数学专业英语第一讲(修改)
数学专业英语
vt.表明，标示，指示; 象征，暗示，预 示; [医]显示需要做…的治疗;
numerical[英][nju:ˈmerɪkl][美][nu:ˈmerɪkl]
adj.数字的，用数字表示的，数值的;
root[英][ru:t][美][rut, rʊt]
n.根，根源; 原因，本质; 祖先; [乐]和弦基音; vt.使生根; 使固定; 根源在于; 欢呼，喝彩; vi.生根; 根除;
n.身份; [逻]同一性; 个性; [数]恒等（式）;
arithmetic[英][əˈrɪθmətɪk][美][əˈrɪθmɪtɪk]
n.算术，计算; 算法;
algebraic[英][ˌældʒɪ'breɪɪk][美][ˌældʒəˈbreɪk]
adj.代数的，关于代数学的;
indicate[英][ˈɪndɪkeɪt][美][ˈɪndɪˌket]
你拿这个或那个都可以。
1. either…or…连接两个成分作主语时，谓语动 词通常与其靠近的主语保持一致。如： Either you or I am to go. 你或我必须有人去。 但在非正式文体中，有时也会一律用复数谓 语。如：If either David or Janet come, they will want a drink.
prep.在...的附近; 在...的周围
解方程意味着求未知项的值，为了求未知项的值，当 然，我们必须移项，直到未知项单独在方程的一边， 令其等于方程的另一边，求得未知项的值，这样我们 就解决了问题。
on one side…on the other side…一方面……另一方面……
脑筋急转弯：
Two little brother,One lives on one side,The other on the other side,They hear what you say But they do not see each other,Who are they?
数学专业英语1
数学专业英语1学专业英语－How to Write Mathematics?How to Write Mathematics?------ Honesty is the Best PolicyThe purpose of using good mathematical language is, of course, to make the u nderstanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the se nse of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss.The emphasis in the preceding paragraph, while perhaps necessary, might see m to point in an undesirable direction, and I hasten to correct a possible misin terpretation. While avoiding pedantry and fuss, I do not want to avoid rigor an d precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way t o get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his s ympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome comp utation.”Here is the sort of the thing I mean by less than complete honesty. At a certa in point, having proudly proved a proposition P, you feel moved to say: “Not e, however, that p does not imply q”, and then, thinking that you’ve done a good expositor y job, go happily on to other things. Your motives may be per fectly pure, but the reader may feel cheated just the same. If he knew all abo ut the subject, he wouldn’t be reading you; for him thenonimplication is, qui te likely, unsupported. Is it obvious? (Say so.) Will a counterexample be suppl ied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean th at you have tried to derive q from p, you failed, a nd you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into y our confidence.There is nothing wrong with often derided “obvious”and “easy to see”, b ut there are certain minimal rules to their use. Surely when you wrote that so mething was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still thi nk that something was obvious? (A few months’ripening always improves ma nuscripts.) When you explained it to a friend, or to a seminar, was the someth ing at issue accepted as obvious? (Or did someone question it and subside, mu ttering, when you reassured him? Did your assurance demonstration or intimida tion?) the obvious answers to these rhetorical questions are among the rules th at should control the use of “obvious”. There is the most frequent source o f mathematical error: make that the “obvious”is true.It should go without saying that you are not setting out to hide facts from the reader: you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them eit her. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the importan t and minimize the trivial. The reason saying that they are obvious is to put t hem in proper perspecti e for the uninitiated. Even if your saying so makes an occasional readerangry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must obey the rules. Don’t le t the reader down; he wants to believe in you. Pretentiousness, bluff, and conc ealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor them selves, but quite properly, the author. Complete honesty makes for greatest clar ity.---------Paul R.Haqlmosvocabularyflashy 一闪的 counter-example 反例unobtrusiveness 谦虚dictum 断言；格言forestall 阻止，先下手deride嘲弄anticipate 预见 subside沉静pedantry 迂腐；卖弄学问 mutter出怨言，喃喃自语fuss 小题大做 intimidation威下reconcilable 使一致的 rhetorical合符修辞学的gloss 掩饰 pretentiousness自命不凡alleviate 减轻，缓和bluff 欺骗implication 包含，含意concealment隐匿notes1. 本课文选自美国数学学会出版的小册子How to write mathematics 中Paul R.Halmos. 的文章第9节2. The purpose is smooth the reader’way, to anticipates his difficulties and to forestall them. Clarit y is what’s wanted, not pedantry; understanding, not fuss.意思是：目的是为读者扫清阅读上的障碍，即预先设想读者会遇到什么困难，并力求避免出现这类困难。
