数学专业英语课后习题2.6
数学专业英语-21页精选文档
the positive integers = the set of all positive integers
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In everyday usage of the English language, the words “sequence” and “series” are synonyms, and they are used to suggest a succession of things or events arranged in some order. 在日常英语中，单词“sequence” 和 “series” 是同义 词，用以表示按某种次序排列的一串东西或事件。
A function f whose domain is the set of all positive integers 1, 2, 3, … is called an infinite sequence. The function value f(n) is called the nth term of the sequence. 以正整数集为定义域的函数称为序列。…
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A sequence { an } is said to have a limit L if, for every
positive number number N ( which may depend on e ) such that
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Very often the dependence on n is denoted by using subscripts, and we write an , xn . 序列各项对 n 的相关性常利用下标来表示，写成 an , xn 。
数学专业英语课文翻译(吴炯圻)第二章2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12
数学专业英语3—A符号指示集一组的概念如此广泛利用整个现代数学的认识是所需的所有大学生。
集是通过集合中一种抽象方式的东西的数学家谈的一种手段。
集，通常用大写字母：A、B、C、进程运行·、X、Y、Z ；由小写字母指定元素：a、 b 的c、进程运行·，若x、y z.我们用特殊符号x∈S 意味着x 是S 的一个元素或属于美国的x如果x 不属于S，我们写xS.≠当方便时，我们应指定集的元素显示在括号内；例如，由符号表示的积极甚至整数小于10 集{2,468} {2，4.6，进程运行·} 作为显示的所有积极甚至整数集，而三个点等的发生。
点的和等等的意思是清楚时，才使用。
上市的大括号内的一组成员方法有时称为名册符号。
涉及到另一组的第一次基本概念是平等的集。
DEFINITIONOFSETEQUALITY。
两组A 和B，据说是平等的（或相同的）如果它们包含完全相同的元素，在这种情况下，我们写A = B。
如果其中一套包含在另一个元素，我们说这些集是不平等，我们写A = B。
EXAMPLE1。
根据对这一定义，由于他们都是由构成的这四个整数2，4.6 和8 两套{2,468} 和{2,864} 一律平等。
因此，当我们用来描述一组的名册符号，元素的显示的顺序无关。
动作。
集{2,468} 和{2,2,4,4,6,8} 是平等的即使在第二组，每个元素2 和4 两次列出。
这两组包含的四个要素2,468 和无他人；因此，定义要求我们称之为这些集平等。
此示例显示了我们也不坚持名册符号中列出的对象是不同。
类似的例子是一组在密西西比州，其值等于{M、我、s、p} 一组单词中的字母，组成四个不同字母M、我、s 和体育3 —B子集S.从给定的集 S，我们可能会形成新集，称为.的子集例如，组成的那些正整数小于 10 整除 4 （集合{8 毫米}）的一组一般是的所有甚至小于 10.整数集的一个子集，我们有以下的定义。
数学专业英语课后答案
数学专业英语课后答案1、He kept walking up and down, which was a sure（）that he was very worried. [单选题] *A. sign(正确答案)B. characterC. natureD. end2、20．Jerry is hard-working. It’s not ______ that he can pass the exam easily. [单选题] * A．surpriseB．surprising (正确答案)C．surprisedD．surprises3、--What’s the weather like today?--It’s _______. [单选题] *A. rainB. windy(正确答案)C. sunD. wind4、Many of my classmates are working _______volunteers. [单选题] *A. as(正确答案)B. toC. atD. like5、Mary is interested ______ hiking. [单选题] *A. onB. byC. in(正确答案)D. at6、We need a _______ when we travel around a new place. [单选题] *A. guide(正确答案)B. touristC. painterD. teacher7、While they were in discussion, their manager came in by chance. [单选题] *A. 抓住时机C. 碰巧(正确答案)D. 及时8、Our campus is _____ big that we need a bike to make it. [单选题] *A. veryB. so(正确答案)C. suchD. much9、57．Next week will be Lisa's birthday. I will send her a birthday present ________ post. [单选题] *A．withB．forC．by(正确答案)D．