Siyavula textbooks: Grade 10 Maths [CAPS] - CNX

1 Siyavula textbooks: Grade 10 Maths [CAPS]Collection Editor:Free High School Science Texts ProjectSiyavula textbooks: Grade 10 Maths [CAPS]Collection Editor:Free High School Science Texts ProjectAuthors:Free High School Science Texts ProjectUmeshree GovenderOnline:< >C O N N E X I O N SRice University, Houston, TexasThisselectionandarrangementofconten tasacollectioniscopyrightedbyFreeHighSch o olScienceTextsPro ( ).Collectionstructurerevised:August3,201 1 PDFgenerated:Octob er29,2012 Forcopyrightandattributioninformationfor themo dulescontainedinthiscollection, .. ductionandkeyconcepts.. ola.. erb olicfunctions.. onentialfunctions..40 Solutions..472 Numb duction..55 Solutions.. ductionandsimpleinterest.. oundinterest..67 Solutions..754 Rationalnumb ers..795 Exp onentials..856 Estimatingsurds..957 Irrationalnumb ersandroundingo ..1018 Pro ductionandrecap.. ducts..113 Solutions.. onentialequations.

2 Dels..140 Solutions.. ductionandStraight-linefunctions.. olicandotherfunctions..156 Solutions.. :Part1.. :Part2..174ivSolutions.. ,linesandangles..200 Solutions.. ofsandconjectures..233 Solutions.. ductionandkeyconcepts..277 Solutions.. etweentwop oints.. ointofaline.. ductionandrecap.. ,errorandmisuse..322 Solutions..328 Glossary.. 333 Index..335 Attributions.. 337 AvailableforfreeatConnexions< >ReviewofPastWork1 Intro ductionThischapterdescrib essomebasicconceptswhichyouhaveseeninear liergradesandlaysthefoundationfortherema inderofthisb o dentwiththecontentinthischapter,b eforemovingonwiththerestoftheb o , o datmaths!Whatisanumb er?Anumb ersarenotsomethingthatyoucantouchorhold, b ,threep encils,threeb o , ,youdonotneedtoseethreeapplesinfrontofyo utoknowthatifyoutakeoneappleaway,therewi llb ersandthenp errepresentsquantityb ecausewecanlo okattheworldaroundusandquantifyitusingnu mb eansweredusingnumb erstotellus howmuch ercanb ewritteninmanydi erentwaysanditisalwaysb esttocho osethemostappropriatewayofwritingthenumb , ahalf mayb esp okenaloudorwritteninwords,butthatmakesma thematicsverydi cultandalsomeansthatonlyp eoplewhosp etterwayofwriting ahalf isasafraction12orasadecimalnumb er0, er, ol,allthenumb erswhichyouwillseearecalledrealnumb ersandmathematiciansusethesymb olRtorepresentthesetofallrealnumb ers,whichsimplymeansalloftherealnumb erscanb erenttyp esofnumb ersaredescrib jectswithawell-de nedcriterionformemb ,thecriterionforb elongingtoasetofapples.

3 Isthattheob jectmustb edividedintoredapplesandgreenapples, < >.AvailableforfreeatConnexions< >12 Nowwecometotheideaofaunion, olforunionis .Here, ,ifxisarealnumb ersuchthat1< x 3or6 x <10,thenthesetofallthep ossiblexvaluesis:(1,3] [6,10)(1)wherethe signmeanstheunion(orcombination) olsdescrib edb edonewithnumb ersisadding,subtracting, ersareadded,subtracted,multipliedordivided,youarep erationscanb ep erformedonanytworealnumb :oneplusoneisequaltotwomathematicianswrite1 + 1 = 2(2)Inearliergrades,placeholderswereusedtoindicatemissingnumb + = 24 = 2 + 3 2 = 2(3)However, ,lettersareusuallyusedtorepresentnumb +x= 24 y= 2z+ 3 2z= 2(4)Theselettersarereferredtoasvariables,sincetheycantakeonanyvaluedep ,x= 1in(4),butx= 26in2 +x= eedoflightinavacuumisalsoaconstantwhichhasb eende nedtob eexactly299792458m s 1(readmetresp ersecond).Thesp eedoflightisabignumb erandittakesupspacetoalwayswritedowntheentirenumb , eedoflight,itisacceptedthatthelettercrepresentsthesp ,letterscanb eusedtodescrib ,thefollowingequationx+y=z(5)canb eusedtodescrib ethesituationof ndinghowmuchchangecanb eexp ,yrepresentsthepriceoftheitemyouarebuying, ,ifthepriceisR10andyougavethecashierR15, + 10 = 15(6)2 ArithmeticisderivedfromtheGreekwordarithmosmeaningnumb < >3 Wewilllearnhowto solve (+)andsubtraction( )arethemostbasicop erationsb etweennumb eingtheopp ositeofaddingsinceaddinganumb erandthensubtractingthesamenumb ,ifwestartwithaandaddb,thensubtractb,wew illjustgetbacktoaagain:a+b b=a5 + 2 2 = 5(7)Ifwelo okatanumb erline, ersareaddeddo esnotmatter,buttheorderinwhichnumb ersaresubtracteddo :a+b=b+aa b6=b aifa6=b(8)Thesign6=means isnotequalto.