数学专业英语(Doc版).Word5
数学专业英语－Differential CalculusHistorical IntroductionNewton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insur mountable problems could be solved by more or less routine methods.The succ essful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch o f calculus,differential calculus.The central idea of differential calculus is the notion of derivative.Like the inte gral,the derivative originated from a problem in geometry—the problem of find ing the tangent line at a point of a curve.Unlile the integral,however,the deriva tive evolved very late in the history of mathematics.The concept was not form ulated until early in the 17th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special func tions.Fermat’s idea,basically very simple,can be understood if we refer to a curve a nd assume that at each of its points this curve has a definite direction that ca n be described by a tangent line.Fermat noticed that at certain points where th e curve has a maximum or minimum,the tangent line must be horizontal.Thus t he problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.This raises the more general question of determining the direction of the tange nt line at an arbitrary point of the curve.It was the attempt to solve this gener al problem that led Fermat to discover some of the rudimentary ideas underlyi ng the notion of derivative.At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of findin g the tangent line at a point of a curve.The first person to realize that these t wo seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they explo ited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.Although the derivative was originally formulated to study the problem of tang ents,it was soon found that it also provides a way to calculate velocity and,mo re generally,the rate of change of a function.In the next section we shall consi der a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.A Problem Involving VelocitySuppose a projectile is fired straight up from the ground with initial velocity o f 144 feet persecond.Neglect friction,and assume the projectile is influenced onl y by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at ti me t we woule have f(t)=144 t.In actual practice,gravity causes the projectile t o slow down until its velocity decreases to zero and then it drops back to eart h.Physical experiments suggest that as the projectile is aloft,its height f(t) is gi ven by the formula(1)f(t)=144t –16 t2The term –16t2is due to the influence of gravity.Note that f(t)=0 when t=0 a nd when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.The problem we wish to consider is this:To determine the velocity of the proj ectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we int roduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient.Change in distance during time interval =f(t+h)-f(t)/hThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positiv e or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute v alue.The limit process by which v(t) is obtained from the difference quotient is wri tten symbolically as follows:V(t)=lim(h→0)[f(t+h)-f(t)]/hThe equation is used to define velocity not only for this particular example bu t,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.The example describe in the foregoing section points the way to the introducti on of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this in terval and introduce the difference quotient[f(x+h)-f(x)]/hwhere the number h,which may be positive or negative(but not zero),is such th at x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to a s the average rate of change of f in the interval joining x to x+h.Now we let h approach zero and see what happens to this quotient.If the quot ient.If the quotient approaches some definite values as a limit(which implies th at the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as “f prime of x”).Thus the formal defi nition of f’(x) may be stated as follows:Definition