數(shù)學(xué)專業(yè)英語(yǔ)(吳炯圻)

1、-New Words & Expressions:algebra代數(shù)學(xué)geometrical幾何的algebraic代數(shù)的identity 恒等式arithmetic算術(shù) , 算術(shù)的measure測(cè)量，測(cè)度axiom公理numerical數(shù)值的 ,數(shù)字的conception概念，觀點(diǎn)operation運(yùn)算constant常數(shù)postulate公設(shè)logical deduction邏輯推理proposition命題division除，除法subtraction減 ，減法formula公式term項(xiàng)，術(shù)語(yǔ)trigonometry三角學(xué)variable變化的，變量2.1數(shù)學(xué)、方程與比例Mat

2、hematics, Equation and Ratio4Mathematicscomes fromman s socialpractice,forexample,industrialandagriculturalproduction,commercialactivities,militaryoperationsandscientific and technological researches.1 AWhat is mathematics數(shù)學(xué)來(lái)源于人類的社會(huì)實(shí)踐，比如工農(nóng)業(yè)生產(chǎn)，商業(yè)活動(dòng)，軍事行動(dòng)和科學(xué)技術(shù)研究。Andin turn,mathematicsservesthepracticea

3、ndplaysa greatrolein all fields.Nomodernscientificandtechnologicalbranchescouldberegularlydeveloped without the application of mathematics.反過(guò)來(lái)，數(shù)學(xué)服務(wù)于實(shí)踐，并在各個(gè)領(lǐng)域中起著非常重要的作用。 沒(méi)有應(yīng)用數(shù)學(xué)，任何一個(gè)現(xiàn)在的科技的分支都不能正常發(fā)展。5-Fromtheearly needof man cametheconceptsofnumbersandforms.Then, geometrydevelopedoutof problemsofmeasur

4、ingland,andtrigonometrycame from problems of surveying. To deal with somemorecomplexpracticalproblems,manestablishedandthensolvedequationwithunknownnumbers,thus algebra occurred.很早的時(shí)候，人類的需要產(chǎn)生了數(shù)和形的概念。接著，測(cè)量土地問(wèn)題形成了幾何學(xué)，測(cè)量問(wèn)題產(chǎn)生了三角學(xué)。為了處理更復(fù)雜的實(shí)際問(wèn)題，人類建立和解決了帶未知數(shù)的方程，從而產(chǎn)生了代數(shù)學(xué)。Before17thcentury,manconfinedhimself

5、totheelementarymathematics,i.e.,geometry,trigonometryandalgebra,inwhichonlytheconstants are considered.17 世紀(jì)前，人類局限于只考慮常數(shù)的初等數(shù)學(xué)，即幾何學(xué)，三角學(xué)和代數(shù)學(xué)。6The rapiddevelopmentof industryin17thcenturypromoted the progress of economics and technologyandrequireddealingwith variablequantities.Theleapfromconstantstovar

6、iablequantitiesbroughtabouttwonewbranchesof mathematics-analyticgeometryandcalculus,which belongto thehighermathematics.17 世紀(jì)工業(yè)的快速發(fā)展推動(dòng)了經(jīng)濟(jì)技術(shù)的進(jìn)步，從而遇到需要處理變量的問(wèn)題。從常量到變量的跳躍產(chǎn)生了兩個(gè)新的數(shù)學(xué)分支 - 解析幾何和微積分，他們都屬于高等數(shù)學(xué)。Now there are many branches in higher mathematics,amongwhicharemathematicalanalysis,higheralgebra,dif

7、ferentialequations,functiontheoryand-so on.現(xiàn)在高等數(shù)學(xué)里面有很多分支， 其中有數(shù)學(xué)分析， 高等代數(shù)，微分方程，函數(shù)論等。7Mathematicians study conceptions and propositions,Axioms,postulates,definitionsandtheoremsareallpropositions.Notationsareaspecialandpowerfultoolofmathematicsandareusedtoexpressconceptions and propositions very often.數(shù)

