数学概念大全（Encyclopediaofmathematicalconcepts）

数学概念大全(Encyclopediaofmathematicalconcepts)

JuniorhighschoolmathematicsconceptDaquan(withECNU

Edition)

Gradeseven:

Positiveandnegative:

Zeroisneitherpositivenornegative.

Positiveinteger,zeroandnegativeintegercalledinteger,

positivefractionandnegativefractionfractioncollectively.

Referredtoasintegerandfractionalrationalnumber.

Putsomenumberstogether,formasetofnumbers,referredto

asthenumberofsets.

Allpositiveintegersandzerosnumbersetiscalledtheset

ofnaturalnumbers.

Axis:

Theprovisionsoftheorigin,isthedirectionandlengthof

thelineiscalledtheaxis.

Thenumbertwothatisontheaxis,thetotalnumberoflarge

numberthantheleft.

Positivenumbersaregreaterthanzero,negativenumbersare

lessthanzero,positivethannegative.

Oppositenumber:

Onlytwothenumberofdifferentsymbolsthatareopposite,the

oppositionofzeroiszero.

Usuallyinfrontofanumberofadd"-"，saidthatthenumber

ofoppositenumber.

Infrontofanumberofaddthatisthenumberitself.

Absolutevalue:

Theabsolutevalueofanumberofitsown,theabsolutevalue

ofzeroiszero.

Theabsolutevalueofanegativenumberistheoppositeofit.

Sizeofarationalnumber:

Twonegative,absolutevalueislargebutsmall.

Rationalnumberaddition:

Thesamenumberofaddingtwonumbersforthesamesymbol,and

thesumofabsolutevalue.

Theabsolutevalueofdifferenceisnotequaltothesumofthe

twonumbers,thelargeraddendsymboltotheabsolutevalueof

absolutevalue,andwiththelargerabsolutevalueofsmaller

reduction.

Oppositenumberthesumoftwonumberszero.

Anumberofzerosum,thisnumberisstillthe.

Arationalnumberbythesymbolandtheabsolutevalueoftwo

parts,additionoperations,andshouldpayattentionto

determinethesymbolsandabsolutevalue.

Rationalnumberadditionstillsatisfiesthecommutativelaw

ofadditionandcombinationoflaw.

Commutativelawofaddition:twonumbers,exchangeaddend

position,andinvariant,(a+B=B+a)

Associativelawofaddition:threenumbers,thefirsttwo

numbers,orafterthefirsttwonumbers,andthesame.

(a+b)+C=a+(B+C)

Subtractionrationalnumber:

Subtractanumber,plustheoppositenumberisequaltothe

number.

Rationalnumbermultiplication:

Themultiplication,ifafactorintoitsoppositenumber,the

productiscontrarytothenumberoforiginalproduct.

Themultiplication,thesamenumberofpositive,negative

difference,andtheabsolutevalueofmultiplication.

Anynumberofzeromultiplication,allzero.

Commutativelawofmultiplication:twonumbermultiplication

factor,exchangeposition,productthesame.(AB=BA)

Associativelaw,multiplythreenumbers,thefirsttwomonths

beforeorafterthefirstnumbermultiplication,multiplytwo

numbers,thesameproduct.

(AB)C=a(BC)

Notequaltothenumbermultipliedbyzero,thenumberof

productsymbolswithnegativefactordecision,whenthe

negativefactorisoddwhentheproductisnegative;whenthe

negativefactorisevenwhentheproductispositive.

Thenumberofmultiplication,afactoriszero,zeroproduct.

Distributivelawofmultiplication:anumberandthetwonumber

andmultiplication,equivalenttothelawsuitrespectivelyand

thetwonumbermultiplication,thesumofproduct.A(B+C)

=AB+AC

Rationaldivision:

Dividedbyanumberequaltothereciprocalofthenumberof

times,(notzerodivisor)

Thenumberwasdivided,withdifferentnumbers,negative,and

theabsolutevalues.

Zerodividedbyanynonzeronumberiszero.

