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1、PartIIntroductionofmathematicsintoalgebra,geometry,andanalysis.SoI.1WhatIsMathematicsAbout?letusbeginwiththis,andthenqualifyitlater.Itisnotoriouslyhardtogiveasatisfactoryanswerto1.1AlgebraversusGeometrythequestion,“Whatismathematics?”TheapproachofMostpeoplewh
2、ohavedonesomehigh-schoolmathe-thisbookisnottotry.Ratherthangivingadefinitionofmaticswillthinkofalgebraasthesortofmathemat-mathematics,theintentionistogiveagoodideaofwhaticsthatresultswhenyousubstitutelettersfornum-mathematicsisbydescribingmanyofitsmostimpor-be
3、rs.Algebrawilloftenbecontrastedwitharithmetic,tantconcepts,theorems,andapplications.Nevertheless,whichisamoredirectstudyofthenumbersthemselves.tomakesenseofallthisinformationitisusefultobeSo,forexample,thequestion,“Whatis3×7?”willbeabletoclassifyitsomehow.tho
4、ughtofasbelongingtoarithmetic,whiletheques-Themostobviouswayofclassifyingmathematicsisbytion,“Ifx+y=10andxy=21,thenwhatistheitssubjectmatter,andthatwillbetheapproachofthisvalueofthelargerofxandy?”willberegardedasabriefintroductorysectionandthelongersectionent
5、i-pieceofalgebra.Thiscontrastislessapparentinmoretledsomefundamentalmathematicaldefinitionsadvancedmathematicsforthesimplereasonthatitis[I.3].However,itisnottheonlyway,andnotevenobvi-veryrarefornumberstoappearwithoutletterstokeepouslythebestway.Anotherapproach
6、istotrytoclas-themcompany.sifythekindsofquestionsthatmathematiciansliketoThereis,however,adifferentcontrast,betweenalge-thinkabout.Thisgivesausefullydifferentviewofthebraandgeometry,whichismuchmoreimportantatansubject:itoftenhappensthattwoareasofmathematicsadva
7、ncedlevel.Thehigh-schoolconceptionofgeometrythatappearverydifferentifyoupayattentiontotheiristhatitisthestudyofshapessuchascircles,trian-subjectmatteraremuchmoresimilarifyoulookatthegles,cubes,andspherestogetherwithconceptssuchkindsofquestionsthatarebeingasked
8、.Thelastsec-asrotations,reflections,symmetries,andsoon.Thus,tionofpartI,entitledthegeneralgoalsofmathe-theobjectsofgeometry,andtheprocessesthattheymaticalresearch[I.4],looksatthesubjectfro