in10、The manager demanded that all employees _____ on time. [单选题] *A. be(正确答案)B. areC. to be11、He has two sisters but I have not _____. [单选题] *A. noneB. someC. onesD. any(正确答案)12、______! It’s not the end of the world. Let’s try it again.（）[单选题] *A. Put upB. Set upC. Cheer up(正确答案)D. Pick up13、My brother usually _______ his room after school. But now he _______ soccer. [单选题] *A. cleans; playsB. cleaning; playingC. cleans; is playing(正确答案)D. cleans; is playing the14、The museum is _______ in the northeast of Changsha. [单选题] *A. sitB. located(正确答案)C. liesD. stand15、______ my great joy, I met an old friend I haven' t seen for years ______ my way ______ town. [单选题] *A. To, in, forB. To, on, to(正确答案)C. With, in, toD. For, in, for16、In the closet（）a pair of trousers his parents bought for his birthday. [单选题] *A. lyingB. lies(正确答案)c. lieD. is lain17、44．—Hi, Lucy. You ________ very beautiful in the new dress today.—Thank you very much. [单选题] *A．look(正确答案)B．watchC．look atD．see18、They all choose me ______ our class monitor.（）[单选题] *A. as(正确答案)B. inC. withD. on19、My mother’s birthday is coming. I want to buy a new shirt ______ her.（）[单选题] *A. atB. for(正确答案)C. toD. with20、--Could you please tell me _______ to get to the nearest supermarket?--Sorry, I am a stranger here. [单选题] *A. whatB. how(正确答案)C. whenD. why21、?I am good at schoolwork. I often help my classmates _______ English. [单选题] *A. atB. toC. inD. with(正确答案)22、He was very excited to read the news _____ Mo Yan had won the Nobel Prize for literature [单选题] *A. whichB. whatC. howD. that(正确答案)23、Becky is having a great time ______ her aunt in Shanghai. （）[单选题] *A. to visitB. visitedC. visitsD. visiting(正确答案)24、In the future, people ______ a new kind of clothes that will be warm when they are cold, and cool when they’re hot.（）[单选题] *A. wearB. woreC. are wearingD. will wear(正确答案)25、This message is _______. We are all _______ at it. [单选题] *A. surprising; surprisingB. surprised; surprisedC. surprising; surprised(正确答案)D. surprised; surprising26、Your homework must_______ tomorrow. [单选题] *A. hand inB. is handed inC. hands inD. be handed in(正确答案)27、Since the war their country has taken many important steps to improve its economic situation. [单选题] *A. 制定B. 提出C. 讨论D. 采取(正确答案)28、How _______ Grace grows! She’s almost as tall as her mother now. [单选题] *A. cuteB. strongC. fast(正确答案)D. clever29、_____ is not known yet. [单选题] *A. Although he is serious about itB. No matter how we will do the taskC. Whether we will go outing or not(正确答案)D. Unless they come to see us30、You cannot see the doctor _____ you have made an appointment with him. [单选题] *A. exceptB.evenC. howeverD.unless(正确答案)。
高等数学双语教材答案
