數(shù)學(xué)專業(yè)英語課后答案

1、.2.1 數(shù)學(xué)、方程與比例詞組翻譯1. 數(shù)學(xué)分支 branches of mathematics，算數(shù) arithmetics，幾何學(xué) geometry，代數(shù)學(xué)algebra，三角學(xué) trigonometry，高等數(shù)學(xué) higher mathematics，初等數(shù)學(xué) elementary mathematics，高等代數(shù) higher algebra，數(shù)學(xué)分析 mathematical analysis，函數(shù)論 function theory，微分方程 differential equation2.命題 proposition，公理 axiom，公設(shè) postulate，定義 definiti

2、on ，定理 theorem，引理 lemma，推論 deduction3. 形 form ，數(shù) number，數(shù)字 numeral，數(shù)值 numerical value，圖形 figure，公式formula，符號 notation(symbol)，記法 / 記號 sign，圖表 chart4. 概念 conception，相等 equality，成立 / 真 true，不成立 / 不真 untrue，等式equation，恒等式 identity ，條件等式 equation of condition，項 / 術(shù)語 term，集 set，函數(shù) function ，常數(shù) constant，方

3、程 equation，線性方程 linear equation，二次方程 quadratic equation5. 運算 operation，加法 addition，減法 subtraction，乘法 multiplication ，除法division ，證明 proof，推理 deduction，邏輯推理 logical deduction6. 測量土地 to measure land，推導(dǎo)定理 to deduce theorems，指定的運算 indicated operation，獲得結(jié)論 to obtain the conclusions，占據(jù)中心地位 to occupy the c

4、entric place漢譯英（ 1）數(shù)學(xué)來源于人類的社會實踐，包括工農(nóng)業(yè)的勞動，商業(yè)、軍事和科學(xué)技術(shù)研究等活動。Mathematics comes from man s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.（ 2）如果沒有運用數(shù)學(xué)，任何一個科學(xué)技術(shù)分支都不可能正常地發(fā)展。No modern scientific and

5、 technological branches could be regularly developed without the application of mathematics.（ 3）符號在數(shù)學(xué)中起著非常重要的作用，它常用于表示概念和命題。Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.（ 4） 17 世紀(jì)之前，人們局限于初等數(shù)學(xué)，即幾何、三角和代數(shù)，那時只考慮常數(shù)。Before 17th ce

6、ntury, man confined himself to the elementary mathematics, i. e. ,geometry, trigonometry and algebra, in which only the constants were considered.（ 5）方程與算數(shù)的等式不同在于它含有可以參加運算的未知量。Equation is different from arithmetic identity in that it contains unknown quantity which can join operations.（ 6）方程又稱為條件等式，

7、因為其中的未知量通常只允許取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it.（ 7）方程很有用，可以用它來解決許多實際應(yīng)用問題。.Equations are of very great use. We can use equations in many mathematical problems. （ 8）解方程時要進(jìn)行一系列移項和同解變形，最后求出它的根，即未知量的值。To solve the e

8、quation means to move and change the terms about without making the equation untrue, until the root of the equation is obtained, which is the value of unknown term.英譯漢1.Algebra has evolved from the operations and rules of arithmetic.The study of arithmetic begins with addition,multiplication,subtrac

9、tion,and division of numbers:4+7,37 682,49×-22,40÷8.In algebra we introduce symbols or letterssuch as a,b,c,d,x,y,z to denotearbitrary numbers and，instead of special cases， we often consider general statements：a+b,cd,x-y,x÷a.代數(shù)是從算術(shù)的運算和規(guī)則當(dāng)中逐漸發(fā)展起來的，算術(shù)的研究是從數(shù)的加減乘除開始的。例如 4+7，37×682，49

10、-22， 40÷8。在代數(shù)學(xué)里 ,我們采用符號或字母。例如a,b,c,d,x,y,z來表示任意的數(shù)字，而不考慮那些特殊情況。我們經(jīng)?？紤]的是一般的表達(dá)式，例如a+b,cd,x-y,x÷a。2.The language of algebra serves a twofold purpose.First,we may use it as a shorthand to abbreviate and simplify long or complicated statements.Second,it proves a convenient means of generalizing

