Mathematics PDF
Preview Mathematics
k^ J;f dthemdtics • • V J By Irving fldler by Hon Illustrated Miller MATHEMATICSWASINVENTEDBYOUREARLIESTANCESTORS TOKEEPTRACKOFTHESEASONSANDTOTALLYBELONGINGS. EUCLIDWASANANCIENTGREEKMATHEMATICIAN WHOSEBOOKISTHEBASISOFTHESTUDYOFGEOMETRY. rldtherndtics By Irving Adler Illustrated by Ron Miller Doubleday NEWYORK LONDON TORONTO SYDNEY AUCKLAND Booksinthisseries: AncientEgyptbyDanielCohen AncientGreecebyDanielCohen DinosaursbyDanielCohen TheHumanBodybyGildaBerger MathematicsbyIrvingAdler PrehistoricAnimalsbyDanielCohen PrehistoricPeoplebyDanielCohen SeasonsbyMelvinBerger SharksbyGildaBerger StarsandPlanetsby ChristopherLampton WeatherbyHowardE.Smith WhalesbyGildaBerger ToJoyce PublishedbyDoubleday,adivisionof BantamDoubledayDell 6NP6ueb6wliFisYfhotirhnkgAvG1er0no1u0ue3p,,NInecw.York, *- flVT-s« h-MWl Doubledayandtheportrayalofananchor withadolphinaretrademarksof Doubleday,adivisionofBantamDoubleday DellPublishingGroup,Inc. LibraryofCongress Cataloging-in-PublicationData Adler,Irving. illMuasttrhaetmedatbiycsRo/nbyMiIlrlveir.ng—A1dslteerd.: p.cm. Includesindex. Summary:Anintroductiontothe scienceofnumbersandspace,discussing basicconcepts,andincludingan introductiontocomputerprogramming, mat1.heMmaatthiecmaaltgiacmse—s,Juvaenndilaectivities. literature. [1.Mathematics.}I.Miller, Ron,1947- .II.Title. QA40.5.A3 1990 510—dc20 89-32712 CIPAC RL:3.1 ISBN0-385-26142-X TIIleSlxButsNtcro0ap-ty3ir8oin5gs-h2ct6o1©p4y3r1-i98g9h0tb(©ylibI1.r9v9bi0dngbg.y)AdRloenrMiller AllRightsReserved PrintedinSingapore September1990 FirstEdition Thefollowingwasreproducedwithpermissionfrom TheMuseumofModernArt:CompositioninWhite, BlackandRedbyPietMondrian,1936.Oiloncanvas, 4N0ex/w\Yxor41k,."GCiofltleocftiAodnv,iTsohreyMCuomsmeiutmteoef.ModernArt, 6 i|Si HPmm The Science of Numbers M and Space athematicsisthescienceinwhichwethinkcarefully aboutnumbersandspace. Itdevelopedalongwith civilizationtohelpmenandwomensolvepractical i problems. Butpeoplealsofoundnumbersandshapesin spaceinterestinginthemselvesandbegantoplaywiththemforfun. Overtenthousandyearsagoallpeoplegottheirfoodbyhuntingand bygatheringwildfruitsandberries. Tokeeptrackoftheirsupplies- theylearnedhowtocount, usingthenaturalnumbers, 1, 2, 3, 4, and soon. Later, whenpeoplebecamefarmersandshepherds, they inventedthecraftsofpotteryandbasketrytomakecontainersfor storingthings, spinningandweavingformakingcloth, andsmelting andforgingofmetalstomaketoolsandweapons. Carpentersand masonsbuilthomesforthepeople, palacesfortheirrulers, andgreat tombslikethepyramidsfortheirdeadkings. Commercedevelopedfor theexchangeofproducts, andmoneywasinvented. Intheseactivitiescountingbecameevenmoreimportant, andpeople alsohadtolearnhowtomeasureandcalculate. Formeasuringthey sometimeshadtousefractions, andtocalculate, theylearnedhowto add, subtract, multiply, anddivide.Arithmeticbegantogrow. Measuringlandmadeitnecessarytolearnaboutthesizesandshapesof fields, andthesubjectofgeometry, whichstudiesspace, wascreated. imlw 34567 7 Climbing the Number Ladder T ocountweusethenaturalnumbers, 1, 2, 3, 4, andsoon. Thenaturalnumbersarelikealadderthathasabottomand notop. Thefirstrungoftheladderisthenumber 1. The nextrungaboveitisthenumber2. Asyouclimbfromrung torung, youreachhigherandhighernumbers. Sincethereisno highestnumber, theladderneverends. Onahandcalculatoryoucandisplaythenaturalnumbersintheir properorder, startingwith 1, withoutusinganynumberkeyexcept 1. Anumberladdercanhelpillustratethis. Pressthekeyfor 1 onthe calculator, anditdisplaysthenumber 1. Thisputsyouonthefirst rungofthenumberladder. Togetto2, youhavetogouponerung. Thismeansyouaddone. Onthecalculatorpressthekeys +, 1, and =. Thecalculatorthendisplaysthenumber2. Togetfrom2to3, you addoneagain, soagainpressthekeys +, 1, =. Ifyourepeatthis