of derivative.The derivative f’(x)is defined by the equationf’(x)=lim(h→o)[f(x+h)-f(x)]/hprovided the limit exists.The number f’(x) is also called the rate of change of f at x.In general,the limit process which produces f’(x) from f(x) gives a way of ob taining a new function f’from a given function f.This process is called differ entiation,and f’is called the first derivative of f.If f’,in turn,is defined on an interval,we can try to compute its first derivative,denoted by f’’,and is calle d the second derivative of f.Similarly,the nth derivative of f denoted by f^(n),is defined to be the first derivative of f^(n-1).We make the convention that f^(0) =f,that is,the zeroth derivative is the function itself.Vocabularydifferential calculus微积分differentiable可微的intergral calculus 积分学differentiate 求微分hither to 迄今 integration 积分法insurmountable 不能超越 integral 积分routine 惯常的integrable 可积的fuse 融合integrate 求积分originate 起源于sign-preserving保号evolve 发展，引出 axis 轴（单数）tangent line 切线 axes 轴（复数）direction 方向 contradict 矛盾horizontal 水平的contradiction 矛盾vertical 垂直的 contrary 相反的rudimentary 初步的，未成熟的composite function 合成函数，复合函数area 面积composition 复合函数intimately 紧密地interior 内部exploit 开拓，开发 interior point 内点inaugurate 开始 imply 推出，蕴含projectile 弹丸 aloft 高入云霄friction摩擦initial 初始的gravity 引力 instant 瞬时rate of change 变化率integration by parts分部积分attain 达到definite integral 定积分defferential 微分indefinite integral 不定积分differentiation 微分法 average 平均Notes1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.意思是：在很大程度上是牛顿和莱伯尼，他们相互独立地把积分学的思想发展到这样一种程度，使得迄今一些难于超越的问题可以或多或少地用通常的方法加以解决。
数学专业英语45753
因此，差商表示PQ与水平线的夹角a的正切。
The real number tana is called the slope of the line through P and Q and it provides a way of measuring the “steepness” of the line.
上一节描述的例子指出了介绍导数概念的方法。
We begin with a function f defined at least on some open interval (a,b) on the x-axis. 我们从一个至少定义在x轴的开区间(a,b)上的函数入 手。
Then we choose a fixed point x in this interval and introduce the difference quotient
一般地, 由f(x)产生f'(x)的极限的过程提供了一种方法, 从一个给定的函数 f 得到一个新的函数f'。这个过程 叫做微分法。 f'叫做f的一阶导数。
If f' , in turn, is defined on an open interval, we can try to compute its first derivative, denoted by f″ and called the second derivative of f.
(P67 第四段第一句话)
在直线运动中，速度的一阶导数称为加速度。
数学专业英语课文翻译
1-A：什么是数学数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
反过来，数学服务于实践并在所有领域扮演一个重要的角色。
没有数学的应用，现代化科学和技术的分支都不能有规律的发展。
从早期人类的需求引出了数和形状。
然后，几何学因测量陆续的发展出来，三角学来自于勘探问题。
为了处理一些更复杂的实践问题，人们建立了方程，通过求解方程的未知数，从而代数学出现了。
17世纪之前，人们局限于初等数学，例如几何、三角和代数，那些只考虑常数。
17世纪工业的迅速发展促进了经济学和科技的发展，并且我们需要处理变量。
从常数到变量的跳跃带来了两个属于高等数学的新的数学分支，解析几何和微积分学。
现在，高等数学中有了许多分支，数学分析、高等代数、微分方程、函数论等。
数学家们研究概念和命题。
公理、公社、定义和定理都是命题。
符号是一种特别并且很重要的数学工具，它常用于表示概念和命题。
公式、图形和表格充满着不同的符号。
阿拉伯数字1,2,3,4,5,6,7,8,9,0和加”+”减”-”乘”*”除”/”等号”=”使我们最熟悉的数学符号。
主要通过逻辑推导和计算来获得数学结论。
在数学史的很长的时期内，逻辑推论一直占据数学方法的中心地位。
现在，自从电子计算机迅速发展和广泛应用，计算的角色越来越重要。
现在，计算不仅用来处理信息与数据，而且用来完成一些在以前只能靠逻辑推理来做的工作，例如证明大多数的几何定理。
1-B：等式等式是关于两个数或数的符号相等的一种陈述。
因此a(a-5)=a^2-5a和x-3=5是等式。
等式有两种，恒等式和条件等式。
算术和代数恒等式是等式。
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。
因此12-2=2+8，(m-n)(m+n)=m^2-n^2是恒等式。
含有字母的恒等式对其中字母的任何一组数值都成立。
因此恒等式x(a+2)=ax+2x变成3(7+2)=21+6或27=27，比如当x=3和a=7。
数学专业英语(Doc版).Word6
数学专业英语－First Order Differential EquationsA differential equation is an equation between specified derivatives of a functio n, itsvalves,and known quantities.Many laws of physics are most simply and naturall y formu-lated as differential equations (or DE’s, as we shall write for short).For this r eason,DE’shave been studies by the greatest mathematicians and mathematical physicists si nce thetime of Newton..Ordinary differential equations are DE’s whose unknowns are functions of a s ingle va-riable;they arise most commonly in the study of dynamic systems and electric networks.They are much easier to treat than partial differential equations,whose unknown functionsdepend on two or more independent variables.Ordinary DE’s are classified according to their order. The order of a DE is d efined asthe largest positive integer, n, for which an n-th derivative occurs in the equati on. Thischapter will be restricted to real first order DE’s of the formΦ(x, y, y′)=0 (1)Given the function Φof three real variables, the problem is to determine all re al functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the follo wing sense.DEFINITION A solution of (1)is a differentiable function f(x) such thatΦ(x. f(x),f′(x))=0 for all x in the interval where f(x) is defined.EXAMPLE 1. In the first-other DEthe function Φis a polynomial function Φ(x, y, z)=x+ yz of three variables i n-volved. The solutions of (2) can be found by considering the identityd(x²+y²)/d x=2(x+yyˊ).From this identity,one sees that x²+y²is a con-stant if y=f(x) is any solution of (2).The equation x²+y²=c defines y implicitly as a two-valued function of x,for any positive constant c.Solving for y,we get two solutions,the(single-valued) functions y=±(c-x²)0.5,for each positive constant c.The graphs of these so-lutions,the so-called solution curves,form two families of scmicircles,which fill t he upper half-plane y>0 and the lower half-plane y>0,respectively.On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solu tionswhich cross the x-axis,except possibly at the origin.This fact is easily overlook ed,because the solution curves appear to cross the x-axis;hence yˊdoes not exist, and the DE (2) is not satisfied there.The preceding difficulty also arises if one tries to solve the DE(2)for yˊ. Div iding through by y,one gets yˊ=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.DEFINITION. A normal first-order DE is one of the formyˊ=F(x,y) (3)In the normal form yˊ=-x/y of the DE (2),the function F(x,y) is continuous i n the upper half-plane y>0 and in the lower half-plane where y<0;it is undefin ed on the x-axis.Fundamental Theorem of the Calculus.The most familiar class of differential equations consists of the first-order DE’s of the formSuch DE’s are normal and their solutions are descried by the fundamental tho rem of the calculus,which reads as follows.FUNDAMENTAL THEOREM OF THE CALCULUS. Let the function g(x)i n DE(4) be continuous in the interval a<x<b.Given a number c,there is one an d only one solution f(x) of the DE(4) in the interval such that f(a)=c. This sol ution is given by the definite integralf(x)=c+∫a x g(t)dt , c=f(a) (5)This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a<x<b in the xy-plane.Second,it describes in precise terms the class of functions g(x) consid ered.And third, it asserts the existence and uniqueness of a solution,given the “initial condition”f(a)=c.We recall that the definite integral∫a x g(t)dt=lim(maxΔt k->0)Σg(t k)Δt k , Δt k=t k-t k-1 (5ˊ)is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral ∫g(x) dx to give meanin g to the definite integral ∫a x g(t)dt,provided only that g(t) is continuous.Such f unctions as the error function crf x =(2/(π)0.5)∫0x e-t²dt and the sine integral f unction SI(x)=∫x∞[(sin t )/t]dt are indeed commonly defined as definite int egrals.Solutions and IntegralsAccording to the definition given above a solution of a DE is always a functi on. For example, the solutions of the DE x+yyˊ=0 in Example I are the func tions y=±(c-x²)0.5,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily describ ed by the equation x²+y²=c,describing a family of circles centered at the origi n.In what sense can such a family of curves be considered as a solution of th e DE ?To answer this question,we require a new notion.DEFINITION. An integral of DE(1)is a function of two variables,u(x,y),whic h assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.In the above example, the function u(x,y)=x²+y²is an integral of the DE x +yyˊ=0,because,upon replacing the variable y by any function ±( c-x²)0.5,we obtain u(x,y)=c.The second-order DEd²x/dt²=-x (2ˊ)becomes a first-order DE equivalent to (2) after setting dx/dx=y:y ( dy/dx )=-x (2)As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE.When th e DE (2ˊ)is interpreted as equation of motion under Newton’s second law,the integrals c=x²+y²represent curves of constant energy c.This illustrates an important prin ciple:an integral of a DE representing some kind of motion is a quantity that r emains unchanged through the motion.Vocabularydifferential equation 微分方程 error function 误差函数ordinary differential equation 常微分方程 sine integral function 正弦积分函数order 阶,序 diameter 直径derivative 导数 curve 曲线known quantities 已知量replace 替代unknown 未知量substitute 代入single variable 单变量strip 带形dynamic system 动力系统 exact differential 恰当微分electric network 电子网络line integral 线积分partial differential equation 偏微分方程path of integral 积分路径classify 分类 endpoints 端点polynomial 多项式 general solution 通解several variables 多变量parameter 参数family 族rigorous 严格的semicircle 半圆 existence 存在性half-plane 半平面 initial condition 初始条件region 区域uniqueness 唯一性normal 正规,正常Riemann sum 犁曼加identity 恒等(式)Notes1. The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs i n the question.