Therationalnumber:

Forseveralofthesamefactorofproductoperation,called

power.

Theresultiscalledpower.

An,calledabase.Theniscalledtheindex.

Anypowerofapositivenumberispositive.

Oddpowernegativeisnegative,negativeevenpoweris

positive.

Scientificnotation:

Usingscientificnotationanumber10,theindexandthe

originalintegerbitless.

Themixedoperationofrationalnumbers:

Thefirstisinvolution,thencalculatethelastadditionand

subtractionmultiplicationanddivision;

Theoperation,inaccordancewiththeorderfromlefttoright;

Iftherearebrackets,bracketsonthefirstcount,thencount

inparentheses,andthenisinbrackets.

Theapproximatenumberandeffectivenumber:

Veryclosetotheactualwidthofthenumber,calledthe

approximatenumber.

Fromthefirstoneontheleftisnot0numbers,tothelast

figuresofar,

Effectivenumberofallthenumberscalledthisnumber.

Columntypealgebra:

Asinglenumberoraletterisalgebraic.(0isanalgebra,

algebrawithoutunit.)

Algebraicvalue:

Thenumericalvaluesofthelettersinthealgebra,algebraic

operationsinrelationtothecalculationresult,called

algebraicvalue.

Zhengshi:

Monomial:algebraconsistsofproductnumbersandletters,(a

singlenumberoraletterisamonomial.)

Thecoefficientfactorinthisdigitalmonomialcalled

monomials.

Amonomial,alllettersindexandthenumberofmonomialsis

called.

Severalmonomialsandpolynomialscalled.Eachitemiscalled

monomialpolynomial.Doesnotcontainletterscalledconstant.

Thedegreeofthepolynomialnumberofallitemsandnot.

Eachincludesasymbolicpolynomialinfrontofit.

Monomialsandpolynomialsareintegral.

Apolynomialofthepositioninaccordancewiththeorderof

sizeindexalettertoorder.

Descendingorder:orderfrombigtosmallorderaccordingto

theXindex.

Ascendingorder;accordingtotheorderfromsmalltolargex

index.

Rearrangethepolynomial,eachonemustmovetogetherwithits

symbol.

Polynomialcontainingtwoormorethantwoletters,oftenin

accordancewithoneletterarrangementorascendingdescending

order.

Adaptivesubtraction:

Typeofadditionandsubtractioncontainedthesameletter,and

thesameletterindexwerealsocalledequalterms,(thesame

letternumberofitemswiththesameisnotmuch.Forexample:

3a2b3and5a3b2)

Theconstanttermisallsimilaritems

Thepolynomialofthemergerofsimilaritemsintoone,called

themergerofsimilaritems.

Thecoefficientofadditiveterms,theresultsasafactor,

lettersandtheindexremainedunchanged.

Infrontofthebracketsisputitinfrontofthebrackets

and"+〃removed,thebracketsdonotchangesign.

Infrontofthebracketsis"-"，thebracketsandinfrontof

it"-"removed,thechangeofsymbolsinbrackets.

Soinfrontofthebracketsisincludingallthesamesymbol

tobrackets.

Soinfrontofthebracketsis"一"，includingallthesymbols

inbracketstochange.

Iftherearebrackets,thengotothebrackets,ifthereare

similaritems,thenthemergerofsimilaritems.

Three-dimensionalgraphics:

Fromthefront,topandside(leftorright)inthreedirections

atanobject,andthendescribethreeseediagram,namelyview.

Fromthefronttoseethegraphics,calledtheview;seefrom

theabovegraph,calledthetopview;seefromthesideofthe

graph,calledthesideview.

Withathree-dimensionalgraphics,developedindifferentways

togetthesurfacedevelopmentisnotthesame.

Thethree-dimensionalgraphicsgraphicissurroundedbythe.

Pointandline:

Usuallyrepresentsanobject'sposition.

The1ineformedbythetwopartytotheunlimitedextensionof

thegraphisastraightline.

Afterthereisastraightline,andonlyastraightline.