8、學(xué)家研究的是概念和命題，公理，公設(shè)，定義和定理都是命題。符號(hào)是數(shù)學(xué)中一個(gè)特殊而有用的工具，常用于表達(dá)概念和命題。Formulas,figures andcharts arefullofdifferentsymbols.Some of the best knownsymbolsofmathematicsaretheArabicnumerals1,2,3,4,5,6,7,8,9,0andthesignsofaddition“+”,subtraction-”“ , multiplication“×”division ,“÷”and equality“ =”.公式，圖形和圖表都是不

9、同的符號(hào) .8The conclusions in mathematics are obtained mainlyby logicaldeductionsandcomputation.For alongperiod ofthehistoryofmathematics,thecentricplace of mathematics methods was occupied by the logical deductions.數(shù)學(xué)結(jié)論主要由邏輯推理和計(jì)算得到。在數(shù)學(xué)發(fā)展歷史的很長(zhǎng)時(shí)間內(nèi)，邏輯推理一直占據(jù)著數(shù)學(xué)方法的中心地位。Now,sinceelectroniccomputersaredevelop

10、edpromptlyandusedwidely,theroleofcomputationbecomesmoreandmoreimportant.In ourtimes,-computationis notonlyusedtodealwitha lotofinformation and data, but also to carry out some workthatmerelycouldbedoneearlierbylogicaldeductions,forexample,theproofofmostofgeometrical theorems.現(xiàn)在，由于電子計(jì)算機(jī)的迅速發(fā)展和廣泛使用，計(jì)算機(jī)

11、的地位越來(lái)越重要。現(xiàn)在計(jì)算機(jī)不僅用于處理大量的信息和數(shù)據(jù)，還可以完成一些之前只能由邏輯推理來(lái)做的工作，例如，證明大多數(shù)的幾何定理。9回顧：1. 如果沒(méi)有運(yùn)用數(shù)學(xué)， 任何一個(gè)科學(xué)技術(shù)分支都不可能正常的發(fā)展 。2. 符號(hào)在數(shù)學(xué)中起著非常重要的作用， 它常用于表示概念和命題。1 AWhat is mathematics10Anequationis a statementof the equalitybetweentwo equal numbers ornumber symbols.1 BEquation等式是關(guān)于兩個(gè)數(shù)或者數(shù)的符號(hào)相等的一種描述。Equation are of two kinds-

12、identities and equationsof condition.An arithmetic or an algebraic identity is an equation.In such an equation either the two members are alike,orbecomealikeon theperformanceof theindicated operation.等式有兩種恒等式和條件等式。算術(shù)或者代數(shù)恒等式都是-等式。這種等式的兩端要么一樣，要么經(jīng)過(guò)執(zhí)行指定的運(yùn)算后變成一樣。11Anidentityinvolvinglettersis trueforanys

13、etofnumerical values of the letters in it.含有字母的恒等式對(duì)其中字母的任一組數(shù)值都成立。An equation which is true only for certain values of a letter in it, or for certain sets of related values of two ormore of its letters, is an equation of condition, or simply an equation. Thus 3x-5=7 is true for x=4 only; and 2x-y=10

14、is true for x=6 and y=2 and for many other pairs of values for x and y.一個(gè)等式若僅僅對(duì)其中一個(gè)字母的某些值成立，或?qū)ζ渲袃蓚€(gè)或者多個(gè)字母的若干組相關(guān)的值成立，則它是一個(gè)條件等式，簡(jiǎn)稱方程。因此 3x-5=7 僅當(dāng) x=4 時(shí)成立，而 2x-y=0 ，當(dāng) x=6,y=2 時(shí)成立，且對(duì) x, y 的其他許多對(duì)值也成立。12Arootofanequationis anynumberornumbersymbol which satisfies the equation.To obtaintherootor rootsofanequ

