英语数学.docx

英语数学.docx

《英语数学.docx》由会员分享，可在线阅读，更多相关《英语数学.docx（15页珍藏版）》请在冰豆网上搜索。

英语数学

Definitions

Article1.InAlgebra,quantitiesarerepresentedbylettersofthealphabet.

2.Quantityisanythingthatiscapableofincreaseordecrease;as,numbers,lines,space,time,etc.

3.Quantityiscalledmagnitude,whenconsideredinanundividedform;as,aquantityofwater.

4.Quantityiscalledmultitude,whenmadeupofindividualanddistinctparts;as,threecents,aquantitycomposedofthreesinglecents.

5.Oneofthesinglepartsofwhichaquantityofmultitudeiscomposed,iscalledtheunitofmeasure;thus,1centistheunitofmeasureofthequantity3cents.

Thevalueormeasureofanyquantityisthenumberoftimesitcontainsitsunitofmeasure.

6.Inquantitiesofmagnitude,wherethereisnonaturalunit,itisnecessarytofixuponanartificialunitasastandardofmeasure;then,tofindthevalueofthequantity,weascertainhowmanytimesitcontainsitsunitofmeasure.Thus,

Tomeasurethelengthofaline,takeacertainassumeddistancecalledafoot,and,applyingitacertainnumberoftimes,say5,itisfoundthatthelineis5feetlong;inthiscase,1footistheunitofmeasure.

7.TheNumericalValueofaquantityisthenumberthatshowshowmanytimesitcontainsitsunitofmeasure.

Thus,thenumericalvalueofaline5feetlong;is5.Thesamequantitymayhavedifferentnumericalvalues,accordingtotheunitofmeasureassumed.

8.AUnitisasinglethingofanorderorkind.

9.Numberisanexpressiondenotingaunit,oracollectionofunits.Numbersareeitherabstractorconcrete.

10.AnAbstractNumberdenoteshowmanytimesaunitistobetaken.

AConcreteNumberdenotestheunitsthataretaken.

Thus,4isanabstractnumber,denotingmerelythenumberofunitstaken;while4feetisaconcretenumber,denotingwhatunitistaken,aswellasthenumbertaken.

Or,aconcretenumberistheproductoftheunitofmeasurebythecorrespondingabstractnumber.Thus,$6equal$1multipliedby6,or$1taken6times.

11.Inalgebraiccomputations,lettersareconsideredtherepresentativesofnumbers.

12.TherearetwokindsofquestionsinAlgebra,theoremsandproblems.

13.InaTheorem,itisrequiredtodemonstratesomerelationorpropertyofnumbers,orabstractquantities.

14.InaProblem,itisrequiredtofindthevalueofsomeunknownquantity,bymeansofcertaingivenrelationsexistingbetweenitandothers,whichareknown.

15.Algebraisageneralmethodofsolvingproblemsanddemonstratingtheorems,bymeansoffigures,letters,andsigns.Thelettersandsignsarecalledsymbols.

ExplanationofSignsandTerms

16.KnownQuantitiesarethosewhosevaluesaregiven;UnknownQuantities,thosewhosevaluesaretobedetermined.

17.Knownquantitiesaregenerallyrepresentedbythefirstlettersofalphabet,asa,b,c,etc.;unknownquantities,bythelastletters,asx,y,z.

18.TheprincipalsignsusedinAlgebraare

＝,＋,－,×,÷,（）,＞,√.

Eachsignistherepresentativeofcertainwords.Theyareusedtoexpressthevariousoperationsintheclearestandbriefestmanner.

19.TheSignofEquality,＝,isreadequalto.Itdenotesthatthequantitiesbetweenwhichitisplacedareequal.Thus,a＝3,denotesthatthequantityrepresentedbyaisequalto3.

20.TheSignofAddition,＋,isreadplus.Itdenotesthatthequantitytowhichitisprefixedistobeadded.

Thus,a＋b,denotesthatbistobeaddedtoa.ifa＝2andb＝3,thena＋b＝2＋3,which＝5.

21.TheSignofSubtraction,－,isreadminus.Itdenotesthatthequantitytowhichitisprefixedistobesubtracted.

Thus,a－b,denotesthatbistobesubtractedfroma.Ifa＝5andb＝3,thena－b＝5－3,which＝2.

22.Thesigns＋and－arecalledthesigns.Theformeriscalledthepositive,thelatterthenegativesign;theyaresaidtobecontraryoropposite.

23.Everyquantityissupposedtobeprecededbyoneofthesesigns.Quantitieshavingthepositivesignarecalledthepositive;thosehavingthenegativesign,negative.

Whenaquantityhasnosignprefixed,itispositive.

24.Quantitieshavingthesamesignaresaidtohavelikesigns;thosehavingdifferentsigns,unlikesigns.

Thus,＋aand＋b,or－aand－b,havelikesigns;while＋cand－dhaveunlikesigns.

25.TheSignofMultiplication,×,isreadinto,ormultipliedby.Itdenotesthatthequantitiesbetweenwhichitisplacedaretobemultipliedtogether.

Theproductoftwoormorelettersinsometimesexpressedbyadotorpoint,butmorefrequentlybywritingtheminclosesuccessionwithoutanysign.Thus,abexpressesthesameasa×bora·b,andabc＝a×b×c,ora·b·c.

26.Factorsarequantitiesthataremultipliedtogether.

Thecontinuedproductofseveralfactorsmeanstheproductofthefirstandsecondmultipliedbythethird,thisproductbythefourth,andsoon.

Thus,thecontinuedproductofa,b,andc,isa×b×c,orabc.Ifa＝2,b＝3,andc＝5,thenabc＝2×3×5＝30.