Alineisdividedintotwoequalsegments,calledthemidpoint

ofthisline.

Angle:

Theangleiscomposedoftwopublicendpointraycomposition

hungrygraphics,canalsoberegardedastheangleofaray

arounditsendpointsandrotategraphics.Theendpointis

calledrayanglevertex,thestartingpositionofrayangleis

calledtheinitialboundary,theendpositioniscalledtheend

sideoftherayangle.

Aroundtheendoftheendsideandrotatetotheangleofthe

initialboundaryinastraightline,thenstraightangle.

Aroundtheendpointtotheendedgeandinitialsiderotation

intimecoincidence,thisiscalledastheangleperigon.

Itwasdividedinto60parts,eachpartis1points,denoted

1’;andthenputthe1pointsinto60equalportions,eachone

is1.Remember,1"1angle=360degrees1degrees1degrees

angle=180=60=T'60〃

Putacornerinanothercorner,makingthemthevertices

coincide,onesideandtheothersidealsocoincide,twocorners

intheedgesofthesameside.

Twocornersareaddedorsubtracted,thesumordifferenceof

angleis.

Araydrawnfromanangleofthevertex,theangleintotwoequal

angle,therayiscalledtheanglebisector.

Twoanglesandequalto90degrees(rectangular),saidthetwo

complementaryangleangle,referredtoascomplementary.

Ifthetwoanglesand180degrees(boxer),equaltosaidthe

twoanglesaresupplementaryangles,referredtoas

complementary.

Anytwovertexof,becausetheyallhavethesameBulu,sothey

areequal,itcansimplysay:theanglesareequal.

Whenthefourcornershavearightangle,theotherthree

cornershavebecomerightangle,atthispoint,thelineAB,

CDperpendiculartoeachother,denotedas“ABanCD"0called

theirintersectionpedal.

Inthesameplane,bylineorlinepoint,thereisonlyone1ine

perpendiculartoagivenline.

Correspondingangles:straightlineonthesameside,bythe

samepartyline.

Alternateangles;straightlineonbothsidesofthelineis

inside.

Theinterioranglesonthesameside:astraightlineonthe

sameside,bythemedialline.

Parallellines:

Twolinesdonotintersectinthesameplanecalledparallel

lines.

Inthesameplane,thetwodoesnotcoincidewiththelinear

positionrelationshiponlytwo:intersectingorparallel.

Afteralittleknownoutsidethestraightline,andonlya

straightlinewithagivenlineinparallel.

Ifthetwolinesandthirdlinesareparallel,sothetwolines

areparalleltoeachother.

Correspondinganglesareequal,thetwoparallellines.

Alternateanglesareequal,thetwoparallellines.

Complementaryinterioranglesonthesameside,twoparallel

lines.

Twoparallellines,correspondinganglesareequal.

Twoparallellines,alternateanglesareequal.

Twoparallellines,complementaryinterioranglesonthesame

side.

Datacollection:

Thefirststep:clearsurveyquestions.

Thesecondstep:determinetheobjectofinvestigation.

Thethirdstep:surveymethod.

Thefourthstep:investigation.

Thefifthstep:recordtheresults.

Stepsixth:conclusion.

Frequency:thenumberoftimeseachobjectofthesaid.

Frequency:ratioofeachobjectandthenumberofthetotal

number(orpercentage)

Thefrequencyandfrequencycanreflectthedegreeofeach

objectappearsfrequently.

Frequency=absolutedata.

Frequency=relativedata.

Solvingalinearequation:

Bothsidesofanequationplusorminusthesamenumberorthe

sametype,invariantsolutionsofequations.

Bothsidesofanequationorbymultipliedbyanon-zeronumber,

invariantsolutionsofequations.

Theequationforsomeitemsinchangingthesign,fromoneside

oftheequationtotheothersideofthedeformationiscalled

transposition.

Containsonlyoneunknown,andcontainingunknownformulais

thenumberofunknownsis1type,andthisequationiscalled

aYuanlinearequation.