15、ationis calledsolving an equation.方程的根是滿足方程的任意數(shù)或者數(shù)的符號(hào)。求方程根的過(guò)程被稱為解方程。Therearevariouskindsof equations.They arelinearequation, quadratic equation, etc.方程有很多種，例如：線性方程，二次方程等。13To solvean equationmeansto findthevalueof theunknownterm.Todothis,wemust,ofcourse,changethetermsaboutuntiltheunknownterm-stands

16、alone on one side of the equation, thus making it equal to something on the other side. We then obtain the value of the unknown and the answer to the question.解方程意味著求未知項(xiàng)的值，為了求未知項(xiàng)的值，當(dāng)然必須移項(xiàng)，直到未知項(xiàng)單獨(dú)在方程的一邊，令其等于方程的另一邊，從而求得未知項(xiàng)的值，解決了問(wèn)題。To solve the equation, therefore, means to move andchange the terms abo

17、ut without making the equation untrue, until only the unknown quantity is left on one side ,no matter which side.因此解方程意味著進(jìn)行一系列的移項(xiàng)和同解變形，直到未知量被單獨(dú)留在方程的一邊，無(wú)論那一邊。14Equations are of very great use. We can use equations in many mathematical problems. We maynoticethatalmosteveryproblemgivesus oneormore stat

18、ements that something is equal to something, this gives us equations, with which we may work if we need to.方程作用很大，可以用方程解決很多數(shù)學(xué)問(wèn)題。注意到幾乎每一個(gè)問(wèn)題都給出一個(gè)或多個(gè)關(guān)于一個(gè)事情與另一個(gè)事情相等的陳述，這就給出了方程，利用該方程，如果我們需要的話，可以解方程。New Words & Expressions:numerical數(shù)值的，數(shù)的position位置，狀態(tài)cuben. 立方體spheren.球cylinder n.柱體cone圓錐geometrical幾何

19、的triangle三角形-surface面， 曲面pyramid菱形plane 平面solid立體，立體的straight line直線line segment直線段broken line折線ray 射線equidistant等距離的curve曲線，彎曲2.2 幾何與三角Geometry and Trigonology1New Words & Expressions:side 邊angle角radius （ radii ） 半徑diameter 直徑endpoint端點(diǎn)circle圓周，圓semicircle半圓arc弧minor arc劣弧major arc優(yōu)弧acute angle

20、銳角right angle直角hypotenuse斜邊adjacent side鄰邊chord弦circumference周長(zhǎng)2Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics.2 AWhy study geometry?許多居于領(lǐng)導(dǎo)地位的學(xué)術(shù)機(jī)構(gòu)承認(rèn)，所有學(xué)習(xí)這個(gè)數(shù)學(xué)分支的人都將得到確實(shí)的受益。This is evidentfromthefactthatthey

21、requirestudyofgeometryasaprerequisitetomatriculationinthose schools.許多學(xué)校把幾何的學(xué)習(xí)作為入學(xué)考試的先決條件，從這一點(diǎn)上可以證明。3Geometry had its origin long ago in the measurement-by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River.幾何學(xué)起源于很久以前巴比倫人和埃及人測(cè)量他們被尼羅河洪水淹沒(méi)的土地。Thegreekwordgeometryisd

22、erivedfromgeo,meaning“ earth”andmetron,meaning“ measure ” .希臘語(yǔ)幾何來(lái)源于 geo ，意思是”土地“，和 metron 意思是”測(cè)量“。4As early as 2000 B.C. we find the land surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry .公元前 2000 年之前，我們發(fā)現(xiàn)這些民族的土地測(cè)量者利用幾何知識(shí)重新確定消失了的土地標(biāo)志和邊界

23、。One of the most important objectives derived from astudy of geometryismakingthestudentbemorecriticalinhislistening,readingandthinking.Instudyinggeometryheis ledawayfrom thepracticeof blind acceptance of statements and ideas and istaughttothinkclearlyandcriticallybeforeformingconclusions.幾何的學(xué)習(xí)使學(xué)生在思考