27.TheSignofDivision,÷,isreaddividedby.Itdenotesthatthequantityprecedingitistobedividedbythatfollowingit.Divisionisoftenerrepresentedbyplacingthedividendasthenumerator,andthedivisorasthedenominatorofafraction.

Thus,a÷b,or

means,thataistobedividedbyb.Ifa＝12andb＝3,thena÷b＝12÷3＝4;or

.

28.TheSignofInequality,＞,denotesthatoneofthetwoquantitiesbetweenwhichitisplacedisgreaterthantheother.Theopeningofthesignistowardthegreaterquantity.

Thus,a＞b,denotesthataisgreaterthanb.Itisread,agreaterthanb.Ifa＝5andb＝3,then5＞3.Also,c＜d,denotesthatcislessthand.Itisread,clessthand.Ifc＝4andd＝7,then4＜7.

29.TheSignofInfinity,∞,denotesaquantitygreaterthananythatcanbeassigned,oroneindefinitelygreat.

30.TheNumeralCoefficientofaquantityisanumberprefixedtoit,showinghowmanytimesthequantityistaken.

Thus,a＋a＋a＋a＝4a;andax＋ax＋ax＝3ax.

31.TheLiteralCoefficientofaquantityisaquantitybywhichitismultiplied.Thus,inthequantityay,amaybeconsideredthecoefficientofy,orythecoefficientofa.

Theliteralcoefficientisgenerallyregardedasaknownquantity.

32.Thecoefficientofaquantitymayconsistofanumberandaliteralpart.Thus,in5ax,5amayberegardedasthecoefficientofx.Ifa＝2,then5a＝10,and5ax＝10x.

Whennonumeralcoefficientisprefixedtoaquantity,itscoefficientisunderstoodtobeunity.Thus,a＝1a,andbx＝1bx.

33.ThePowerofaquantityistheproductarisingfrommultiplyingthequantitybyitselfoneormoretimes.

Whenthequantityistakentwiceasafactor,theproductiscalleditssquare,orsecondpower;whenthreetimes,thecube,orthirdpower;whenfourtimes,thefourthpower,andsoon.

Thus,a×a＝aa,isthesecondpowerofa;a×a×a＝aaa,isthethirdpowerofa;a×a×a×a＝aaaa,isthefourthpowerofa.

AnExponentisafigureplacedattheright,andalittleaboveaquantity,toshowhowmanytimesitistakenasafactor.

Thus,

;

;

;

.

Whennoexponentisexpressed,itisunderstoodtobeunity.Thus,aisthesameasa1,eachexpressingthefirstpowerofa.

34.Toraiseaquantitytoanygivenpoweristofindthatpowerofthequantity.

35.TheRootofaquantityisanotherquantity,somepowerofwhichequalsthegivenquantity.Therootiscalledthesquareroot,cuberoot,fourthroot,etc.,accordingtothenumberoftimesitistakenasafactortoproducethegivenquantity.

Thus,aisthesecondorsquarerootof

since

.So,

isthethirdorcuberootof

since

.

36.Toextractanyrootofaquantityistofindthatroot.

37.TheRadicalSign,

placedbeforeaquantity,indicatesthatitsrootistobeextracted.

Thus,

or

denotesthesquarerootofa;

denotesthecuberootofa;

denotesthefourthrootofa.

38.Thenumberplacedovertheradicalsigniscalledindexoftheroot.Thus,2istheindexofthesquareroot,3ofthecuberoot,4ofthefourthroot,andsoon.Whentheradicalhasnoindexoverit,2isunderstood.

39.Everyquantityorcombinationofquantitiesexpressedbymeansofsymbols,iscalledanalgebraicexpression.

Thus,3aisthealgebraicexpressionfor3timesthequantitya;3a－4b,for3timesa,diminishedby4timesb;

fortwicethesquareofa,increasedby3timestheproductofaandb.

40.AMonomial,orTerm,isanalgebraicexpression,notunitedtoanyotherbythesign＋or－.

Amonomialissometimescalledasimplequantity.Thus,a,3a,

aremonomials,orsimplequantities.

41.APolynomialisanalgebraicexpression,composedoftwoormoreterms.

Thus,c＋2d-bisapolynomial.

42.ABinomialisapolynomialcomposedoftwoterms.

Thus,a＋b,a－b,and

arebinomials.

AResidualQuantityisabinomial,inwhichthesecondtermisnegative,asa－b.

43.ATrinomialisapolynomialconsistingofthreeterms.Thus,a＋b＋c,anda－b－c,aretrinomials.

44.TheNumericalValueofanalgebraicexpressionisthenumberobtained,bygivingparticularvaluestotheletters,andthenperformingtheoperationsindicated.

Inthealgebraicexpression2a＋3b,ifa＝4,andb＝5,then2a＝8,and3b＝15,andthenumericalvalueis8＋15=23.

45.Thevalueofapolynomialisnotaffectedbychangingtheorderoftheterms,providedeachtermretainsitsrespectivesign.Thus,

.Thisisself-evident.

46.Eachoftheliteralfactorsofanysimplequantityortermiscalledadimensionofthatterm.Thedegreeofatermdependsonthenumberofitsliteralfactors.

Thus,

consistsoftwoliteralfactors,a,andx,andisoftheseconddegree.Thequantity

containsthreeliteralfactors,a,a,andb,andisofthethirddegree.

contains5literalfactors,a,a,a,x,andx,andisofthefifthdegree;andsoon.

47.Apolynomialissaidtobehomogeneous,wheneach