4 Forexample,2 + 3 = 5and3 + 2 = 5,but5 3 = 2and3 5 = 2. 2isanegativenumb er,whichisexplainedindetailin"NegativeNu mb ers"(Section:NegativeNumb ers).CommutativityforAdditionThefactthat a+b=b+a,isknownasthecommutativeprop ,multiplication( , )anddivision( ,/)areopp erandthendividingbythesamenumb ergetsusbacktothestartagain:a b b=a5 4 4 = 5(9)Sometimesyouwillseeamultiplicationof lettersasadotorwithoutanysymb 'tworry, cientandliketowritethingsintheshortest,n eatestwayp b ca b c=a b c(10)Itisusuallyneatertowriteknownnumb erstotheleft, ,itlo oksb ,the 4 isaconstantthatisreferredtoastheco e ,b othadditionandmultiplicationaredescrib edascommutativeop < > ortantasyoucangetdi erentanswersdep :(5 5) + 20 = 45(11)whereas5 (5 + 20) = 125(12)Iftherearenobrackets,youshouldalw aysdomultiplicationsanddivisions ,forexample:a b+c d=(a b) + (c d)5 5 + 20 4 = (5 5) + (20 4)(13)Ifyouseeamultiplicationoutsideabra cketlikethisa(b+c)3 (4 3)(14)thenitmeansyouhavetomultiplyeachpa rtinsidethebracketbythenumb eroutsidea(b+c) =ab+ac3 (4 3) = 3 4 3 3 = 12 9 = 3(15) ,intheab oveexample,itwouldhaveb eensmartertohavedonethis3 (4 3) = 3 (1) = 3(16)Itcanhapp enwithlettersto o3 (4a 3a) = 3 (a) = 3a(17)DistributivityThefactthata(b+c) =ab+acisknownasthedistributiveprop ,thenyoucandoitonestepatatime:(a+b) (c+d) =a(c+d) +b(c+d)=ac+ad+bc+bd(a+ 3) (4 +d) =a(4 +d) + 3 (4 +d)=4a+ad+ 12 + 3d(18)3 Sometimesp eoplesay parentheses insteadof brackets.

5 4 Multiplyinganddividingcanb ep erformedinanyorderasitdo esn' esn' b eforeany+ .AvailableforfreeatConnexions< >5 NegativeNumb ersWhatisanegativenumb er?Negativenumb erscanb everyconfusingtob eginwith,butthereisnothingtob ersareknownasp ositivenumb erisanumb ,ifweweretotakeap ositivenumb eraandsubtractitfromzero,theanswerwouldb a= a(19)Onanumb erline,anegativenumb erapp earstotheleftofzeroandap ositivenumb erapp :Onthenumb erline,numb ersapp eartotherightofzeroandnegativenumb ersapp ersWhenyouareaddinganegativenumb er,itisthesameassubtractingthatnumb erifitwerep ,ifyousubtractanegativenumb er,itisthesameasaddingthenumb erifitwerep ersareeitherp ositivenumb erhasap ositivesign(+)andanegativenumb erhasanegativesign( ).Subtractionisactuallythesameasaddingan egativenumb ,aandbarep ositivenumb ers,but bisanegativenumb era b=a+ ( b)5 3 = 5 + ( 3)(20)So,thismeansthatsubtractionissimpl yashort-cutforaddinganegativenumb erandinsteadofwritinga+ ( b),wewritea b+aisthesameasa ,whichdoyou ndeasiertoworkout?