11、many specific statements.代數(shù)的語言有兩個作用。第一個是使用它作為一種速記法去縮減和減化那些又長又復(fù)雜的表達(dá)。第二，它被證明是一種概括許多具體的表達(dá)方式的便捷途徑。3.Many expressions involve two or more operations.Grouping symbols tell us which operation is to be done first.The common grouping symbols are parentheses,(),brackets.,and the fraction bar, .For example,in

12、the expression2(3+4),we do the addition first and then we do the multiplication:2(3+4)=2(7)=14. 許多數(shù)學(xué)表達(dá)式包含兩個或更多的運算。分組符號告訴我們哪一個運算先做。常見的分組符號是圓括號，方括號和分?jǐn)?shù)線。例如，在數(shù)學(xué)表達(dá)公式 2（3+4）里。我們先做加法再做乘法 2（3+4） =2（7）=142.2 幾何與三角詞組翻譯1.學(xué)會 institution ，建筑師 architect, 機械師 machinist, 制圖員 draftsman, 測量者 surveyor, 木匠 carpenter2.點

13、 point, 端點 endpoint, 線 line, 直線 straight line, 線段 line segment, 曲線 curved line, 折線 broken line, 射線 ray , 平面 plane,曲面 curved surface3.立體 solid, 柱體 cylinder, 立方體 cube,球 sphere, 棱錐 pyramid,圓錐 cone ,4.圓 circle,圓心 center, 直徑 diameter, 半徑 radius, 半圓 semicircle, 弦 chord, 弧 arc, 優(yōu)弧 major arc, 劣弧 minor arc5.

14、角 angle, 邊 side, 三角形 triangle, 直角三角形 right triangle,斜邊 hypotenuse, 直角邊 right-angle side6.長度 length,寬度 breadth/width,厚度 thickness, 位置 position.7.幾何的 geometrical,立體的 three-dimensional , 彎曲的 curved,等距離的 equidistant ,無限的 infinite8.培養(yǎng)創(chuàng)造力 train originality, 必須的毅力 necessary perseverance , 提高鑒賞力 raise/impro

15、ve the appreciation ability9.消失了的邊界 vanishing boundaries/landmarks,有序性和優(yōu)美感 orderliness and sense of beauty, 幾何圖形大量存在 geometric forms abound in , 定理成立的先決條件 a prerequisite to a theorem漢譯英（ 1）許多專家都認(rèn)為數(shù)學(xué)是學(xué)習(xí)其他科學(xué)技術(shù)的必備基礎(chǔ)和先決條件。Many experts recognize that mathematics is the necessary foundation and prerequisi

16、te of studying other science technology.（ 2）西方國家的專家認(rèn)為幾何起源于巴比倫和埃及人的土地測量技術(shù)，其實中國古代的數(shù)學(xué) 家對幾何做了許多出色的研究。The western experts think that geometry had its origin in the measurements by the Babylonians and Egyptians of their lands. Infect, the ancient Chinese mathematicians made much remarkable study for geome

17、try.（ 3）幾何的學(xué)習(xí)使學(xué)生在思考問題時更周密和審慎，他們將不會盲目接受任何結(jié)論。In studying geometry, the student is taught to think clearly and critically and he is led away from the practice of blind acceptance of any conclusions.（ 4）數(shù)學(xué)培養(yǎng)學(xué)生的分析問題的能力，使他們能應(yīng)用毅力、創(chuàng)造性和邏輯推理來解決問題。Studying mathematics can develop the students abilityzeproblems

18、toanalyand utilizing perseverance, originality, and logical reasoning in solving the problem.（ 5）幾何主要不是研究數(shù)，而是形，例如三角形，平行四邊形和圓，雖然它也與數(shù)有關(guān)。Geometry mainly studies hot numbers but figures such as triangles, parallelograms and circles, though it is related with numbers.（ 6）一個立體（圖形）有長、寬和高；面（曲面或平面）有長和寬，但沒有厚度

