c 48

10 12 14 16 18 4 4 5

0 2 4 6 8 ac )

12 y 2

f 4(1)(12) ( x 8 2 ) 5

4 x x 2

2 5 6 4 2 x -axis sotheequationhastwocomplexsolutions x 1

1 2 12 √ 10 __ 2 . i and 12 14 x

5 2 16 2 2 18 √ __ 2 E i . x quations and Chapter 15 .

Summary .

. Ifthe

1129 15 15 1130

Chapter 15

Real NumberSystems