Siyavula textbooks: Grade 10 Maths [CAPS]

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.

2 Ducts..113 Solutions.. onentialequations.. 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.