Twoyuanaequationanditssolution:

Eachequationwithtwounknowns,andthenumberofunknownitems

are1,likethisequation,wecallittwoyuanaequation.The

twotwoyuanaequationtogethertoformatwoyuanaequation

group.

Twounknownsmaketwoequationstwoyuanaequationgroupin

theleftandrightsidesofthevalueequaltothevalueoftwo

yuan,iscalledasolutionofequations.

Twoyuanforasolutionofequations;

Throughthe“substitution"eliminateanunknownquantity,the

equationsaretransformedintoalinearequationsolution,this

methodiscalledsubstitutionmethod,referredtoasthe

substitutionmethod.

Thetwoequations(additionorsubtraction)eliminatean

unknownquantity,theequationsaretransformedintoalinear

equationsolution,thismethodiscalledandelimination,

referredtoastheadditionandsubtraction.

Understandinginequality:

WiththesignVorformula,calledinequality.

Theinequalityholdsthevalueoftheunknown,called

inequalities.

Allsolutionsofasetofsolutionscomposedofinequality,the

inequality,referredtoasthesolutionsetofthisinequality.

Thenatureofinequality1ifa>b,then:a+C+Ca->b,

C>b

Onbothsidesoftheinequalityplus(orless)withanumber

orawhole,samedirectionofaninequality.

Thepropertiesofinequality2ifa>b,C>0andAC>be.,

then:

Thepropertiesofinequality3ifa>b,

Andthen:ACC<0,<be.

Onbothsidesoftheinequality(multipliedbyordividedby

thesameconstant)positivedirectionofaninequality;

inequalityonbothsides(ordivide)multipliedbythesame

negative,changethedirectionofaninequality.

Asystemoflinearinequalities:

Thetwooneinequalityandtogether,togetasystemoflinear

inequalities.

Triangle：

Thetriangleisthreenotinthegraphiconthesameline

segmentsareserialconnected,thethreelineisthesideof

atriangle.

Thetriangleanglesofthesideandtheothersideofthereverse

extensionwhichiscalledtrianglecornerangle.

Apieceoftriangleisequaltothetwoanglesanditisnot

adjacentto.

Apieceoftriangleisgreaterthananyofitsneighborsand

notinside

Takeasumfromthetwocornerswitheachangleadjacent

respectively,andarecalledtrianglelinesandcorners.

Andwiththecornersandanglesofapolygon:

Ifthepolygonedgesareequal,theanglesareequal,thenit

iscalledpositiveformultilateral.

Thediagonal1ineconnectingtwoverticesnotadjacentpolygons

calledpolygon.

Inthenedgeshapeand180degrees(n-2).

Axialsymmetryinlife;

Ifalinealongthefold,twofoldpartiscompletelycoincident,

wecallthispatternastheaxisofsymmetry.Thislineiscalled

theaxisofsymmetryofthegraph.

Agraphalongastraightlinefoldinthepast,ifitcanbe

withanothergraphicsoverlap,thenwesaythatthetwofigure

intotheaxialsymmetry,thislineistheaxisofsymmetry,the

correspondingpointsoftwopatterns(i.e.twographics

coincideeachothercoincidentpoint)iscalledsymmetric

point.

Axisofsymmetry(oronacertainlineoftwosymmetrical

graphics)alongtheaxisofsymmetrytwopartafterfoldingis

completelycoincident,soitscorrespondingsegments(1ine

overlapfolded)equalsthecorrespondingangle(angleoverlap

folded).

Understandingofaxialsymmetry:

Theverticalstraightlinesegmentsandsplitalinecalledthe

lineperpendicularbisector.

Theverticalbisectorofthelinepointtoadistanceequalto

thelineoftwopoints.

Ifagraphistheaxisofsymmetry,thenthelineconnecting

thesymmetrypointoftheperpendicularbisectoroftheline

graphistheaxisofsymmetry.

Ifthegraphiscomposedofstraightlinesegments,orrays,

thendrawitona1ineofsymmetry,justdrawspecialpoints

inthegraph(suchastheendpointofthesegment,vertexangle

etc.)symmetric,thenlinkthesymmetricalpoints,candrawon

thesymmetricfigurestraightline.