高等数学双语教材答案Chapter 1: Limits and ContinuitySection 1: Introduction to Limits and ContinuityThe concept of limits and continuity is fundamental in higher mathematics. In this section, we will introduce the basic definitions and properties associated with limits and continuity.1.1 Definitions of LimitsIn order to understand limits, we need to define what it means for a function to approach a particular value. Let f(x) be a function defined on an open interval containing c, except possibly at c. We say that the limit of f(x) as x approaches c is L, denoted bylim (x→c) f(x) = L, if for every ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - c| < δ.1.2 Basic Limit LawsOnce we have a clear understanding of limits, we can explore some basic laws that govern their behavior. These laws include the sum law, constant multiple law, product law, quotient law, and the power law.1.3 ContinuityA function f(x) is said to be continuous at a point c if three conditions are met: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c). We can also discuss continuity on an interval or at infinity.Chapter 2: DifferentiationSection 1: Introduction to DifferentiationDifferentiation is an important concept in calculus that allows us to find the rate at which a function is changing at any given point. In this section, we will introduce the concept of differentiation and its applications.2.1 Derivative DefinitionThe derivative of a function f(x) at a point c is defined as the limit of the difference quotient as h approaches 0. Mathematically, this can be written as f'(c) = lim (h→0) [(f(c + h) - f(c))/h].2.2 Differentiation RulesThere are several rules that allow us to find the derivative of a function quickly. These rules include the constant rule, power rule, sum rule, difference rule, product rule, quotient rule, and chain rule.2.3 Applications of DifferentiationDifferentiation has many applications in various fields, such as physics, economics, and engineering. It can be used to find maximum and minimum values, determine rates of change, and solve optimization problems.Chapter 3: IntegrationSection 1: Introduction to IntegrationIntegration is the reverse process of differentiation. It enables us to find the area under a curve and solve various mathematical problems. In this section, we will introduce the concept of integration and its applications.3.1 Indefinite IntegralsThe indefinite integral of a function f(x) is the collection of all antiderivatives of f(x). It is denoted by ∫ f(x) dx and represents a family of functions rather than a single value.3.2 Integration TechniquesThere are various techniques for finding antiderivatives and evaluating definite integrals. These techniques include basic integration rules, substitution, integration by parts, and trigonometric substitution.3.3 Applications of IntegrationIntegration has numerous applications, such as finding the area between two curves, calculating the length of curves, determining volumes of solids, and solving differential equations.ConclusionIn conclusion, the study of high-level mathematics, particularly limits, continuity, differentiation, and integration, is crucial for a comprehensive understanding of advanced mathematical concepts. This article has provided a brief overview of these topics, highlighting their definitions, properties, and applications. By mastering these concepts, students can develop strong problem-solving skills and apply them in various academic and real-world scenarios.。