19、；線（直線 或曲線）有長度，但既沒有寬度，也沒有厚度；點只有位置，卻沒有大小。A solid (figure) has length, width and height. A surface (curved surface or plane surface) has length and width, but no thickness. A line (straight line or curved line) has length, butno width and thickness. A point has position, but no dimension.（ 7）射線從某個點出發(fā)無限

20、延伸；兩條從同一點出發(fā)的射線構(gòu)成了角。這兩條射線稱為這個 角的兩邊，當(dāng)這兩邊位于同一直線上且方向相反時，所得的角是平角。A ray starts from a point and extends infinitely far. Two rays starting from one point form an angle, which are called two edges of the angle. When two edges lie in the same line and have opposite direction named plane angle.（ 8）平面上的閉曲線當(dāng)其中每

21、一點到一個固定點的距離均相等時叫做圓。這個固定點稱為圓 心，經(jīng)過圓心且其兩個端點在圓周上的線段稱為這個圓的直徑，直徑的一半叫做半徑，這條 曲線的長度叫做周長。A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point. The fixed point called the center. A diameter of a circle is a line segment through the center of the circle with endp

22、oints on the circle. Half of the diameter is called radius. The length of the circle is called circumference.英譯漢 1.In geometry an angle is defined as the set of points determined by two rays l1 and l 2 having the same endpoint O. 在幾何學(xué)里從同一點 O 出發(fā)引出的兩條射線 l 1 和 l 2 所組成的點的集合叫做角。2.In trigonometry we often

23、 interpret angles as rotations of rays.To obtain an angle we may start with a fixed ray l 1 having endpoint O,and rotate it about O,in a plane,to a position specified by ray l 2 .We call l 1 the initial side, l 2 the terminal side,and O thevertex of angle. 在三角學(xué)里，我們經(jīng)常解釋角就是射線的旋轉(zhuǎn)。在平面上，我們許會從端點是 O 的射線 l

24、1 開始讓它繞著端點 O 旋轉(zhuǎn)，轉(zhuǎn)到一個位置，由射線 l 2 標(biāo)注。我們把 l 1 叫做角的始邊， l 2 叫做角的終邊， O 叫做角的頂點。3.A right angle is a 90 angle . An angleis acute if 0 <<90 or obtuse if90 < <180 .A straight angle is a 180 angle .Two acute angles are complementary if their sum is 90 .Two positive angles are supplementary if their

25、sum is 180. 直角就是一個 90 的角。如果 0 < <90 把它叫做銳角，如果 90 < <180 叫做鈍角。平角就是一個 180 的角。如果兩個銳角的和是 90 ，那么這兩個角互為余角。如果兩個正角的和是 180 ，那么這兩個角是互為補角。2.3 集合論的基本概念單詞、詞組1.1 集 set，子集 subset，真子集 proper subset，全集 universal subset，空集 void/ empty set，基地集 the underlying set1.2正數(shù) positive number，偶數(shù) even integer，圖形 diag

26、ram，文氏圖 Venn diagram，啞標(biāo) dummy index，大括號 brace1.3可以被整除的 be divisible by，兩兩不同的 distinct from each other，確定的definite，無關(guān)緊要的 irrelevant/inessential1.4一樣的結(jié)論 the same conclusion，等同的效果 equivalent effect，用大括號表示集sets are designated by braces，把這個圖形記作 A ： this diagram is designated byletter A，區(qū)別對象 to distinguis

27、h between objects，證明定理 to prove theorems，把結(jié)論可視化 to visualize conclusions/consequences漢譯英（ 1）由小于10 且能被3 整除的正整數(shù)組成的集是整數(shù)集的子集。The set consisting of those positive integers less than 10 which are divisible by 3 is a subset of the set of all integers.（ 2）如果方便，我們通過在括號中列舉元素的辦法來表示集。.When convenient, we shall

28、designate sets by displaying the elements in braces.（ 3）用符號表示集的包含關(guān)系，也就是說，式子AB 表示 A 包含于B。The relationis referred to as set inclusion; AB means that A is contained in B.（ 4）命題AB 并不排除BA 的可能性。The statement AB does not rule out the possibility that BA.（ 5）基礎(chǔ)集可根據(jù)使用場合不同而改變。The underlying set may vary from

