Chapter 1 Number and operations review数与运算

Chapter1Numberandoperationsreview1.Propertiesofintegers•Youwillneedtoknowthefollowinginformationforsomequestionsinthemathematicssection:Integersconsistofthewholenumbersandtheirnegatives(includingzero)....,-3,-2,-1,0,1,2,3,4,...Integersextendinfinitelyinbothnegativeandpositivedirections.PositiveandNegative•Therearethreerulesregardingthemultiplicationofpositiveandnegativenumbers.negativenegativepositivepositivenegativenegativepositivepositivepositiveNotethatpositiveintegersgetbiggerastheymoveawayfrom0,whilenegativeintegersgetsmaller.2.Oddnumbers•...-5,-3,-1,1,3,5,...3.Evennumbers•...-4,-2,0,2,4,6,...Theintegerzero(0)isanevennumber.Oddnumbersareintegersthathavearemainderwhendividedby2.Evennumbersareintegersthatcanbedividedby2leavingnoremainder.4.Consecutiveintegers•Integersthatfollowinsequence,wherethedifferencebetweentwoconsecutiveintegersis1,areconsecutiveintegers.Herearethreeeaxmplesofsomeconsecutiveintegers:-3,-2,-1,0,1,2,3,41000,1001,1002,1004-15,-14,-13,-12,-11,-10Thefollowingisanexpressionrepresentingconsecutiveintegers:whereisanyinteger.,,3,2,1,nnnnn5.Additionofintegers•even+even=even•odd+odd=even•odd+even=oddAddingzero(0)toanynumberdosen'tchangethevalue.6.MultiplicationofintegersevenevenoddoddoddoddevenevenevenMultiplyinganynumberbyone(1)doesn'tchangethevalue.Thereareonlysixoperations:Addition,subtraction,multiplication,division,raisingtoapower,findingasquareroot7.Arithmeticwordproblems•Example1Ms.GriffenismakingbagsofHalloweentreats.Ifsheputs3treatsineachbag,shewillmake30bagsoftreatsandhavenotreatsleftover.Ifinsteadsheputs5treatsineachbags,howmanybagsoftreatscanshemake?7.Arithmeticwordproblems•Example2Jorgebought5pencilsfromthestore.Hegavethecashierafive-dollarbillandgotback＄0.75inchange.Jorgesawthathehadgottentoomuchchange,andhegave＄0.25backtothecashier.Whatwastheprice,indollars,ofeachpencil?8.Numberlines•Anumberlineisusedtographicallyrepresenttherelationshipsbetweennumbers:integers,fractionsordecimals.Numbersonanumberlinealwaysincreaseasyoumovetotheright,andtickmakesarealwaysequallyspaced.Negativenumbersarealwaysshownwithanegativesign(-).Forpositivenumbers,theplussign(+)isusuallynotshown.Numberlinesaredrawntoscale.Youwillbeexpectedtomakereasonableapproximationofpositionsbetweenlabeledpointsontheline.Numberlinequestionsgenerallyrequireyoutofigureouttherelationshipamongnumbersplacesontheline.Numberlinequestionsmayask:•Whereanumbershouldbeplacedinrelationtoothernumbers;•Thedifferenceorproductoftwonumbers;•Thelengthsandtheratiosofthelengthsoflinesegmentsrepresentedonthenumberline.Example•Onthenumberlineabove,theratioofACtoAGisequaltotheratioofCDtowhichofthefollowings?•A.ADB.BDC.CGD.DFE.EG9.Squaresandsquareroots:•1)SquaresofintegersYourknowledgeofcommonsquaresandsquarerootsmayspeedupyoursolutiontosomemathproblems.Themostcommontypesofproblemsforwhichthisknowledgewillhelpyouwillbethoseinvolving:Factoringand/orsimplifyingexpressions;ProblemsinvolvingthePythagoreantheorem();Areasofcirclesorsquares;222cba•Rememberthatifapositivefractionwithavaluelessthan1issquared,theresultisalwayssmallerthantheoriginalfraction:2)SquaresoffractionsnnthennIf2,10Tryit.Whatarethevaluesofthefollowingfractions?23229110.FractionsandrationalnumbersYoushouldknowhowtodobasicoperationswithfractions:1)Adding,subtracting,multiplyinganddividingfractions;2)Reducingtolowestterms;3)Findingtheleastcommondenominator;4)Expressingavalueasamixednumber()andasanimproperfraction;5)Workingwithcomplexfractions-onesthathavefractionsintheirnumeratorsordenominators.31237•Youshouldknowthatarationalnumberthatcanberepresentedbyafractionwhosenumeratoranddenominatorarebothintegers(andthedenominatormustbenonzero)1)Decimalfractionequivalents•Youmayhavetoworkwithdecimalfractionequivalents.Thatis,youmayhavetobeabletorecognisecommonfractionsasdecimalsandviseversa.•Tochangeanyfractiontoadecimal,dividethenumeratorbythedenominator.2)Reciprocals•Thereciprocalofanumberis1dividedbythatnumber.•Theproductofanumberanditsreciprocalisalways1.•Notethatyoucanfindthereciprocalofanynonzerofractionbyswitchingitsnumeratoranddenominator.•Notethatthereciprocalofanegativenumberisnegative.•Thenumberzero(0)hasnoreciprocal.•Thenumber1isitsownreciprocal.Also,thenumber-1isitsownreciprocal.3)Placevalueandscientificnotation•Thenumber123canbewrittenas100+20+3oras.Thedigit1stands1times100;thedigit2standsfor2times10;andthedigit3standsfor3times1.Wesaythesedigitshavethefollowingplacevalues:13102101121isinthehundredsplace.2isinthetensplace.3isintheunits(ones)place.Everydigitinadecimalnumberhasaplacevalue.Sometimes,usingtheconceptofplacevaluecanletyouwriteaverybigorsmallnumberinamuchshorterform.•Becausesuchnumbersoftenoccurinscientificcalculations,writinganumberastheproductofapowerof10andanumbergreaterthanto1andlessthan10iscalledscientificnotation.10121077,000,000,000.0103.2000,000,000,300,211.Elementarynumbertheory•Factors:Thefactorsofanumberarepositiveintegersthatcanbedividedevenlyintothenumber-thatis,withoutremainder.1)Factors,multiplesand