数学专业英语
第二章精读课文----入门必修2.1数学方程与比例（Mathematics，Equation and Ratio）一、词汇及短语：1. change the terms about 变形2. full of ：有许多的充满的例The streets are full of people as on a holiday(像假日一样，街上行人川流不息)3. in groups of ten…4. match something against sb. “匹配”例Long ago ,when people had to count many things ,they matched them against their fingers. 古时候，当人们必须数东西时，在那些东西和自己的手指之间配对。
5. grow out of 源于由…引起例Many close friendships grew out of common acquaintance6. arrive at 得出（到达抵达达到达成）例we both arrived at the same conclusion(我们俩个得出了相同的结论)7. stand for “表示，代表”8. in turn “反过来，依次”9. bring about 发生导致造成10. arise out of 引起起源于11. express by“用…表示”12. occur 发生，产生13. come from 来源于，起源于14. resulting method 推论法15. be equal to 等于的相等的例Twice two is equal to four(2乘以2等于4)16. no matter 无论不管17. mathematical analysis 数学分析18. differential equation 微分方程19. higher mathematics 高等数学higher algebra 高等代数20. equation of condition 条件等式二句型及典型翻译1.For a long period of the history of mathematics, the centric place of mathematical methods was occupied by the logical deductions“在数学史的很长的时期内，是逻辑推理一直占据数学方法的中心地位”2.An equation is a statement of the equality between two equal numbers or number symbols. equation ：“方程”“等式”等式是关于两个数或数的符号相等的一种陈述3.In such an equation either the two members are alike, or become alike on performance of the indicated operation. 这种等式的两端要么一样，要么经过执行指定的运算后变成一样。
人教A版高中数学必修第二册课后习题 第6章 平面向量及其应用 6.2.2 向量的减法运算
6.2.2 向量的减法运算课后训练巩固提升1.如图,D,E,F 分别是△ABC 的边AB,BC,CA 的中点,则AF ⃗⃗⃗⃗⃗ −DB⃗⃗⃗⃗⃗ 等于( )A .FD ⃗⃗⃗⃗⃗B .FC ⃗⃗⃗⃗ C .FE ⃗⃗⃗⃗D .DF⃗⃗⃗⃗⃗DB ⃗⃗⃗⃗⃗ =AD ⃗⃗⃗⃗⃗ ,即AF ⃗⃗⃗⃗⃗ −DB ⃗⃗⃗⃗⃗ =AF ⃗⃗⃗⃗⃗ −AD ⃗⃗⃗⃗⃗ =DF ⃗⃗⃗⃗⃗ .2.如图,已知六边形ABCDEF 是一个正六边形,点O 是它的中心,其中OA ⃗⃗⃗⃗⃗ =a,OB ⃗⃗⃗⃗⃗ =b,OC ⃗⃗⃗⃗⃗ =c,则EF ⃗⃗⃗⃗ 等于( )A.a+bB.b-aC.c-bD.b-c=CB ⃗⃗⃗⃗⃗ =OB ⃗⃗⃗⃗⃗ −OC ⃗⃗⃗⃗⃗ =b-c.3.(多选题)下列各式化简结果为PQ⃗⃗⃗⃗⃗ 的是( )A .QC ⃗⃗⃗⃗⃗ −QP ⃗⃗⃗⃗⃗ +CQ ⃗⃗⃗⃗⃗B .AB ⃗⃗⃗⃗⃗ +(PA ⃗⃗⃗⃗⃗ +BQ ⃗⃗⃗⃗⃗ ) C.(AB ⃗⃗⃗⃗⃗ +PC ⃗⃗⃗⃗ )+(BA ⃗⃗⃗⃗⃗ −QC ⃗⃗⃗⃗⃗ )D .PA ⃗⃗⃗⃗⃗ +AB ⃗⃗⃗⃗⃗ −BQ⃗⃗⃗⃗⃗项,QC ⃗⃗⃗⃗⃗ −QP ⃗⃗⃗⃗⃗ +CQ ⃗⃗⃗⃗⃗ =PC ⃗⃗⃗⃗ +CQ ⃗⃗⃗⃗⃗ =PQ ⃗⃗⃗⃗⃗ ;B 项,AB ⃗⃗⃗⃗⃗ +(PA ⃗⃗⃗⃗⃗ +BQ ⃗⃗⃗⃗⃗ )=PA ⃗⃗⃗⃗⃗ +AB ⃗⃗⃗⃗⃗ +BQ ⃗⃗⃗⃗⃗ =PB ⃗⃗⃗⃗⃗ +BQ ⃗⃗⃗⃗⃗ =PQ ⃗⃗⃗⃗⃗ ;C 项,(AB ⃗⃗⃗⃗⃗ +PC ⃗⃗⃗⃗ )+(BA ⃗⃗⃗⃗⃗ −QC ⃗⃗⃗⃗⃗ )=(AB ⃗⃗⃗⃗⃗ +BA ⃗⃗⃗⃗⃗ )+(PC ⃗⃗⃗⃗ −QC ⃗⃗⃗⃗⃗ )=0+PQ ⃗⃗⃗⃗⃗ =PQ ⃗⃗⃗⃗⃗ ;D 项,PA ⃗⃗⃗⃗⃗ +AB ⃗⃗⃗⃗⃗ −BQ ⃗⃗⃗⃗⃗ =PB ⃗⃗⃗⃗⃗ −BQ ⃗⃗⃗⃗⃗ ≠PQ ⃗⃗⃗⃗⃗ .故选ABC.4.已知OA ⃗⃗⃗⃗⃗ =a,OB ⃗⃗⃗⃗⃗ =b,|OA ⃗⃗⃗⃗⃗ |=5,|OB ⃗⃗⃗⃗⃗ |=12,∠AOB=90°,则|a-b|等于( ) A.7B.17C.13D.8|a-b|=|OA ⃗⃗⃗⃗⃗ −OB ⃗⃗⃗⃗⃗ |=|BA ⃗⃗⃗⃗⃗ |, 由勾股定理,得AB=13,所以|a-b|=13.5.平面上有三点A,B,C,设m=AB ⃗⃗⃗⃗⃗ +BC ⃗⃗⃗⃗⃗ ,n=AB ⃗⃗⃗⃗⃗ −BC ⃗⃗⃗⃗⃗ ,若m,n 的长度恰好相等,则有( )A.A,B,C 三点必在同一条直线上B.