24、問(wèn)題時(shí)更周密、審慎，他們將不會(huì)盲目接受任何結(jié)論 .5Asolidisathree-dimensionalfigure.Commonexamplesofsolids arecube,sphere,cylinder,coneand pyramid.2 BSome geometrical terms立體是一個(gè)三維圖形，立體常見(jiàn)的例子是立方體，球體，柱體，圓錐和棱錐。-A cube has six faces which are smooth and flat. These faces are called plane surfaces or simply planes.立方體有 6 個(gè)面，都是光滑的

25、和平的，這些面被稱為平面曲面或者簡(jiǎn)稱為平面。6A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop isan example of a plane surface.平面曲面是二維的，有長(zhǎng)度和寬度，黑板和桌子上面的面都是平面曲面的例子。A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point called th

26、e center.平面上的閉曲線當(dāng)其中每點(diǎn)到一個(gè)固定點(diǎn)的距離均相當(dāng)時(shí)叫做圓。固定點(diǎn)稱為圓心。7A line segment drawn from the center of the circle toa point on the circle is a radius of the circle. The circumference is the length of a circle.經(jīng)過(guò)圓心且其兩個(gè)端點(diǎn)在圓周上的線段稱為這個(gè)園的直徑，這條曲線的長(zhǎng)度叫做周長(zhǎng)。Oneofthemostimportantapplicationsoftrigonometry is the solution of tr

27、iangles. Let us now take up the solution to right triangles.三角形最重要的應(yīng)用之一是解三角形，現(xiàn)在我們來(lái)解直角三角形。8A triangle is composed of six parts three sides and three angles. To solve a triangle is to find the parts notgiven.一個(gè)三角形由6 個(gè)部分組成，三條邊和三只角。解一個(gè)三角-形就是要求出未知的部分。A triangle may be solved if three parts (at least one

28、oftheseis a side) are given.Arighttrianglehasoneangle,therightangle,alwaysgiven.Thus arighttrianglecanbesolvedwhentwosides,oronesideand an acute angle, are given.如果三角形的三個(gè)部分（其中至少有一個(gè)為邊）為已知，則此三角形就可以解出。直角三角形的一只角，即直角，總是已知的。因此，如果它的兩邊，或一邊和一銳角為已知，則此直角三角形可解。New Words & Expressions:brace大括號(hào)roster名冊(cè)consequ

29、ence結(jié)論，推論roster notation枚舉法designate標(biāo)記，指定rule out排除，否決diagram圖形，圖解subset子集distinct互不相同的theunderlyingset基礎(chǔ)集distinguish 區(qū)別，辨別 divisible 可被除盡的 dummy 啞的，啞變量even integer 偶數(shù) irrelevant 無(wú)關(guān)緊要的空集universal set全集validity有效性visual可視的visualize 可視化 void set(empty set)2.3 集合論的基本概念Basic Concepts of the Theory of Se

30、ts1The concept of a set has been utilized so extensivelythroughoutmodernmathematicsthatanunderstanding of it is necessary for all college students. Sets are a means by which mathematicians-talk of collections of things in an abstract way.3 ANotations for denoting sets集合論的概念已經(jīng)被廣泛使用，遍及現(xiàn)代數(shù)學(xué)，因此對(duì)大學(xué)生來(lái)說(shuō)，理解

31、它的概念是必要的。集合是數(shù)學(xué)家們用抽象的方式來(lái)表述一些事物的集體的工具。Sets usuallyaredenotedby capitalletters;elementsare designated by lower-case letters.集合通常用大寫字母表示，元素用小寫字母表示。2We use the special notationto meanthat“ x is an element of S” or“ x belongs to S” . If x donot belong to S, we write.我們用專用記號(hào)來(lái)表示x 是 S 的元素或者x 屬于 S。如果 x不屬于 S，我

32、們記為。When convenient,weshall designatesetsbydisplayingtheelementsinbraces;for example,theset of positiveevenintegersless than10is displayedas 2,4,6,8whereastheset ofallpositiveevenintegers is displayed as 2,4,6, the three dots takingthe place of“ and so on.”如果方便，我們可以用在大括號(hào)中列出元素的方式來(lái)表示集合。例如，小于 10 的正偶數(shù)的集