29、one application to another according to using occasions.（ 6）為了避免邏輯上的困難，我們必須把元素 x 與僅含有元素 x 的集 x 區(qū)別開來。To avoid logical difficulties, we must distinguish between the element x and the set x whose only element is x.（ 7）圖解法有助于將集合之間的關(guān)系形象化。Diagrams often help using visualize relationship between sets.（ 8）定

30、理的證明僅僅依賴于概念和已知的結(jié)論，而不依賴于圖形。The proofs of theorems rely only on the definitions of the concepts and known result, not on the diagrams.英譯漢1.If A is the set of all the letters of the alphabet，then listing each of elements would be tedious. So we write A=a,b,c, ,z.如果 A 是所有字母的集合，那么把每一個其中的字母列舉出來將是很冗長乏味的，因此

31、我們寫出 A= a，b，c， ,z。2.In the set A， the last element is z. Many sets do not have last elements . Two important sets are N , the set of natural numbers , and W , the set of whose numbers . To list all the elements in these sets would be impossible because they go on forever . So we use three dots and w

32、rite N=1,2,3, ,W=0,1,2,3, .在集合 A 里,最后一個元素是 z，許多集合沒有最后一個元素，兩個重要的集合是N，自然數(shù)集合，和 W，整數(shù)的集合。把這兩個集合里所有的元素列舉出來是不可能的，因為它們是永遠(yuǎn)持續(xù)下去的，所以我們用三個點來表示，集合N 寫成N=1,2,3, ，集合 W 寫成 W=0,1,2,3 ， 。3.The whole numbers have many important subsets . A whole number is said to be even if it is divisible by 2 ；2,6,and 18 are examples

33、 of even numbers. A whole number is said to be odd if it is not divisible by 2 ; 1,7,and 13 are examples of odd numbers . The natural numbers greater than 1 are called prime or composite , A number is prime if it is divisible only by 1 and itself , A number is composite if it is divisible by a natur

34、al number other than 1 and itself.整數(shù)有許多重要的子集。如果一個整數(shù)能被2 除開就是偶數(shù)； 2， 6， 18 就是偶數(shù)的例子。一個整數(shù)如果不能被2 整除就是奇數(shù)； 1，7， 13 就是奇數(shù)的例子。大于1的自然數(shù)叫做素數(shù)或者合數(shù)，如果一個自然數(shù)只能被1 和它本身整除，那么這個數(shù)就是素數(shù)（質(zhì)數(shù)），如果一個自然數(shù)除了能被1 和它本身整除外，還可以被其他的自然數(shù)整除，就叫做合數(shù)。.2.4 整數(shù)、有理數(shù)與實數(shù)1.單詞1）Integer rational number irrational number， real number，negative number， the

35、negative，real line， real axis，scale，to the left/right of 2）sum， difference，product， quotient，power，inequality3）axiom，the field axiom，the order axiom4）ordered， entirely complete，Euclidean， appropriate，distinguished，illuminating5）can be deduced formula，formula interchangeably，using a set of formulas，

36、corresponding to an object， proof by induction， the two set to be distinguished2、漢譯英（ 1）嚴(yán)格說，這樣描述整數(shù)是不完整的，因為我們并沒有說明 “依此類推 ”或“反復(fù)加 1”的含義是什么。Strictly speaking, this description of the positive integers is not entirely complete becausewe have not explained in detail what we mean by the expressions“ and so

37、 on ” , or“ repeated addition of 1” .（ 2）兩個整數(shù)的和、差或積是一個整數(shù)，但是兩個整數(shù)的商未必是一個整數(shù)。The sum, difference, or product of two integers is an integer, but the quotient of two integers need not be an integer.（ 3） 這種用幾何來表示實數(shù)的辦法對于幫助我們更好地發(fā)現(xiàn)與理解實數(shù)的性質(zhì)是非常有價值 的。This device for representing real numbers geometrically is a v

38、ery worthwhile aid that helps us to discover and understand better certain properties of real numbers.（ 4）幾何經(jīng)常為一些特定的定理提供證明思路（建議） ，而且，有時幾何的論證比純分析的 （完全依賴于實數(shù)公理的）證明更清晰。The geometry often suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a