△ABC 必为等腰三角形,且∠B 为顶角C.△ABC 必为直角三角形,且∠B=90°D.△ABC 必为等腰直角三角形,因为m,n 的长度相等,所以|AB ⃗⃗⃗⃗⃗ +BC ⃗⃗⃗⃗⃗ |=|AB ⃗⃗⃗⃗⃗ −BC ⃗⃗⃗⃗⃗ |,即|AC ⃗⃗⃗⃗⃗ |=|DB⃗⃗⃗⃗⃗ |, 所以四边形ABCD 是矩形,故△ABC 是直角三角形,且∠B=90°.6.已知四边形ABCD 是边长为1的正方形,则|AB ⃗⃗⃗⃗⃗ −AD ⃗⃗⃗⃗⃗ |= .|AB ⃗⃗⃗⃗⃗ −AD ⃗⃗⃗⃗⃗ |=|DB ⃗⃗⃗⃗⃗ |=√2, 所以|AB ⃗⃗⃗⃗⃗ −AD ⃗⃗⃗⃗⃗ |=√2. √27.如图,在梯形ABCD 中,AD ∥BC,AC 与BD 交于点O,则BA ⃗⃗⃗⃗⃗ −BC ⃗⃗⃗⃗⃗ −OA ⃗⃗⃗⃗⃗ +OD ⃗⃗⃗⃗⃗ +DA ⃗⃗⃗⃗⃗ = .BA ⃗⃗⃗⃗⃗ −BC ⃗⃗⃗⃗⃗ −OA ⃗⃗⃗⃗⃗ +OD ⃗⃗⃗⃗⃗ +DA ⃗⃗⃗⃗⃗ =CA ⃗⃗⃗⃗⃗ −OA ⃗⃗⃗⃗⃗ +OA ⃗⃗⃗⃗⃗ =CA ⃗⃗⃗⃗⃗ .⃗⃗⃗8.如图,已知O 为平行四边形ABCD 内一点,OA ⃗⃗⃗⃗⃗ =a,OB ⃗⃗⃗⃗⃗ =b,OC ⃗⃗⃗⃗⃗ =c,则OD ⃗⃗⃗⃗⃗ = .AD ⃗⃗⃗⃗⃗ =BC ⃗⃗⃗⃗⃗ ,则OD ⃗⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ +AD ⃗⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ +BC ⃗⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ +OC ⃗⃗⃗⃗⃗ −OB⃗⃗⃗⃗⃗ =a+c-b.9.如图,在正六边形ABCDEF 中,O 是正六边形中一点,若已知OA ⃗⃗⃗⃗⃗ =a,OF ⃗⃗⃗⃗⃗ =b,EO ⃗⃗⃗⃗⃗ =c,DO ⃗⃗⃗⃗⃗ =d,试用向量a,b,c,d 表示ED ⃗⃗⃗⃗⃗ ,AD ⃗⃗⃗⃗⃗ ,DB⃗⃗⃗⃗⃗ .⃗⃗⃗ =EO ⃗⃗⃗⃗⃗ +OD ⃗⃗⃗⃗⃗ =EO ⃗⃗⃗⃗⃗ −DO ⃗⃗⃗⃗⃗ =c-d. AD ⃗⃗⃗⃗⃗ =AO ⃗⃗⃗⃗⃗ +OD ⃗⃗⃗⃗⃗ =-OA ⃗⃗⃗⃗⃗ −DO ⃗⃗⃗⃗⃗ =-a-d.DB ⃗⃗⃗⃗⃗ =DC ⃗⃗⃗⃗⃗ +CB ⃗⃗⃗⃗⃗ =FA ⃗⃗⃗⃗⃗ +EF ⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ −OF ⃗⃗⃗⃗⃗ +OF ⃗⃗⃗⃗⃗ −OE ⃗⃗⃗⃗⃗ =OA ⃗⃗⃗⃗⃗ +EO ⃗⃗⃗⃗⃗ =a+c. 10.已知△ABC 是等腰直角三角形,∠ACB=90°,M 是斜边AB 的中点,CM ⃗⃗⃗⃗⃗⃗ =a,CA ⃗⃗⃗⃗⃗ =b,求证: (1)|a-b|=|a|; (2)|a+(a-b)|=|b|.,在等腰直角三角形ABC 中,|CA ⃗⃗⃗⃗⃗ |=|CB ⃗⃗⃗⃗⃗ |,由M 是斜边AB 的中点,得|CM ⃗⃗⃗⃗⃗⃗ |=|AM⃗⃗⃗⃗⃗⃗ |.(1)在△ACM 中,AM ⃗⃗⃗⃗⃗⃗ =CM ⃗⃗⃗⃗⃗⃗ −CA ⃗⃗⃗⃗⃗ =a-b. 由|AM ⃗⃗⃗⃗⃗⃗ |=|CM ⃗⃗⃗⃗⃗⃗ |,得|a-b|=|a|. (2)在△MCB 中,因为MB ⃗⃗⃗⃗⃗⃗ =AM ⃗⃗⃗⃗⃗⃗ =a-b, 所以CB ⃗⃗⃗⃗⃗ =MB ⃗⃗⃗⃗⃗⃗ −MC ⃗⃗⃗⃗⃗⃗ =a-b+a=a+(a-b). 由|CB ⃗⃗⃗⃗⃗ |=|CA ⃗⃗⃗⃗⃗ |,得|a+(a-b)|=|b|.。
数学专业英语第2章课后答案上课讲义
数学专业英语第2章课后答案上课讲义2.12.比:ratio 比例:proportion 利率:interest rate 速率:speed 除:divide 除法:division 商:quotient 同类量:like quantity 项:term 线段:line segment 角:angle 长度:length 宽:width高度:height 维数:dimension 单位:unit 分数:fraction 百分数:percentage3.(1)一条线段和一个角的比没有意义,他们不是相同类型的量.(2)比较式通过说明一个量是另一个量的多少倍做出的,并且这两个量必须依据相同的单位.(5)为了解一个方程,我们必须移项,直到未知项独自处在方程的一边,这样就可以使它等于另一边的某量.4.(1)Measuring the length of a desk, is actually comparing the length of the desk to that of a ruler.(3)Ratio is different from the measurement, it has no units. The ratio of the length and the width of the same book does not vary when the measurement unit changes.(5)60 percent of students in a school are female students, which mean that 60 students out of every 100 students are female students.2.22.初等几何:elementary geometry 三角学:trigonometry 余弦定理:Law of cosines 勾股定理/毕达哥拉斯定理:Gou-Gu theorem/Pythagoras theorem 角:angle 锐角:acute angle 直角:right angle 同终边的角:conterminal angles 仰角:angle of elevation 俯角:angle of depression 全等:congruence 夹角:included angle 三角形:triangle 三角函数:trigonometric function直角边:leg 斜边:hypotenuse 对边:opposite side 临边:adjacentside 始边:initial side 解三角形:solve a triangle 互相依赖:mutually dependent 表示成:be denoted as 定义为:be defined as3.