6 Mostp eople ndthatthe rstwayisabitmoredi ,mostp eople nd12 3aloteasiertoworkoutthan 3 + 12, b,whichlo rstcolumnshowsthesignofthe rstnumb er,thesecondcolumngivesthesignofthesecon dnumb erandthethirdcolumnshowswhatsigntheanswe rwillb bora b++++ + +AvailableforfreeatConnexions< >6 Table1:Tableofsignsformultiplyingordivid ingtwonumb erbyap ositivenumb eralwaysgivesyouanegativenumb er,whereasmultiplyingordividingnumb erswhichhavethesamesignalwaysgivesap ositivenumb ,2 3 = 6and 2 3 = 6,but 2 3 = 6and2 3 = ersworksslightlydi erently(seeTable2).The rstcolumnshowsthesignofthe rstnumb er,thesecondcolumngivesthesignofthesecon dnumb erandthethirdcolumnshowswhatsigntheanswe rwillb +b++++ ? +? Table2:Tableofsignsforaddingtwonumb ositivenumb ersyouwillalwaysgetap ositivenumb er,butifyouaddtwonegativenumb ersyouwillalwaysgetanegativenumb ershaveadi erentsign,thenthesignoftheanswerdep erLineThenumb erlineinFigure1isago o dwaytovisualisewhatnegativenumb ersare,butitcangetveryine cienttouseiteverytimeyouwanttoaddorsubtr actnegativenumb ,wewillwritedownthreetipsthatyoucanuseto makeworkingwithnegativenumb erswhichmayb enegative, ersTip1 Ifyouaregivenanexpressionlike a+b,thenitiseasiertomovethenumb ersaroundsothattheexpressionlo ,wehaveseenthataddinganegativenumb ertoap ositivenumb eristhesameassubtractingthenumb erfromthep ositivenumb , a+b=b a 5 + 10 = 10 + ( 5)=10 5=5(21) ,aquestionlike Whatis 7 + 11?

7 Lo oksalotmorecomplicatedthan Whatis11 7? , ersTip2 Whenyouhavetwonegativenumb erslike 3 7,youcancalculatetheanswerbysimplyadding togetherthenumb ersasiftheywerep ositiveandthenputtinganegativesigninfron t. c d= (c+d) 7 2 = (7 + 2) = 9(22)AvailableforfreeatConnexions< >7 NegativeNumb ersTip3 InTable2wesawthatthesignoftwonumb ersaddedtogetherdep erawayfromthelargeroneandrememb ertogivetheanswerthesignofthelargernumb , F= (F e)2 11 = (11 2) = 9(23)Youcanevencombinethesetips:forexamp le,youcanuseTip1on 10 + 3toget3 10andthenuseTip3toget (10 3) = :(a)( 5) ( 3)(b)( 4) + 2(c)( 10) ( 2)(d)11 ( 9)(e) 16 (6)(f ) 9 3 2(g)( 1) 24 8 ( 3)(h)( 2) + ( 7)(i)1 12(j)3 64 + 1(k) 5 5 5(l) 6 + 25(m) 9 + 8 7 + 6 5 + 4 3 + 2 +or (a) 5 + 6(b) 5 + 1(c) 5 5(d) 5 5(e)5 5(f )5 5(g) 5 5(h) 5 5(i)5 5(j)5 5 Table4 Clickhereforthesolution6 RearrangingEquationsNowthatwehavedescrib edthebasicrulesofnegativeandp ositivenumb ersandwhattodowhenyouadd,subtract,multip lyanddividethem,wearereadytotacklesomere almathematicsproblems!

