中英文对照 数学常用短语与从句

数学常用短语与从句(Phrasesorclausesfrequentlyusedinmathematics)

常见的涉及运算的短语或句子

1.Differentiatebothsidesoftheequationandweget...

Integrate

bothsidesoftheequationandweget...2.Differentiatingbothsidesoftheequation,weget...Integratingbothsidesoftheequation,weget...3.

Add(A)to(B)andwehave…

(其中(A)、(B)为某些表达式，如不等式、等式、方程等，下同)4．Subtract

(B)

from

(A)

and

wehave…5.Multiplyingeachtermoftheequationby…,weobtain…6.Dividingtheequationthroughby...,wehave...7.

(A)and(B)

together

give

…

(A)and(B)

together

yield…

(A)and(B)

together

imply…

8.

Comparing(A)with(B),itiseasytoseethat…9.

Substituting(A)into(B),weobtain…10.Eliminating(theparameter)tfrom(A)and(B),wehave…11.Byintroducinganewvariable…,wecanthenrewrite(A)asfollows

Byintroducinganewvariable…,wecanthenrewrite(A)inthefollowingform12.Byasimplecalculation,weobtainfrom(A)…

定理证明过程中常见的短语和句子

1.下面的句型可用来表达“根据什么即可得到什么”的意思Accordingtodefinition,itfollows…

Accordingtohypothesis,itfollows…

Accordingtoassumptions,itfollows…

Accordingtotheorem(N),itfollows…

Accordingtolemma(A),itfollows…

Accordingtocorollary(B),itfollows…

Accordingtotheremark,itfollows…

Accordingtothefactthat…,itfollows…

(可以把上面的“accordingto”换成“by”)Since…,itfollows…2.如果一个论断可以通过一些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可以自行推算，因而只需直接写出论断来，这时可用下面句型：(1)

Itiseasytoseethat…

Itiseasytoshowthat…

Itiseasytoprovethat…

Itiseasytoverifythat…

Itiseasytocheckthat…(2)

Itcaneasilybeseenthat…Itcaneasilybeshownthat…Itcaneasilybeprovedthat…Itcaneasilybeverifiedthat…Itcaneasilybecheckedthat…

3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：

(1)

Itisclearthat…Itisobviousthat…Itisevidentthat…Itiswell-knownthat…(2)

Clearly,…

Obviously,…

Evidently,…

4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：

Letusfirstdefinethefunction…Letusintroduceanewfunction…

Letusconsiderthefunction…Letusfirst

investigatethefunction…Let…Set…Define…Put…Consider…

5.在一个定理中，有几个结论需要证明，其中有些结论比较明显，可不用证明，仅需证明余下结论即可，这时可用下面句型：

Since(A)and(B)areobvious,weneedonlyprove(C).Since(A)and(B)aretrivial,weneedonlyprove(C).

Since(A)and(B)aretrivial,itsufficestoprove(C)

6.为了证明一个定理，有时我们并不是直接去证明，而是证明一个新的论断，一旦新的论断得到证明，已给定理不难由此而得证，这时可用下面句型：

以下各句用于新的论断被证明之前：Thetheoremwillbeprovedifwecanshow…Theresultwillbeprovedifwecanshow…Thetheoremwillbeprovedbyshowingthat…Ifwecanprove…thenthetheoremfollowsimmediately.以下各句用于新的论断被证明之后：Thetheoremisnowadirectconsequenceofwhatwehaveproved.Thetheoremfollowsimmediatelyfromwhatwehaveproved.Thetheoremisnowevidentfromwhatwehaveproved.Itisevidenttoseethatthetheoremholds.

7.在证明过程中，有时要用到一些早已学过的知识或技巧，这时可用下面句子，以提醒读者：

Recallthat…Noticethat…Notethat…Observethat…Inordertoprovethetheorem,weneedtheknowledgeof…Inordertoobtainthefollowingequation,weneed…

8.如果需要证明的定理的假设条件是一般条件，但是，只要定理在特殊条件下成立，就不难推出定理在一般条件下也成立，这时仅需要在特殊情况下去证明定理就够了，为此可用下面句型：

Withoutlossofgenerality,wemayconsider…

Withoutlossofgenerality,wemayassume…

Withoutlossofgenerality,wemayprovethetheoreminthecase…

Itsufficestoprovethetheoreminthecase…

Weneedonlyconsiderthecase…

Forsimplicity,wemaytake…

9.如果待证的论断可用以前用过的相似的方法或步骤进行证明，则可用下面句型：

Thistheoremcanbeprovedinthesamewayasshownbefore.

Thisstatementcanbeprovedinasimilarwayasshownbefore.

Thistheoremcanbeprovedbythesamemethodasemployedinthelastsection.

Thistheoremcanbecompletedbythemethodanalogoustothatusedabove.

UsingthesameargumentasintheproofoftheoremN,wecaneasilycarryouttheproofofthistheorem.

WenowproceedasintheproofoftheoremN.

WeshalladoptthesameprocedureasintheproofoftheoremN.

10.如果我们用的是反证法，则其开头及结尾可用下面句型：

Ifthestatement(orassertion,conclusion)werefalse(ornottrue,notright)then…

Iftheassertionwouldnothold,then…

Thisiscontraryto…

Thiscontradictsthefactthat…

Thisleadstoacontradiction.

11.表示定理已证毕或者把前面所证的总结为一结论

Wehavethusprovedthetheorem.

Thiscompletestheproof.

Theproofofthetheoremisnowcompleted.

Itisnowobviousthatthetheoremholds.

Thuswehavederivedthat…

Consequently,weinferthat…

Thusweconcludethat…

Thusweareledtotheconclusionthat…