39、purely analytic proof (one depending entirely on the axioms for the real numbers).（ 5）一個由實數(shù)組成的集若滿足如下條件則稱為開區(qū)間（open interval） 。If a set consisting of real numbers satisfies the following conditions we call it an open interval.（ 6）實數(shù)a 是-a 的相反數(shù)，它們的絕對值相等，且當(dāng)a 0時，其符號不同。The real number a is the negative num

40、ber ofa and their absolute values are equal. When a 0, their notations are different.（ 7）每個實數(shù)剛好對應(yīng)著實軸上的一點，反之，對實軸上的每一點，有且只有一個實數(shù)與之 對應(yīng)。Each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number.（ 8）在幾何上，實數(shù)之間的次序關(guān)系可以在數(shù)

41、軸上清楚地表示出來。In geometry, the ordering relation among the real numbers can be expressed clearly in real axis.3. 英譯漢.1）一個常見的錯誤是認(rèn)為 x 是一個負(fù)數(shù)。但 x 可以為正數(shù)、 0 或負(fù)數(shù)，這取決于 x 的值2）我們在最后一節(jié)中介紹的每個屬性是唯一的操作，例如，ab =ba和 0 + a= a，我們現(xiàn)在考慮一個連接加法和乘法的屬性。它被稱為分配律或者乘法對加法的分配，用下面的公式表所示： a（ b + c) = ab + ac；（ b + c) =ba + ca。3）考慮小數(shù)按自然數(shù)

42、順序?qū)懗鰹椋?123456789101112131415,由于自然數(shù)數(shù)字不會中止或重復(fù)，這是一個無限不循環(huán)小數(shù)，不能轉(zhuǎn)換為兩個整數(shù)的比的小數(shù)叫做無理數(shù)，這組數(shù)字是指符號 H，H=×：x 是無限不循環(huán)小數(shù) 2.5 笛卡兒幾何學(xué)的基本概念1. 翻譯單詞、詞組、短語(1)解析幾何 analytic geometry笛卡兒幾何 Cartesian geometry三維的 three-dimensional, 坐標(biāo) coordinate,坐標(biāo)系 coordinate system, 坐標(biāo)原點 the origin, 橫坐標(biāo) abscissa,坐標(biāo)軸 coordinate axis,縱坐標(biāo) or

43、dinate，象限 quadrant,有序?qū)?ordered pair, 尺度 scale,單位長度 the unit distance(2)向量 vector,線段 line segment,垂直的 perpendicular,水平的 horizontal,豎直的 vertical, 相交 intersect，交點 a point(3)三角形 triangle, 直角三角形 right triangle, 斜邊 hypotenuse, 直角邊 leg,區(qū)域 area/region, 多邊形的 polygonal, 多邊形區(qū)域 polygonal region,拋物線的 parabolic,

44、拋物線弓形 parabolic segment circular, 圓的 circular，圓域 circular region(4)積分的計算 integralcalculation,整數(shù)的性質(zhì) integral quality, 微積分的基本定理 basic theorem of calculusp(5)對符號做適當(dāng)認(rèn)定an appropriate regardfor signs,把一個問題轉(zhuǎn)化為另一個問題to reduce a question to another question,把條件翻譯成表達(dá)式to translate these conditionsintoexpression

45、s ，緊密融合在一起inntimately intertwined ，刻畫了該曲線的特征to characterize the curve in question2.漢譯英（ 1）計算圖形的面積是積分的一種重要應(yīng)用。The calculation of figure area is the important application of the integral.（ 2）在 x-軸上 O 點右邊選定一個適當(dāng)?shù)狞c，并把它到 O 點的距離稱為單位長度。On the x-axis a convenient point is chosen to the right of O and its dist

46、ance from O is called the unit distance.（ 3）對 xy-平面上的每一個點都指定了一個數(shù)對，稱為它的坐標(biāo)。Each point in the xy-plane is assigned a pair of numbers, called its coordinates.（ 4）選取兩條互相垂直的直線，其中一條是水平的，另一條是豎立的，把它們的交點記作 O， 稱為原點。Two perpendicular reference lines are chosen, one horizontal, the other vertical. Their point of