Anisoscelestriangle：

Isoscelestriangle,equalonbothsidesofthewaistandthe

othersideiscalled,calledthebase,anangleoftwowaist

iscalledvertexangle,waistandbottomedgecornercalledthe

hungry.

Twoareequalisoscelestriangle,(abbreviatedas''equivalence

equilateralangle")

Thehighoverlapbetweentheisoscelestriangleanglebisector,

themedianandthehem,referredtoasthe“threelinesone”.

Thevariousanglesoftheequilateraltriangleareequal,and

everyoneisequalto60degrees.

Ifatrianglehastwoequal,thenthetwocornersoftheedge

areequal,(abbreviatedas"theequiangularequilateral").

Maydetermine:

Ineveryexperimentwilleventastheinevitableeventinevery

experimentmustnotoccurastheimpossibleevent.Thesetwo

eventsarehappeningintheexperimentwhetherwecandetermine

inadvance,socalledforthedeterminationofevent.

Unabletodetermineinadvancewillhappeninoneexperiment,

wecallthemuncertaineventsorrandomevents.

Gradeeight:

Squareandcuberoots:

Ifanumberisequaltothesquareofa,thenthisnumberis

calledthesquarerootofA.

Ifthereisapositivesquareroot,thentheremustbetwo,they

aretheoppositenumber.Ifweknowthatthetwosquareroot

ofone,thenimmediatelygetanothersquarerootofit.

Positiveisthesquarerootofa,calledthearithmeticsquare

root,denotedasa,pronounced"Reagana”;

Anothersquarerootisitsoppositenumber,namely.The

positivesquarerootofacanbedenotedasa+,calledthe

radicand.

Becausethesquarenumber0isequalto0.Andanyothernumber

ofsquareisnotequalto0,sothesquarerootof0isonly

one,thatis0.Isusuallydenotedas=0.

Thesquarerootofanonnegativenumberofoperations,called

square.

Ifanumberofcubicequaltoa,thenthisnumberiscalledthe

cuberootofa

Anynumberof(positive,negativeorzero)ofthecuberootif

itexists,itisonlyone.

Thenumberofa*,written,read“threetimesthesquareroot

ofa”.Acalledtheradicand,3indexfinger.Findanumberof

cuberootoperations,calledsubtriplicate.

Withtherealaxis:

Infinitedecimalscalledirrationalnumber.Referredtoas

rationalnumbersandirrationalnumbersarereal.

Thepowerofoperation:

Am*an=am+n(m,nisapositiveinteger).

Thesamebasepowermultiplication,thebaseindexsum

unchanged.

(AM)=amn(m,nisapositiveinteger).

Thepowerofthepowerbase,constant,exponential

multiplication.

(AB)n=anbn(nispositiveinteger).

Theproductofpower,equaltoeveryfactorofproductarethen

obtainedbymultiplyingpower,power.

Am/an=am-n

Thesamebasepowersdividingthebaseunchanged,subtraction

index.

Typeofmultiplication:

Polynomialandpolynomialmultiplication,firstwitha

polynomialofeacharemultipliedbyanotherpolynomialofeach,

thenthesummationoftheproduct.

Multiplicationformula:

(a+b)(a-b=A2)-B3(squaredifferenceformula)

Thenumberandthisdifferenceisequaltotheproductoftwo

numbers,twothenumberofsquaredifference.

(a+b)2=A2+2Ab+B2or(a-b)2=a2-2Ab+B2

Thenumberandthesquare,equalto2timesthenumberoftheir

productandaddthesquare.

Typeofdivision:

Thefirstpolynomialdividedbyamonomialandpolynomialof

eachdividedbythemonomial,thenputthesumofbusiness

income.

Factorization:

Apolynomialoftheformofsometypeofproduct,calledthe

factorizationofpolynomials.