(1)Trigonometric function of the acute angle shows the mutually dependent relations between each sides and acute angle of the right triangle.(3)If two sides and the included angle of an oblique triangle areknown, then the unknown sides and angles can be found by using the law of cosines.(5)Knowing the length of two sides and the measure of the included angle can determine the shape and size of the triangle. In other words, the two triangles made by these data are congruent.4.(1)如果一个角的顶点在一个笛卡尔坐标系的原点并且它的始边沿着x轴正方向，这个角被称为处于标准位置.(3)仰角和俯角是以一条以水平线为参考位置来测量的,如果正被观测的物体在观测者的上方,那么由水平线和视线所形成的角叫做仰角.如果正被观测的物体在观测者的下方,那么由水平线和视线所形成的的角叫做俯角.(5)如果我们知道一个三角形的两条边的长度和对着其中一条边的角度,我们如何解这个三角形呢?这个问题有一点困难来回答,因为所给的信息可能确定两个三角形,一个三角形或者一个也确定不了.2.32.素数:prime 合数:composite 质因数:prime factor/prime divisor 公倍数:common multiple 正素因子: positive prime divisor 除法算式:division equation 最大公因数:greatest common divisor(G.C.D) 最小公倍数: lowest common multiple(L.C.M) 整除:divide by 整除性:divisibility 过程:process 证明:proof 分类:classification 剩余:remainder辗转相除法:Euclidean algorithm 有限集:finite set 无限的:infinitely 可数的countable 终止:terminate 与矛盾:contrary to3.(1)We need to study by which integers an integer is divisible, that is , what factor it has. Specially, it is sometime required that an integer is expressed as the product of its prime factors.(3)The number 1 is neither a prime nor a composite number;A composite number in addition to being divisible by 1 and itself, can also be divisible by some prime number.(5)The number of the primes bounded above by any given finite integer N can be found by using the method of the sieve Eratosthenes.4.(1)数论中一个重要的问题是哥德巴赫猜想,它是关于偶数作为两个奇素数和的表示.(3)一个数,形如2p-1的素数被称为梅森素数.求出5个这样的数.(5)任意给定的整数m和素数p,p的仅有的正因子是p和1,因此仅有的可能的p和m的正公因子是p和1.因此,我们有结论:如果p是一个素数,m是任意整数,那么p整除m,要么(p,m)=1.2.42.集:set 子集:subset 真子集:proper subset 全集:universe 补集:complement 抽象集:abstract set 并集:union 交集:intersection 元素:element/member 组成:comprise/constitute包含:contain 术语:terminology 概念:concept 上有界:bounded above 上界:upper bound 最小的上界:least upper bound 完备性公理:completeness axiom3.(1)Set theory has become one of the common theoretical foundation and the important tools in many branches of mathematics.(3)Set S itself is the improper subset of S; if set T is a subset of S but not S, then T is called a proper subset of S.(5)The subset T of set S can often be denoted by {x}, that is, T consists of those elements x for which P(x) holds.(7)This example makes the following question become clear, that is, why may two straight lines in the space neither intersect nor parallel.4.(1)设N是所有自然数的集合,如果S是所有偶数的集合,那么它在N中的补集是所有奇数的集合.(3)一个非空集合S称为由上界的,如果存在一个数c具有属性:x<=c对于所有S中的x.这样一个数字c被称为S的上界.(5)从任意两个对象x和y，我们可以形成序列（x，y），它被称为一个有序对，除非x=y，否则它当然不同于（y，x）.如果S和T是任意集合，我们用S*T表示所有有序对（x,y),其中x术语S，y属于T.在R.笛卡尔展示了如何通过实轴和它自己的笛卡尔积来描述平面的点之后，集合S*T 被称为S和T的笛卡尔积.2.52.