8 Earlierinthischapter,wewroteageneralequa tionforcalculatinghowmuchchange(x)wecane xp ectifweknowhowmuchanitemcosts(y)andhowmu chwehavegiventhecashier(z).Theequationis :x+y=z(24)So,ifthepriceisR10andyougaveth ecashierR15, + 10 = 15(25)5 < >8 Nowthatwehavewrittenthisequationdown,how exactlydowegoab out ndingwhatthechangeis?Inmathematicalterms ,thisisknownassolvinganequationforanunkn own(xinthiscase).Wewanttore-arrangethete rmsintheequation,sothatonlyxisonthelefth andsideofthe= ortantthingtorememb ,whateverisdonetoonesidemustb ,youmustdothesamethingtob ,ifyouadd,subtract,multiplyordividetheon eside,youmustadd,subtract,multiplyordivi detheothersideto d:RearrangingEquationsYoucanadd,subtract ,multiplyordivideb othsidesofanequationbyanynumb eryouwant,aslongasyoualwaysdoittob othsidesx+y=zx+y y=z yx=z yx= 15 10=5(26) ,thechangeshouldb ; erfromb othsidesofanequation,itlo okslikeyoujustmovedap ositivenumb erfromonesideanditb ecameanegativeontheother,whichisexactlyw hathapp ,ifyoumoveamultipliednumb erfromonesidetotheother,itlo ecauseyoureallyjustdividedb othsidesbythatnumb erandanumb erdividedbyitselfisjust1a(5 +c) =3aa(5 +c) a=3a aaa (5 +c) =3 aa1 (5 +c) =3 15 +c=3c= 3 5 = 2(27)However,youmustb ecarefulwhendoingthis, < >9 ThefollowingistheWRONG thingtodo5a+c= 3a5 +c=3(28)Canyouseewhyitiswrong?

9 Itiswrongb +c=3a5 +c a=3c a= 3 5 = 2(29) (2r 5) = 27,then2r 5 =.. ,5 (x 8) = 0,2x+ 2n= 3 (n+ 2) +AkttoA= :1ax+1bx= 1 Clickhereforthesolution11 FractionsandDecimalNumb ersAfractionisonenumb erdividedbyanothernumb erdividedbyanotherone,suchasa b, rstwayofwritingafractionisveryhardtowork with, eronthetop(left)thenumeratorandthenumb erontheb ottom(right) ,inthefraction1/5or15, calofafractionisthefractionturnedupsided own,inotherwordsthenumeratorb ecomesthedenominatorandthedenominatorb ,therecipro calisalwaysequalto1andcanb ewrittenab ba= 1(30)Thisisb ecausedividingbyanumb eristhesameasmultiplyingbyitsrecipro nition-MultiplicativeInverseTherecipro calofanumb erisanumb oint,whichiswrittenasacommainSouthAfrica nscho er314100canb ewrittenmuchmoreneatlyas3, < >10 Allrealnumb erscanb ewrittenasadecimalnumb ,somenumb erswouldtakeahugeamountofpap er(andink)towriteoutinfull!

10 Somedecimalnumb erswillhaveanumb erwhichwillrep eatitself,suchas0, nitenumb erof3' ovetherep eatingnumb er,so0, 3 = 0, eatingnumb erssuchas0, eatednumb ers0, 1 2 = 0, ,051160,06251100,1180,125160,16 6150,2120,5340,75 Table5 cNotationInscienceoneoftenneedstoworkwit hverylargeorverysmallnumb ewrittenmoreeasilyinscienti cnotation,whichhasthegeneralforma 10m(31)whereaisadecimalnumb erb etween0and10thatisroundedo ositiveitrepresentshowmanyzerosshouldapp ,thenitrepresentshowmanytimesthedecimalp laceinashouldb ,2 103represents32000and3,2 10 3represents0, ermustb econvertedintoscienti cnotation,weneedtoworkouthowmanytimesthe numb ermustb emultipliedordividedby10tomakeitintoanum b erb etween1and10( onentm)andwhatthisnumb eris(thevalueofa).Wedothisbycountingthen umb erofdecimalplacesthedecimalp ,writethesp eedoflightwhichis299792458m s 1inscienti cnotation, ,determinewherethedecimalp ointmustgofortwodecimalplaces(to nda)andthencounthowmanyplacesthereareaft erthedecimalp ,thedecimalp ointmustgoafterthe rst2,butsincethenumb erafterthe9isa7,a= 3, eris3,00 10m,wherem= 8,b ecausethereare8digitsleftafterthedecimal p ,thesp eedoflightinscienti cnotationtotwodecimalplacesis3,00 108m s 1 Asanotherexample,thesizeoftheHIvirusisar ound1,2 10 ,2 0,0000001m,whichis0, ,like0,12 AvailableforfreeatConnexions< >11 RealNumb ersNowthatwehavelearntab outthebasicsofmathematics,wecanlo okatwhatrealnumb ersanditisseenthateachnumb eriswritteninadi erentway.