33、合表示為2,4,6,8 ，而所有正偶數(shù)的集合表示為 2,4,6, ,三個(gè)圓點(diǎn)表示“等等”。3The dots are used only when the meaning of“ and soon ” is clear. The method of listing the members of aset within braces is sometimes referred to as the rosternotation.只有當(dāng)省略的內(nèi)容清楚時(shí)才能使用圓點(diǎn)。在大括號(hào)中列出集合元素的方法有時(shí)被歸結(jié)為枚舉法。The first basic concept that relates one set

34、 to another-is equality of sets:聯(lián)系一個(gè)集合與另一個(gè)集合的第一個(gè)基本概念是集合相等。4DEFINITION OF SET EQUALITY Two sets A and B are said to be equal (or identical) if they consist of exactly thesame elements, in which case we write A=B. If one ofthe sets contains an element not in the other, we saythe sets unequal and we wr

35、ite A B.集合相等的定義如果兩個(gè)集合 A 和 B 確切包含同樣的元素 ,則稱二者相等，此時(shí)記為A=B 。如果一個(gè)集合包含了另一個(gè)集合以外的元素，則稱二者不等，記為A B。5EXAMPLE 1. Accordingto thisdefinition,the twosets2,4,6,8 and2,8,6,4are equalsincetheybothconsist of the four integers 2,4,6 and 8. Thus, when weuse the roster notation to describe a set, the order inwhich the el

36、ements appear is irrelevant.根據(jù)這個(gè)定義，兩個(gè)集合2,4,6,8和2,8,6,4是相等的，因?yàn)樗麄兌及怂膫€(gè)整數(shù)2,4,6,8 。因此， 當(dāng)我們用枚舉法來(lái)描述集合的時(shí)候，元素出現(xiàn)的次序是無(wú)關(guān)緊要的。6EXAMPLE 2. The sets 2,4,6,8 and 2,2,4,4,6,8 are equal even though, in the second set, each of the elements 2 and 4 is listed twice. Both sets contain thefour elements 2,4,6,8 and no oth

37、ers; therefore, the definition requires that we call these sets equal.例 2. 集合 2,4,6,8 和2,2,4,4,6,8 也是相等的， 雖然在第二個(gè)集合中， 2 和 4 都出現(xiàn)兩次。兩個(gè)集合都包含了四個(gè)元素2,4,6,8 ，沒(méi)有其他元素，因此，依據(jù)定義這兩個(gè)集合相等。 This example shows that we do not insist that theobjectslistedintherosternotationbedistinct.A-similarexampleis thesetoflettersi

38、nthewordMississippi, which is equal to the set M,i,s,p, consisting of the four distinct letters M,i,s, and p.這個(gè)例子表明我們沒(méi)有強(qiáng)調(diào)在枚舉法中所列出的元素要互不相同。一個(gè)相似的例子是，在單詞 Mississippi 中字母的集合等價(jià)于集合 M,i,s,p, 其中包含了四個(gè)互不相同的字母 M,i,s, 和 p.7FromagivensetS wemayformnew sets,calledsubsetsofS. For example,thesetconsistingofthosepos

39、itiveintegersless than10whicharedivisibleby 4(the set 4,8) is a subset of the set of all even integerslessthan10.Ingeneral,we havethefollowingdefinition.3 BSubsets一個(gè)給定的集合 S 可以產(chǎn)生新的集合，這些集合叫做 S 的子集。例如，由可被 4 除盡的并且小于 10 的正整數(shù)所組成的集合是小于 10 的所有偶數(shù)所組成集合的子集。一般來(lái)說(shuō)，我們有如下定義。8In all our applicationsof set theory,weh