EverypolynomialMa+MB+MCinallcontainthesamefactor

M,wecallthecommonfactor.Bringforwardthecommonfactor,

polynomialMa+MB+MCcanbedecomposedintotwofactors(M

anda+B+C)oftheproduct.Suchmethodscalledextracting

commonfactormethod.

Factorizationofpolynomialfactorization,thismethodis

calledformulamethod.

ThePythagoreantheorem:

ThePythagoreantheoremofrighttrianglesquaretworight

anglesideandisequaltothesquareofthehypotenuse.

Ifthethreesideoftrianglea,B,C:a2+b2=C2,thenthe

triangleisarighttriangle.

Translation:

Thecorrespondingsegmenttranslationafterthegraphicswith

theoriginalgraphicsparallelandequal,thecorresponding

anglesareequal,theshapeandsizeofthegraphicsarenot

changed.

Inthetranslationprocess,thecorrespondingsegmentpointmay

eveninastraightline.

Rotation:

Theplanewillbeadirectiontorotateacertainangle,this

figureiscalledtherotatingmotion.

Thegraphicsineachpointaroundthecenterofrotationinthe

samerotatingdirectionoftherotatingangleofthesamesize,

correspondingtothesamedistancefromthecenterofrotation,

thecorrespondinganglesareequal,theshapeandsizeofthe

graphicsarenotchanged.

Afigurearoundafixedpointalongadirectionofrotationand

itscanoverlap,suchagraphiscalledrotationalsymmetric

graph.

Centerofsymmetry:

Afigurearoundthecenterpointandcanrotate180degrees

aftertheircoincidence,wecallthisafigureofcentral

symmetry,thecenterpointiscalledthecenterofsymmetry.

Inthetwographicsymmetricalin1inelinksymmetrypoint

throughthecenterofsymmetry,andisacenterofsymmetry

split.

Ifthelinesegmentcorrespondingpointsofthetwographsare

connectedthroughapoint,andthepointsaresplit,thenthe

twographicsmustonthispointsymmetrical.

Congruenceoffigures:

Cantwographicsoverlapcompletelycongruentfigures(called

congruentpolygons).

Twocongruentpolygons,thetransformationandcoincidence,

Overlapbetweenverticescalledcorrespondingvertices,

overlappingedgecalledthecorrespondingedgeoverlapping

angle,calledthecorrespondingangle.

Thecorrespondingcongruentpolygonedges,correspondingangle

respectively,(edgeandanglerespectivelycorrespondingto

theequaltwocongruentpolygons.)

Thecorrespondingcongruenttriangleedges,corresponding

anglerespectively,(twotriangles,respectively

correspondingtotheequal,thenthe22congruenttriangles.

Thenatureoftheparallelogram;

Therearetwogroupsoffoursidesparallellinecalledthe

parallelogram.

Twogroupsofparallelsides,isoneofthemainpropertiesof

theparallelogram.

Theoppositesidesoftheparallelogramareequal,diagonalare

equal.

Thediagonaloftheparallelogrambisecteachother.

Thedistancebetweentheparallel1inesequaleverywhere.

Propertiesofrectangularanddiamondandsquare:

Thefourangleisrightanglerectangular.

Therectanglediagonalareequalandequallywitheachother.

Diamondfoursidesareequal.

Thediamonddiagonalperpendiculartoeachotherequally,and

eachgroupofdiagonaldiagonalsplit.

Square;foursidesareequal,thefouranglesarequadrilateral

atrightangles.

Agroupofadjacentsidesareequal,aparallelogramangleis

rightangle.

Agroupofadjacentedgesofequalrectangular.

Arightangleisdiamond.

Hypotenuseisequaltothemedianhypoteneusehalf.

Anisoscelestrapezoidwithtwoequalanglesonthebottom.

Thetwodiagonalsequalisoscelestrapezoid.

Thefractionanditsbasicproperties:

Like(A,BandBintype,containingtheletters,B=0)formula,

calledfractional.TheAiscalledthefractionofmolecules,

thedenominatoriscalledthefractionofB.

Integralandfractionalcollect