数学专业英语(1)

MathematicalEnglish1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity1MathematicalEnglishDr.XiaominZhangEmail:[e-mail]1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity2§2.1Mathematics,EquationandRatioTEXTAWhatismathematicsMathematicscomesfromman’ssocialpractice,forexample,industrialandagricultureproduction,commercialactivities,militaryoperationsandscientificandtechnologicalresearches.Andinturn,mathematicsservesthepracticeandplaysagreatroleinallfields.Nomodernscientificandtechnologicalbranchescouldberegularlydevelopedwithouttheapplicationofmathematics.Fromtheearlyneedofmancametheconceptsofnumbersandforms.Then,geometrydevelopedoutofmeasuringland,andtrigonometrycamefromproblemsofsurveying.Todealwithsomemorecomplexpracticalproblems,manestablishedandthensolvedequationswithunknownnumbers,thusalgebraoccurred.Before17thcentury,manconfinedhimselftotheelementarymathematics,i.e.geometry,MathematicalEnglish1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity3trigonometryandalgebra,inwhichonlytheconstantswereconsidered.Therapiddevelopmentofindustryin17thcenturypromotedtheprogressofeconomicsandtechnologyandrequireddealingwithvariablewithvariablequantities.Theleapfromconstantstovariablequantitiesbroughtabouttwonewbranchesofmathematics--analyticgeometryandcalculus,whichbelongtothehighermathematics.Nowtherearemanybranchesinhighermathematics,amongwhicharemathematicalanalysis,higheralgebra,differentialequations,functiontheoryandsoon.Mathematiciansstudyconceptsandpropositions.Axioms,postulates,definitionsandtheoremsareallpropositions.Notationsareaspecialandpowerfultoolofmathematicsandareusedtoexpressconceptionsandpropositionsveryoften.Formulasfiguresandchartsarefullofdifferentsymbols.SomeofthebestknowsymbolsofmathematicsaretheArabicnumerals1,2,3,4,5,6,7,8,9,0,andsignsofaddition“+”,MathematicalEnglish1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity4subtraction“-”,multiplication“×”,division“÷”andequality“=”.Theconclusionsinmathematicsareobtainedmainlybylogicaldeductionsandcomputations.Foralongperiodofthehistoryofmathematics,thecentricplaceofmathematicswasoccupiedbythelogicaldeductions.Now,sinceelectroniccomputersaredevelopedpromptlyandusedwidely,theroleofcomputationbecomesmoreandmoreimportant.Inourtimes,computationisnotonlyusedtodealwithalotofinformationanddata,butalsotocarryoutsomeworkthatmerelycouldbedoneearlierbylogicaldeductions,forexample,theproofofmostofgeometricaltheorems.MathematicalEnglish1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity5NotationsmathematicalanalysisTheareaofmathematicsgenerallytakentoincludethosetopicsthatinvolvetheuseoflimitingprocess.Thusdifferentialcalculusandintegralcalculuscertainlycomeunderthisheading.Besidesthese,thereareothertopics,suchasthesummationofinfiniteseries,whichinvolveinfiniteprocessesofthissort.Thetermanalysishasalsocometobeusedtoindicatearathermorerigorousapproachtothetopicsofcalculus,andtothefoundationsoftherealnumbersystem.logicaldeductionsThelogico-deductivemethod,asystemofinferencewhereconclusions(newknowledge)followfrompremises(oldknowledge)throughtheapplicationofsoundarguments.inductionTheprocessofderivinggeneralprinciplesfromparticularfactsorinstances.axioms,postulatesThebasicassumptionsunderlyingagivenbodyofdeductiveknowledge.Theyareacceptedwithoutdemonstration.AllMathematicalEnglish1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity6otherassertions(theorems,ifwearetalkingaboutmathematics)mustbeprovenwiththeaidofthebasicassumptions.Axiom,inclassicalterminology,referredtoaself-evidentassumptioncommontomanybranchesofscience.Atthefoundationofthevariousscienceslaycertainbasichypothesesthathadtobeacceptedwithoutproof.Suchahypothesiswastermedapostulate.Thepostulatesofeachscienceweredifferent.Theirvalidityhadtobeestablishedbymeansofreal-worldexperience.TheclassicalapproachiswellillustratedbyEuclid'selements,whereweseealistofaxiomsandpostulates.A1Thingswhichareequaltothesamethingarealsoequaltooneanother.A2Ifequalsbeaddedtoequals,thewholesareequal.A3Ifequalsbesubtractedfromequals,theremaindersareequal.A4Thingswhichcoincidewithoneanotherareequaltooneanother.A5Thewholeisgreaterthanthepart.MathematicalEnglish1:Mathematics,EquationandRatioDr.XiaominZhang:MathematicsDepartment,SchoolofScience,NingboUniversity7P1Itispossibletodrawastraightlinefromanypointtoanyotherpoint.P2Itispossibletoproduceafinitestraightlinecontinuouslyinastraightline.P3Itispossibletodescribeacirclewithanycentreanddistance.P4Itistruethatallrightanglesareequaltooneanother.P5Itistruethat,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.signsof+,