40、avea fixedset S given in advance, and we are concerned onlywith subsets of this given set. The underlying set S mayvaryfromoneapplicationtoanother;itwillbereferred to as the universal set of each particular discourse. （35 頁(yè)第二段）當(dāng)我們應(yīng)用集合論時(shí)，總是事先給定一個(gè)固定的集合S，而我們只關(guān)心這個(gè)給定集合的子集。基礎(chǔ)集可以隨意改變，可以在每一段特定的論述中表示全集。9It i

41、s possible for a set to contain no elements whatever.-This set is called the empty set or the void set, and willbedenotedbythesymbol. Wewillconsiderto be a subset of every set.（ 35 頁(yè)第三段）一個(gè)集合中不包含任何元素，這種情況是有可能的。這個(gè)集合被叫做空集，用符號(hào)表示。空集是任何集合的子集。Some peoplefindit helpful to thinkofaset asanalogoustoa containe

42、r(suchas abagora box)containingcertainobjects,itselements.Theemptyset is then analogous to an empty container.一些人認(rèn)為這樣的比喻是有益的，集合類似于容器（如背包和盒子）裝有某些東西那樣，包含它的元素。10To avoidlogicaldifficulties,wemustdistinguishbetweenthe elementsx andthe set x whose onlyelementis x. Inparticular, theemptysetis notthe same

43、as the set. （35頁(yè)第四段）為了避免遇到邏輯困難，我們必須區(qū)分元素x 和集合 x ，集合 x 中的元素是 x。特別要注意的是空集和集合是不同的。In fact, the empty set contains no elements, whereasthe sethas one element. Sets consisting ofexactlyoneelementaresometimescalledone-element sets.事實(shí)上，空集不含有任何元素，而有一個(gè)元素。由一個(gè)元素構(gòu)成的集合有時(shí)被稱為單元素集。11Diagrams often help us visualize

44、relations between sets. For example, we may think of a set S as a regionin the plane and each of its elements as a point. Subsets of S may then be thought of the collections ofpoints within S. For example, in Figure 2-3-1 the shaded portion is a subset of A and also a subset of B.-（ 35 頁(yè)第五段）圖解有助于我們將

45、集合之間的關(guān)系形象化。例如，可以把集合 S 看作平面內(nèi)的一個(gè)區(qū)域， 其中的每一個(gè)元素即是一個(gè)點(diǎn)。那么 S 的子集就是 S 內(nèi)某些點(diǎn)的全體。例如，在圖2-3-1 中陰影部分是 A 的子集，同時(shí)也是 B 的子集。12Visual aids of this type, called Venn diagrams, are useful for testing the validity of theorems in set theoryor for suggesting methods to prove them. Of course, the proofs themselves must rely o

46、nly on the definitions of the concepts and not on the diagrams.這種圖解方法，叫做文氏圖，在集合論中常用于檢驗(yàn)定理的有效性或者為證明定理提供一些潛在的方法。當(dāng)然證明本身必須依賴于概念的定義而不是圖解。New Words & Expressions:conversely反之geometricinterpretation幾何意義correspond對(duì)應(yīng)induction歸納法deducible可推導(dǎo)的proofbyinduction歸納證明difference差inductiveset歸納集distinguished著名的ine

47、quality不等式entirely complete完整的integer 整數(shù)Euclid 歐幾里得interchangeably可互相交換的Euclidean歐式的intuitive直觀的the field axiom域公理irrational無(wú)理的2.4 整數(shù)、有理數(shù)與實(shí)數(shù)-Integers, Rational Numbers and Real Numbers 1New Words & Expressions:irrational number無(wú)理數(shù)rational 有理的the order axiom序公理rational number有理數(shù)ordered有序的reasonin

48、g推理product積scale尺度，刻度quotient商sum 和2There exist certain subsets of R which are distinguishedbecause theyhave special propertiesnot shared byall real numbers. Inthis section we shall discuss suchsubsets, the integers and the rational numbers.4 AIntegers and rational numbers有一些 R 的子集很著名， 因?yàn)樗麄兙哂袑?shí)數(shù)所不具備的特殊性質(zhì)。在本節(jié)我們將討論這樣的子集，整數(shù)集和有理數(shù)集。3To introducethe p