竖直线:vertical line 水平线:horizontal line 数对:pairs of numbers 有序对:ordered pairs 纵坐标:ordinate 横坐标:abscissas 一一对应:one-to-one 对应点:corresponding points圆锥曲线:conic sections 非空图形:non vacuous graph 直立圆锥:right circular cone 定值角:constant angle 母线:generating line 双曲线:hyperbola 抛物线:parabola 椭圆:ellipse退化的:degenerate 非退化的:nondegenerate 任意的:arbitrarily 相容的:consistent 在几何上:geometrically 二次方程:quadraticequation 判别式:discriminant 行列式:determinant3.(1)In the planar rectangular coordinate system, one can set up aone-to-one correspondence between points and ordered pairs of numbers and also a one-to-one correspondence between conic sections and quadratic equation.(3)The symbol can be used to denote the set of ordered pairs （x，y）such that the ordinate is equal to the cube of the abscissa.（5）According to the values of the discriminate，the non-degenerate graph of Equation （iii）maybe known to be a parabola， a hyperbolaor an ellipse.4.（1）在例1，我们既用了图形，也用了代数的代入法解一个方程组（其中一个方程式二次的，另一个是线性的）。
数学专业英语翻译2-6
It may be done by a formula as the 18th century mathematics presumed but it can equally well be done by a tabulation such as a statistical chart, or by some other form of description.
EXAMPLE 1. The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x. That is, F=cx, where c is a number independent of x called the spring constant.
这个公式是在17世纪中叶被胡克发现的，叫做胡克定 律，它用来表示力关于位移的函数。
EXAMPLE 2. The volume of a cube is a function of its edge-length. If the edges have length x, the volume V is given by the formula V=x3.
Today, the meaning of function is essentially this: Given two sets, say X and Y, a function is a correspondence which associates with each element of X one and one only element of Y.
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1.1）function，domain，range，the identity function，the absolute-value function，the real-valued function，real variable
2）cube，volume，edge-length，prime，totality
3）Hooke's law，stretch，displacement，spring，constant，proportional
4）schematic representation，plot，image，output，input
5）it is not difficult to imagine，the idea was much too limited
2.（1）常用英语字母和希腊字母来表示函数。
Letters of the English and Greek alphabets are often used to denote functions.
（2）若 f 是一个给定的函数，x 是定义域里的一个元素，那么记号f(x)用来表示由 f 确定的对应于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）该射线将两个坐标轴的夹角分成两个相等的角。
The ray makes equal angles with the coordinates axes.
（4）可以用许多方式给出函数思想的图解说明。
The function idea may be illustrated schematically in many ways.
（5）容易证明，绝对值函数满足三角不等式。
It is easy to proof that the absolute-value function satisfies the triangle inequality.
（6）对于实数x>0，函数g(x)表示不超过x 的素数的个数。
For a given real number x>0, the function g(x) is defined by the number of primes less than or equal to x.
（7）函数是一种对应，它未必可以表示成一个简单的代数公式。
A function is a correspondence. It is not necessary to be expressed by a simple algebraic formula. （8）在函数的定义中，关于定义域和值域中的对象，没对其性质做出任何限制。
The function idea places no restriction on the nature of the objects in the domain X and in the range Y.
3.。