2006-680: THE ROADMAP OF ARITHMETIC: SUMMING IT UP Andrew Grossfield, Vaughn College of Aeronautics Throughout his career Dr. Grossfield, has combined an interest in engineering design and mathematics. He earned a BSEE at the City College of New York. During the early sixties, he obtained an M.S. degree in mathematics at night while working full time during the day, designing circuitry for aerospace/avionics companies. He is licensed in New York as a Professional Engineer and is a member of ASEE, IEEE, SIAM and MAA. [email protected] is his e-mail address. Page 11.1323.1 Page © American Society for Engineering Education, 2006 The Roadmap of Arithmetic: Summing it up Abstract Noticethis problemofhumanconsciousness.Ideasandconceptsenterourmindoneatatime. Usuallyteachersandauthorshavetheentirecoursecontentintheirminds beforethey begina courseorwriteatext.However,astudentcanonlyconfrontoneideaatatimeandsothecourse contentisarrangedsequentially,onetopicafteranother,onewordafteranother.Itisthenleftto thestudenttoconstructinhismindthe bestarrangementofthematerial.Wemustacceptthat coursesandtextsaresequentiallyordered.Pagesarenumbered.Astudent,whenconfrontedby a particularlydifficult partofthetext,maysetthetextasidenevertoseetheremainderofthe course.The problemis:whatisthe bestarrangementforcoursematerialsothatastudentcan visualizethemajorcomponentsofacourseatthe beginningandsubsequentlyfillinthedetails? This paper presentsaroadmapintheformofatreestructure(Seethechartattheconclusion.) thatwillallowyoungstudentstotreattheirstudyofarithmeticasaresearchproject.The roadmapwill provideanavigationalaidtoassistintheexplorationoftheworldofnumbers. Historically,thediscoveryofthe periodictableguidedscientistsinthestudyofchemistry, indicatingwhatwasknownandwheregapsexistedandraisingquestionsaboutwhatremainedto beexplored.The periodictablehas beenimmenselyvaluable,eventhoughitsfinalformmay differfromthetablethatwasoriginally proposed.Thisroadmapwillservestudentssimilarlyin theirstudyofarithmetic. Ineverywell-plannedcourse,onlyonethingisstudied.Inarithmetic,thesubjectisnumbersand sotheword'numbers'is placedatthetopoftheroadmap.Attheendofthecourse,astudent shouldknowthekinds,forms,operations,propertiesandusesofnumbers.Inoursociety,this studytakesafewyearsandisoftendisorganized.Itshouldnot besurprisingthatayoungstudent might becomeconfusedanddisenchanted.Attheendofthecourse,thestrategyshouldbe reviewedsothatthestudentcanseewhathas beenlearnedaboutnumbersandhownumbersare usedandwhatremainsforfuturestudy. Thisroadmapsuggeststhatincludinginthecourse presentation,atreestructure,givingequal importancetothekinds,operations,formsandpropertiesofnumbers,hasconceptualadvantages. Thetreestructureseparatestheseconcepts buthighlightstheirinteractioninthesolutionof quantitative problems.Thevisualizationofthetreestructuretogetherwithanunderstandingof theimportantconceptofformsmayaidstudentsinmasteringthesubject. Thestudyofnumbersisunendingandsothisroadmapcannot beexhaustive.Stillitcanprovide aframeworkforintroducingthestudyofnumbers. Page 11.1323.2 Page Whatarenumbers? Numbersaresymbols,whichcanconveyinformationaboutsize,orderor position.Theircrudest useisasidentifierssuchasareseenonthe backsoffootball players.Inthisuse,theyarenever added,subtractedorcompared.Numbersusedtoidentifyhousesasstreetaddressesnotonly identify butalsoprovideinformationabouttheorderofthehousesonthestreetandalsoindicate onwhichsideofthestreethousesaresituated.Numbersarecategorizedbytheiralgebraicor topological(distance) properties. Kindsofnumbers Thenumbers,whichareusedtoindicatethesizeofsetsof discrete (separateanddistinct) objects,arethenaturalor countingnumbers (1,2,3etc).Childrenlearnthecountingnumbers andtheirorderinthesamewaytheylearnthealphabet,bymemorization.Latertheylearnto add,subtractandmultiplythecountingnumbers.Asweknow,inthesetofcountingnumbers,it isnotmeaningfultosubtractalargernumberfromasmallernumber.Wedescribethissituation bysayingthatthesetofcountingnumbersis closed underadditionandmultiplicationbutisnot closedwithrespecttosubtraction. Thenumbers,whichindicatenotonlyadiscreteamount butalsoadirection,arecalled integers andarethesignedwholenumbers(0, ±1, ±2, ±3etc).Theintegersmay beadded,multiplied andsubtracted,andtheresultoftheoperationwillalwaysyieldanotherinteger.Thatis;theset ofintegersisclosedundertheoperationsofaddition,multiplicationandsubtraction.Buttheset ofintegersisnotclosedunderdivision.Forexample,thereisnointegerwhosevalueequalsto5 dividedby3. Therealnumbersystemiscomposedofallthenumbers,eachofwhichcanbe plottedona straightline.One pointisdesignatedaszero.Thecountingnumbers compriseasubsetofthe realnumbersthatareconventionallyspaceduniformlytotherightofzeroextendingindefinitely. Theintegers are plotteduniformlywiththe positivenumberstowardtherightandthenegative