From zero to infinity: what makes numbers interesting ; fiftieth anniversary edition PDF
Preview From zero to infinity: what makes numbers interesting ; fiftieth anniversary edition
(cid:1) (cid:1) (cid:1) (cid:1) Whatmathematiciansandteacherswriteabout FROMZEROTOINFINITY “I read From Zero to Infinity when I was a schoolboy in Oxford, England,andmyonlyregretisthatIwaswellintomyteens(17) before it happened. Just last week I gave away my most recent copytothe12-year-olddaughterofafriend. Iwillbegettingan- othercopyformyselfassoonasIcan.” —JohnB.Cosgrave,St.Patrick’sCollege,Dublin,Ireland “AfterreadingFromZerotoInfinity,Iwashooked. Thisbookdis- cussedmanybeautifulideasandfactsabouttheintegersandposed severalinterestingproblems. Itriedtosolvethem. Ifailed. Itried to construct counter examples. I made my own conjectures and proved some related results. By the time I graduated from high schoolIhadfilledtwo2-inchnotebookswithmyownideas, re- sults,andcalculations.” —NathanielDean,MathematicsDepartment, BellCommunicationsResearch “Constance Reid’s book From Zero to Infinity was translated into Japanese, and I found it when I was a junior high school boy. I wasreallyimpressedbyReid’sbook,andIreaditrepeatedly. In- spiredbyit, IeventriedtosolveFermat’sLastTheorem. NowI amworkinginanalyticnumbertheory,andIthinkoneoftherea- sonsformychoiceisasentenceinReid’sbook:‘Analyticnumber theoryissaidtobetechnicallythemostdifficultinthewholeof mathematics.’” —KohjiMatsumoto,studentatNagoyaUniversity (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) “Yesterday,aboardaflightfromDenver,Ihadaniceconversation withagentleman,JohnMoulter.WhenIlearnedthatMr.Moulter is a retired Los Angeles mathematics teacher, I mentioned that I hadthegoodfortunetoknowConnieReid.Mr.Moulterabruptly demanded,‘DoyoumeanConstanceR-E-I-D?’WhenIanswered affirmatively, he told me that you had changed his life. About fiftyyearsago,thenahigh-schoolhistoryteacher,hepickedupa copyofEsquiremagazineinabarbershopcontainingareviewof FromZerotoInfinity.Thereviewpromptedhimtobuyyourbook. And your book inspired him to switch from teaching history to teachingmathematics!” —Letterfromafriendwithappendednote fromJohnMoulter:“Thankyou.Thankyou.” “I was the sort of child who always carried a book wherever he went.InfifthandsixthgradethatbookwasmostfrequentlyFrom ZerotoInfinity.Iwasindeedborntobeamathematician,butFrom ZerotoInfinityhelpedmetorealizethatIwaspartofacommunity of number-people. There can be few pleasures more satisfying than having the chance, as an adult, to help bringone’s favorite childhoodbookbackintoprint.” —BruceReznick,UniversityofIllinoisUrbana “Iwanttothankyouforhavingwrittensuchawonderfulbook.It waspitchedonjusttherightlevelforayoungteenagerbut,more tothepoint,itexpressedtherightmixofbeautyandwonder.Ijust hadtolearnmore.Iverymuchbelievethatthissmallbook,which stilloccupiesanimportantplaceinmypersonallibrary,enriched mylifeimmeasurably. Itisveryrarethatwefindwhatwereally wanttodoinlife,andIamverygratefulthatyourbookledmein therightdirection.” —HughWilliams,UniversityofCalgary,Canada, authorofEdouardLucasandPrimalityTesting (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) · · FROM ZERO TO INFINITY (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) · · FIFTIETH ANNIVERSARY EDITION From Zero to Infinity What Makes Numbers Interesting Constance Reid AKPeters,Ltd. Wellesley,Massachusetts (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) Editorial,Sales,andCustomerServiceOffice AKPeters,Ltd. 888WorcesterStreet,Suite230 Wellesley,MA02482 www.akpeters.com Copyright(cid:1)c2006byAKPeters,Ltd. Allrightsreserved. Nopartofthematerialprotectedbythiscopyright noticemaybereproducedorutilizedinanyform,electronicormechani- cal,includingphotocopying,recording,orbyanyinformationstorageand retrievalsystem,withoutwrittenpermissionfromthecopyrightowner. Firstpublishedin1955byThomasY.CrowellCo. Secondedition,1960byThomasY.CrowellCo. Thirdedition,1964byThomasY.CrowellCo. Fourthedition,1992byTheMathematicalAssociationofAmerica. Fifthedition,2006 LibraryofCongressCataloging-in-PublicationData Reid,Constance. Fromzerotoinfinity:whatmakesnumbersinteresting/ ConstanceReid.–5thed. p.cm. ISBN-13:978-1-56881-273-1(alk.paper) ISBN-10:1-56881-273-6(alk.paper) 1.Numerals.2.Numbertheory.I.Title. QA93.R422006. 510--dc22 2005027860 PrintedintheUnitedStatesofAmerica 1009080706 10987654321 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) · · I N M E M O R Y O F JuliaBowmanRobinson 1919–1985 and Raphael MitchelRobinson 1911–1995 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) · · CONTENTS ACKNOWLEDGMENTS xi AUTHOR’S NOTE xiii 0 ZERO 1 Onlyaplace-holderuntilfinallyrecognized asfirstofthenaturalnumbers. 1 ONE 15 Anumberthatmakesmathematicsdifferent fromalltheothersciences. 2 TWO 27 Aprimitivenumbersystemcomesintoitsown withtheelectroniccomputer. 3 THREE 39 Firstamongtheoddnumbersdivisibleonlyby themselvesandone. 4 FOUR 53 Numbersmultipliedbythemselvesprovide beautifullydifficulttheorems. ·vii· (cid:1) (cid:1) (cid:1) (cid:1)
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