processoverandover, thecalculatorwilldisplaythenaturalnumbers intheirproperorder. Press 1+1=+1=+I=+1= andsoonforever. Insteadofrepeatingthesequence +1= overandoveragainforever, wecangivethesameinformationinashortwaybywriting +1= only onceandthendrawinganarrowfromthe = tothe + toshowthat yougobacktorepeatthatsequence: 1 + 1 = <5* <5 — Wecanalsoinstructacomputertocountbyprintingthenatural numbersonthescreenintheirproperorder. Bylearninghowtodo thisyouwillbegintolearncomputerprogramminginthecomputer languagecalledBASIC. ToprepareacomputerforusingBASIC, turnitonandputinthe discwiththeoperatingsystem. Thenwhenthecomputerannounces thatitisready, typeBASIC, andpresstheENTERkey, thekeythat hasacurvedarrowonit. (SkipthisstepinanApplecomputer.) Acomputerprogramisasetofinstructionstothecomputerthatit storesinitsmemoryandthenreadsandfollowswhenitistoldtodo so. Theinstructionsarenumberedsothatthecomputerknowsthe orderinwhichtheinstructionsaretobefollowed. Butinsteadof numberingtheinstructions 1, 2, 3, andsoon, itisusualtonumber thesteps 10, 20, 30, etc. Thismakesitpossibletoinsertastep withoutrenumberingthewholeprogram. Suppose, forexample, you decidethatyouneedanewstepbetweenstep 10andstep20. The computerwillautomaticallyputthenewstepinitsrightplaceifyou callitstep 15. Thefirstthingwehavetodoisgiveanametothenumbersthatwe wantthecomputertoprint. Weusealetterofthealphabetforthis name. Let’susetheletterN. Wealsohavetotellthecomputerthefirst numberthatthisletterwillstandfor. Sincethefirstnumberwewant printedis 1, thefirststepoftheprogramis: 10N=1 Togetthecomputertostorethisinstructioninitsmemory, we presstheENTERkey. PresstheENTERkeyafteryoufinishtyping eachoftheremainingnumberedstepsintheprogram. Nowthatthe computerknowswhatnumberwewantittoprintfirst, wehavetotell ittoprintit. Sothenextinstructionis: 20PRINTN Nowwewantthecomputertostartclimbingthenumberladder. Wewantittochangethenumberinitsmemorytothenext-higher number. Fromwhatwehavedonewiththehandcalculator, weknow thatwemusttellittoadd 1. Wedothisbytyping: 30N=N+1 ThistellsthecomputerthatthenewvalueofNisobtainedbyadding 1 totheoldvalueofN. NowwewantthecomputertoprintthenewvalueofN. Insteadof sayingPRINTNagain, wetakeadvantageofthefactthatwegavethis instructioninanearlierstep. Wetype: 40GOTO20 Thistellsthecomputer, whenitrunstheprogram, thatafterstep40it goestostep20, andthengoesontostep30, etc. Afteryouhavestoredthisprogram, togetittorun, typeRUNand presstheENTERkey. Youwillseeimmediatelythatweareintrouble. Thenumberladderhasnotop. Thecomputerisprintingthenumbers asweaskeditto, butitwillgoonforever. Weforgottotellitwhen tostop. Togetittostopnow, presstheBREAKkey. Togettheprogramon thescreen, typeLISTandpresstheENTERkey. Wehavetoinsert anothersteptotellthecomputerwhentostop. Type: N = 35 IFN=21THENSTOP 1 PresstheENTERkey. Thecomputerwillputthisstepbetweenstep * ”| 30andstep40. Thissteprequiresthecomputertomakeadecision. It r 1 mustcomparethenumberthatNstandsforwiththenumber21. If theyarenotthesame, itskipstheSTOPinstructionandthenkeeps PRINT N going. Iftheyarethesame, itwillstop. Step35 allowsittoprintonly thenumbersfrom 1 to20. Aconvenientwayofshowingwhatthecomputerhastodointhis pcarrorgiredamouitstisopdurtawintaofalobwocxh.artA.sItnepatfhlaotwcrheaqruti,reesaachdesctiespiotnhaatbmouusttbe add 1 to N whattodonextisputintoadiamond. Theprogramhastwobranches goingoutofthediamond, showingthetwopossibledecisions. The decisionthatsaysGOTO20isshownbythearrowthatgoesbackto thePRINTNinstruction. looTp.heArpertourgnrafmrowmitahlaatelroosptewpiltloraunnefaorlrieevreronuenlfesosrmasswthepatisisprcoavllieddeda STOP forbreakingoutoftheloop, likeourstep35. Hereistheprogramforprintingthenaturalnumbersfrom 1 to20: N= 10 1 20PRINTN 30N=N+1 35 IFN = 21THENSTOP 40GOTO20
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