这是另一种定义句型,请参看附录IV.此外要注意nth derivative 之前用an 不用a .2. This chapter will be restricted to real first order differential equations of the formΦ(x,y,yˊ)=0意思是;文章限于讨论形如Φ(x,y,yˊ)=0的实一阶微分方程.有时可以用of the type代替of the form 的用法.The equation can be rewritten in the form yˊ=F(x,y).3. Dividing through by y,one gets yˊ=-x/y,…划线短语意思是:全式除以y4. As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE这里x²+y²=c 因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.5. Their solutions are described by the fundamental theorem of the calculus,which reads as follows.意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下.句中reads as follows 就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.ExerciseⅠ.Translate the following passages into Chinese:1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫c M(x,y) dx +N(x,y) dy is the same for all paths of int egration c in D, which have the same endpoints.Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.2. For any normal first order DE yˊ=F(x,y) and any initial x0 , the initial valve problem consists of finding the solution or solutions of the DE ,for x>x0 which assumes a given initial valve f(x0)=c.3. To show that the initial valve problem is well-set requires proving theorems of existence (there isa solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on t he initial value).Ⅱ. Translate the following sentences into English:1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有dч(x)/dx=f (x,ч(x))2) 两边从x0到x取定积分得ч(x)-ч(x0)=∫x0x f(x,ч(x)) dx x0<x<x0+h3) 把y0=ч(x0)代入上式, 即有ч(x)=y0+∫x0x f(x,ч(x)) dx x0<x<x0+h4) 因此y=ч(x) 是积分方程y=y0+∫x0x f (x,y) dx定义于x0<x<x0+h 的连续解.Ⅲ. Translate the following sentences into English:1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数f (x, dy/dx) 有连续的偏导数.2) 引入参数dy/dx=p, 则已给方程变为y=f (x,p).3) 在y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为y=f (x,ч(x,c)).6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解 x=ψ(p,c)y=f(ψ(p,c)p)其中p是参数,c是任意常数.。
数学专业英语
Mathematical English Dr. Xiaomin ZhangEmail:第1 页§2.4 Integers, Rational Numbers and Real numbersTEXT A Integers and rational numbersThere exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss two such subsets, the integers and the rational numbers.To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1, 2, 3, …, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers. Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in details what we mean by the expressions “and so on〞, or “repeated addition of 1〞. Although the intuitive meaning of expressions may seem clear, in a careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set. DEFINITION OF AN INDUCTIVE SET A set of real numbers is called an inductive set if it has the第2 页following two properties:(a) The number 1 is in the set.(b) For every x in the set, the number x+1 is also in the set.For example, R is an inductive set. So is the set R+. Now we shall define the positive integers to be those real numbers which belong to every inductive set.DEFINITION OF POSITIVE INTEGERS A real number is called a positive integer if it belongs to every inductive set.Let P denote the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P belong to every inductive set, we refer to P as the smallest inductive set. This property of the set P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this Introduction.The negatives of the positive integers are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply the set of integers.第3 页In a thorough treatment of the real-number system, it would be necessary at this stage to prove certain theorems about integers. For example, the sum, difference, or product of two integers is an integer, but the quotient of two integers need not be an integer. However, we shall not enter into the details of such proofs.Quotients of integers a/b(where b0) are called rational number. The set of rational numbers, denoted by Q, contains Z as a subset. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. Real numbers that are not in Q are called irrational.NotationsField axioms A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra, where division algebra, also called a "division ring" or "skew field," means a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative.Order axioms A total order (or "totally ordered set," or "linearly ordered set") is a set plus a relation on第4 页the set (called a total order) that satisfies the conditions for a partial order plus an additional condition known as the comparability condition. A relation is a total order on a set S (" totally orders S") if the following properties hold.1. Reflexivity: a a for all a S.2. Antisymmetry: a b and b a implies a=b.3. Transitivity: a b and b c implies a c.4. Comparability (trichotomy law): For any a, b S, either a b or b a.The first three are the axioms of a partial order, while addition of the trichotomy law defines a total order.第5 页TEXT B Geometric interpretation of real numbers as points on a lineThe reader is undoubtedly familiar with the geometric representation of real numbers by means of points on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 2-4-1. This choice determines the scale. If one adopts an appropriate set of axioms for Euclidean geometry, then 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. For this reason the line is often called the real line or the real axis, and it is customary to use the words real number and point interchangeably. Thus we often speak of the point x rather than the point corresponding to the real numbers.The ordering relation among the real numbers has a simple geometric interpretation. If x<y, the point x lies to the left of the point y as shown in Figure 2-4-1. Positive numbers lie to the right of 0 and negative numbers to the left of 0. If a<b, a point x satisfies the inequalities a<x<b if and only if x is between a and b.This device for representing real numbers geometrically is a very worthwhile aid that helps us to第6 页discover and understand better certain properties of real numbers. However, the reader should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any reference to geometry. This does not mean that one should not make use of geometry in studying properties of real numbers. On the contrary, the geometry often suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof (one depending entirely on the axioms for the real numbers). In this book, geometric arguments are used to a large extent to help motivate or clarify a particular discuss. Nevertheless, the proofs of all the important theorems are presented is analytic form.第7 页SUPPLEMENT Prime NumberA prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24=233), making 24 not a prime number. Positive integers other than 1 which are not primes are called composite numbers.Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers n whose divisors are trivial and given by exactly 1 and n.The number 1 is a special case which is considered neither prime nor composite. With 1 excluded, the smallest prime is therefore 2 and since 2 is the only even prime, it is also somewhat special. Note also that while 2 is considered a prime today, at one time it was not.第8 页The n th prime number is commonly denoted p n, so p1=2, p2=3, and so on, and may be computed in Mathematica as Prime[n]. The set of primes is sometimes denoted P, represented in Mathematica as Primes.Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate". In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision".第9 页Large primes include the large Mersenne primes, Ferrier's prime. The largest known prime as of Feb. 2005 is the Mersenne prime 2-1 (Weisstein 2005).Prime numbers can be generated by sieving processes (such as the sieve of Eratosthenes), and lucky numbers, which are also generated by sieving, appear to share some interesting asymptotic properties with the primes. Prime numbers satisfy many strange and wonderful properties.The function that gives the number of primes less than or equal to a number n is denoted (n) and is called the prime counting function. The theorem giving an asymptotic form for (n) is called the prime number theorem.The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of primes. Euclid's second theorem demonstrated that there are an infinite number of primes. However, it is not known if there are an infinite number of primes of the form n2+1, whether there are an infinite number of twin primes (the twin prime conjecture), or if a prime can always be found between n2 and (n+1)2. The latter two of these are two of Landau's problems.第10 页Primes consisting of consecutive digits (counting 0 as coming after 9) include 2, 3, 5, 7, 23, 67, 89, 4567, 78901, .... Primes consisting of digits that are themselves primes include 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, ..., which is one of the Smarandache sequences. Because a prime number p has the trivial factors 1 and p, Bill Gates accidentally referred to a trivial operation when he stated "Because both the system's privacy and the security of digital money depend on encryption, a breakthrough in mathematics or computer science that defeats the cryptographic system could be a disaster. The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers [emphasis added]" (Gates 1995, p. 265).NotationsMersenne prime A Mersenne prime is a Mersenne number, i.e., a number of the form M n=2n-1 that is prime. In order for M n to be prime, n must itself be prime. This is true since for composite n with factors r and s, n=rs. Therefore, 2n-1 can be written as 2rs-1, which is a binomial number that always has a factor 2r-1.第11 页The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... corresponding to indices n=1, 3, 5, 7, 13, 17, 19, 31, 61, 89, ....Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number. L. Welsh maintains an extensive bibliography and history of Mersenne numbers.It has been conjectured that there exist an infinite number of Mersenne primes.However, finding Mersenne primes is computationally very challenging. For example, the 1963 discovery that M11213 is prime was heralded by a special postal meter design, illustrated above, issued in Urbana, Illinois.G. Woltman has organized a distributed search program via the Internet known as GIMPS (Great Internet Mersenne Prime Search) in which hundreds of volunteers use their personal computers to perform pieces of the search. On November 17, 2003, a GIMPS volunteer reported discovery of the 40th Mersenne prime, a discovery that was subsequently confirmed. Almost exactly six months later,第12 页discovery of the 41st known Mersenne prime by a GIMPS volunteer was announced. The 42nd known Mersenne prime was announced on Feb. 18, 2005 and its exponent was released on Feb. 26. The efforts of GIMPS volunteers make this distributed computing project the discoverer of all eight of the largest known Mersenne primes. In fact, as of Feb. 2005, GIMPS participants have tested and double-checked all exponents below 9889900 and tested all exponents below 15130000 at least once (GIMPS).The table below gives the index p of known Mersenne primes M p, together with the number of digits, discovery years, and discoverer. A similar table has been compiled by C. Caldwell. Note that the region after the 38th known Mersenne prime has not been completely searched, so identification of "the" 40th Mersenne prime is tentative (GIMPS).#p digits year discoverer (reference)121antiquity231antiquity352antiquity第13 页473antiquity51341461Reguis (1536), Cataldi (1603) 61761588Cataldi (1603)71961588Cataldi (1603)831101750Euler (1772)961191883Pervouchine (1883), Seelhoff (1886) 1089271911Powers (1911)11107331913Powers (1914)12127391876Lucas (1876)13521157Jan. 30, 1952Robinson14607183Jan. 30, 1952Robinson151279386Jan. 30, 1952Robinson162203664Jan. 30, 1952Robinson172281687Jan. 30, 1952Robinson第14 页183217969Sep. 8, 1957Riesel1942531281Nov. 3, 1961Hurwitz2044231332Nov. 3, 1961Hurwitz2196892917May 11, 1963Gillies (1964)2299412993May 16, 1963Gillies (1964)23112133376Jun. 2, 1963Gillies (1964)24199376002Mar. 4, 1971Tuckerman (1971)25217016533Oct. 30, 