numberstowardtheleft.The realnumbersystem isclosedunderanycombinationofthe basic fourarithmeticoperations;addition,subtraction,multiplicationand/ordivision(exceptfor divisionbyzero).Sometimesthereareadvantagestothinkingofthenumbersas pointsonaline butthereareothertimeswhenthedistinctionbetweennumbersandpointsshouldbemaintained. Thesmallestsubsetoftherealnumbersystem,whichcontainstheintegersandisclosedunder the basicfouroperations(exceptdivisionbyzero)isthesetof rationalnumbers ;thatis,those m 2 numbers,whichcanbedescribedbytheratio wheremandnrepresentintegers(e.g. ± ).It n 3 canbeshownthataninfinitenumberofrationalnumberscanbefoundbetweenanytwodifferent rationalnumbersandthereforearationalnumberhasnoimmediate predecessororsuccessor. Therationalnumbersarenotdescribedasdiscrete.Thesetofrationalnumbersisdescribedas dense inthesetofrealnumbers. 11.1323.3 Page Sincetherationalnumbersaredense,atonetimeitwas believedthatalltherealnumberswere rational.Butitwasshownnottobeso.Inparticular,thediagonalofasquarewhosesideshave lengthequalto1cannot beexpressedasaratioofintegersandthereforecannot berational. Polynomialequationsinasinglevariablemayhaverealsolutionsthatarenotrational.The solutionsor roots ofthese polynomialequationsarecalled algebraic numbers.Thesetofall realalgebraicnumbersisclosedunderthe basicfouroperations(exceptdivisionbyzero).The setof positivealgebraicnumbersisalsoclosedundertheoperationoftakingroots.Thesetof algebraicnumbers,whichincludestherationalnumbers,isalsodenseinthesetofrealnumbers. Thesetofrationalnumbersisalsoclosedundertheoperationofinteger powers(exceptwhen the baseiszero) butnotfractional powers(orroots).Examplesarethatwhilethenumbers 1 1 2 2 =4and2 –2= arerational,thevalues2 (1/2) = 2 and 2 ( –1/2) = are algebraic 4 2 numbers. Students are usually introduced to algebraic numbers, denoted by radicals. Compared to radical notation, fractional exponent notation has conceptual and computational advantages. Compare: 5 2 5 32 = 3 = 9 (2/5) 5 (2/5)*5 2 to (3 ) = 3 = 3 = 9. 2 The multiplication *5 in the exponent produces the answer naturally. 5 Mathematicians have shown that there exist real numbers that are neither integers, nor rational nor algebraic; that is, numbers which are not the solutions of polynomial equations. One such number is the number that represents the ratio of the circumference of every circle to its diameter. This number is called π. Another is the number, denoted by e, which appears in the description of natural growth and decay processes. These non-algebraic numbers, which are needed to complete the real number line, are called the transcendental numbers. In summary, the real number system contains as a dense proper subset the algebraic numbers, which contain as a dense proper subset the rational numbers, which in turn, contain the uniformly spaced integers, which contain the counting numbers. Page 11.1323.4 Page Numbersystem Closurefeatures Countingnumbers closedunderaddition,multiplication,countingnumber exponent Integers aboveoperations plussubtraction Rationalnumbers aboveoperations plusintegerexponents,excludingdivision byzero Algebraicnumbers While positivealgebraicnumberscanberepresentedwith rationalexponents,negativenumbersdonothaveeven rootsintherealnumbersystem. Transcendentalnumbers Alloftheremainingrealnumbers. Againnote: Negativenumbersdonothaveevenrootsand divisionbyzeroisalwaysmeaninglessandexcluded. Becausethenumbersthatareseenincommonuseareeitherintegers,fractionsortruncated numbersindecimalformorscientificnotation,someonecouldreasonablyconcludethat transcendentalnumbersarerare.Suchisnotthecase.Whilecomparisonofinfinitesetsisnot obvious,Cantordevisedaconstructionthatshowedthatifanattemptweremadetomatch algebraicnumbersone-to-onewiththetranscendentalnumbers,thealgebraicnumberswouldbe depletedwhiletherewerestilltranscendentalnumbersremaining.Inacourseonthetheoryof measure,acomparisonofthesetsofthealgebraicandtranscendentalnumbersintheinterval betweenzeroandone;wouldrevealthatthesetofalgebraicnumbershavemeasureequaltozero whilethemeasureofthesetoftranscendentalnumbersisone.Transcendentalnumbersarenot rare.Thetranscendentalnumbersarenotobservedincommonuse becauseitisimpossibleto writethemexactly.Likeπande,theymust beapproximatedwithrationalnumbers. Propertiesofnumbers Intextbooks,the propertiesofnumbersaredescribedinthelaws.The propertiesofnumbers involvingtheoperationsofaddition,multiplicationandpowersandtheinversesofthese operationsarecalledthe algebraic properties.The propertiesofnumbersconcerningthe relations>, ≥,<,and ≤arecalledthe order properties.The propertiesofnumbersinvolving distance betweennumbersarecalledthe topological properties.Mathematiciansstudysetsthat strictlyexhibitonlythealgebraic properties(groupsandfields)ororder properties(ordertypes