47、 intersection, denoted by O, is called the origin.（ 5）當(dāng)我們用一對數(shù)（ a, b）來表示平面的點時，商定要把橫坐標(biāo)寫在第一個位置上。When we write a pair of numbers such as (a, b) to represent a point, we agree that the abscissa or x-coordinate, a, is written first.（ 6）微積分與解析幾何在它們的發(fā)展史上已經(jīng)互相融合在一起了。Throughout their historical development, ca

48、lculus and analytic geometry have been intimately intertwined.（ 7）如果想拓展微積分的范圍與應(yīng)用，需要進(jìn)一步研究解析幾何，而這種研究需用到向量的 方法。A deeper study of analytic geometry is needed to extend the scope and applications of calculus, and this study will be carried out using vector methods.（ 8）今后我們要對三維解析幾何做詳細(xì)研究，但目前只限于考慮平面解析幾何。We

49、shall discuss three-dimensional Cartesian geometry in more detail later on; for the present we confine our attention to plane analytic geometry.2.6 函數(shù)的概念與函數(shù)思想1.1）function ，domain，range，the identity function ，the absolute-value function， the real-valued2）cube，volume，edge-length， prime，totality3)Hooke

50、's law，stretch，displacement，spring， constant，proportional4）schematic representation， plot，image，output，input5）it is not difficult to imagine ，the idea was much too limited2.漢譯英（ 1）常用英語字母和希臘字母來表示函數(shù)。Letters of the English and Greek alphabets are often used to denote functions.（ 2）若 f 是一個給定的函數(shù)， x 是

51、定義域里的一個元素，那么記號 f(x) 用來表示由 f 確定的 對應(yīng)于 x 的值。If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function f.（ 3）該射線將兩個坐標(biāo)軸的夾角分成兩個相等的角。The ray makes equal angles with the coordinates axes.（ 4）可以用許多方式給

52、出函數(shù)思想的圖解說明。The function idea may be illustrated schematically in many ways.（ 5）容易證明，絕對值函數(shù)滿足三角不等式。It is easy to proof that the absolute-value function satisfies the triangle inequality.（ 6）對于實數(shù)x>0，函數(shù) g(x)表示不超過x 的素數(shù)的個數(shù)。For a given real number x>0, the function g(x) is defined by the number of pr

53、imes less than or equal to x.（ 7）函數(shù)是一種對應(yīng)，它未必可以表示成一個簡單的代數(shù)公式。A function is a correspondence. It is not necessary to be expressed by a simple algebraic formula.（ 8）在函數(shù)的定義中，關(guān)于定義域和值域中的對象，沒對其性質(zhì)做出任何限制。.The function idea places no restriction on the nature of the objects in the domain X and in the range Y.2

54、.7 序列及其極限序列及其極限（ 1）序列各項對n 的相關(guān)性常利用下標(biāo)來表示，寫成如下形式：a n , x n 等。The dependence of every team of sequence on n is denoted by using subscript, and wewrite a n , x n and so on.（ 2）以正整數(shù)集為定義域的函數(shù)稱為序列。A function whose domain is the set of all positive integers is called an infinite sequence.（ 3）一個復(fù)值序列收斂當(dāng)且僅當(dāng)它的實部和

55、虛部分別收斂。A complex-valued sequence converges if and only if both the real part and the imaginarypart converge separately.（ 4） 一個序列 a n 若滿足：對任意正數(shù)， 存在另一個正數(shù)N （N 可能與有關(guān)）使得 a n - L <對所有n N成立，就稱 a n 收斂于 L 。A sequence a n is said to have a limit L if, for every positive numberhere is , tanother positive nu

56、mber N (which may depend on ) such that In this case, we say thesequence a n converges to L. an ? L < for all n N.（ 5） 重要的是， 該集的每一個成員都用一個正整數(shù)標(biāo)上記號。 這樣一來， 就可以談?wù)摰谝豁棥?第二項和一般項，即第 n 項。The important thing is that each member of the set has been labeled with an integer so that we may speak of the first te