48第一章
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AfterthedeclineofancientGreekcivilization,thedevelopmentofmathematicsenteredatroughinEurope,whiletheancientChinese,Indian,andArabiancivilizationscontinuedtopromotetheprogressofmathematics.AncientChinesemathematicsfocusedonpracticalarithmeticandalgorithmthinking.TheancientmathematicalclassicTheNineChaptersontheMathematicalArtsystematicallysummarizedthealgorithmsoffieldmeasurement,graintransportation,engineeringcalculation,andequationsolvinginancientChina,formingacompleteancientarithmeticsystem.Chinesemathematicianssuccessivelymadeworld-leadingachievementsinfractionoperation,positiveandnegativenumberoperation,remaindertheorem,andhigh-orderequationsolution.IndianmathematiciansinventedthemodernArabicnumeralsystem0-9,perfectedthedecimalcountingmethod,andlaidthefoundationforglobalunifiednumericalcalculation.ArabianmathematiciansinheritedandintegratedthemathematicalachievementsofGreece,India,andChina,sortedoutalgebraictheories,andpreservedandspreadthemathematicalclassicsofancientcivilizationsduringthemedievalEuropeandarkages,becominganimportantbridgefortheinheritanceanddevelopmentofmathematics.
TheRenaissanceinthe16thcenturyusheredinthegoldenageofmodernmathematics.Withtheliberationofhumanthoughtandthedevelopmentofastronomy,physics,andnavigation,mathematicsfacednewdevelopmentdemands.Inthe17thcentury,Descartesfoundedanalyticgeometry,unifiedgeometryandalgebraforthefirsttime,andrealizedthetransformationfromgeometricfigureresearchtoalgebraicequationresearch,openingupanewfieldofmathematicalanalysis.Subsequently,NewtonandLeibnizindependentlyinventedcalculus,whichsolvedthecoremathematicalproblemsofvariablemotion,instantaneousvelocity,andcurvedareacalculationthatcouldnotbesolvedbyclassicalmathematics.Thebirthofcalculusmarkedtheentryofmathematicsintotheeraofvariablemathematicsfromtheeraofconstantmathematics,andgreatlyexpandedtheboundaryofmathematicalresearch.
The18thand19thcenturieswitnessedthesystematizationandrefinementofmodernmathematics.MathematicianssuchasEuler,Lagrange,andLaplaceperfectedthetheoreticalsystemofcalculusandestablisheddisciplinessuchasdifferentialequations,analyticmechanicsmathematics,andprobabilitytheory.Atthesametime,numbertheory,projectivegeometry,non-Euclideangeometry,andgrouptheoryweresuccessivelyestablished,makingmathematicsbranchmorediversifiedandtheoreticalsystemmorerigorous.Inthelate19thcentury,theaxiomatizationmovementofmathematicsrose,andmathematicianssortedoutthelogicalfoundationofvariousmathematicalbranches,makingmodernmathematicsformacompleteandstandardizedtheoreticalsystem.
Inthe20thcentury,mathematicsenteredtheeraofmodernmathematicswithrapidexplosion.Withtheprogressofquantummechanics,relativity,computertechnology,andinformationscience,mathematicalresearchexpandedtomoreabstractandcutting-edgefields.Topology,functionalanalysis,abstractalgebra,differentialgeometry,mathematicallogic,andotheremergingdisciplinesdevelopedrapidly.Atthesametime,theintersectionofmathematicsandotherdisciplinesbecameincreasinglyclose,andinterdisciplinarymathematicssuchasmathematicalphysics,mathematicalbiology,financialmathematic