1978Noll and Nickel (1980)26232096987Feb. 9, 1979Noll (Noll and Nickel 1980) 274449713395Apr. 8, 1979Nelson and Slowinski (Slowinski 1978-79) 288624325962Sep. 25, 1982Slowinski2911050333265Jan. 28, 1988Colquitt and Welsh (1991) 3013204939751Sep. 20, 1983Slowinski3121609165050Sep. 6, 1985Slowinski第15 页32756839227832Feb. 19, 1992Slowinski and Gage33859433258716Jan. 10, 1994Slowinski and Gage34378632Sep. 3, 1996Slowinski and Gage35420921Nov. 12, 1996Joel Armengaud/GIMPS36895832Aug. 24, 1997Gordon Spence/GIMPS (Devlin 1997)37909526Jan. 27, 1998Roland Clarkson/GIMPS38Jun. 1, 1999Nayan Hajratwala/GIMPS39Nov. 14, 2001Michael Cameron/GIMPS (Whitehouse 2001, Weisstein 2001) 40Nov. 17, 2003Michael Shafer/GIMPS (Weisstein 2003)41May 15, 2004Josh Findley/GIMPS (Weisstein 2004)42Feb. 18, 2005Martin Nowak/GIMPS (Weisstein 2005)Ferrier's prime According to Hardy and Wright (1979), the 44-digit Ferrier's prime is+1)/17=24593863921(2148第16 页determined to be prime using only a mechanical calculator, is the largest prime found before the days of electronic computers. Mathematica can verify primality of this number in a (small) fraction of a second, showing how far the art of numerical computation has advanced in the intervening years.lucky numbers Write out all the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number greater than 1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, .... Numbers remaining after this procedure has been carried out completely are called luckynumbers.The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, .... Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/lnN, just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjectureto hold. It therefore appears that the sieving process accounts for also seemsmany properties of the primes.第17 页第 18 页prime number theorem It gives an asymptotic form for the prime counting function (n), which countsthe number of primes less than some integer n.Legendre (1808) suggested that for large n,with B=-1.08366 (where B is sometimes called Legendre's constant).In 1792, when only 15 years old, Gauss proposed thatGauss later refined his estimate to(n)~Li(n), where is the logarithmic integral.Gauss did not publish this result, which he first mentioned in an 1849 letter to Encke. It was subsequently posthumously published in 1863. For small n, it had been checked and always found that (n)<Li(n). As a result, many prominent mathematicians, including no less than both Gauss and Riemann, conjectured that the inequality was strict. To everyone's surprise, this conjecture was refuted when Littlewood (1914) proved that the inequality reverses infinitely often for sufficiently large , a number now34)16)10((10occurs before Li(n)=0 -(n)first crossing of . Skewes then showed that the n第 19 页known as the Skewes number. The upper bound for the crossing has subsequently been reduced to 167 decimalreversals occur for numbers with 1166 or 1500 10. Lehman (1966) proved that at least 37110digits.Twin primes Twin primes are pairs of primes of the form (p, p+2). The first few twin primes are n 1 for n=4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, .... Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), .... All twin primes except (3, 5) are of the form6n 1.It is conjectured that there are an infinite number of twin primes (this is one form of the twin primeconjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the number obtained byadding the reciprocals of the odd twin primes,converges to a definite number ("Brun's constant"), which expresses the scarcity of twin primes, even if there are infinitely many of them. The following table gives the first few p for the twin primes (p, p+2),cousin primes (p, p+4), sexy primes (p, p+6), etc.pair first member(, )3, 5, 11, 17, 29, 41, 59, 71, ...(, )3, 7, 13, 19, 37, 43, 67, 79, ...(, )5, 7, 11, 13, 17, 23, 31, 37, ...(, )3, 5, 11, 23, 29, 53, 59, 71, ...(, )3, 7, 13, 19, 31, 37, 43, 61, ...(, )5, 7, 11, 17, 19, 29, 31, 41, ...Goldbach Conjecture Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes". Note that here Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this第20 页conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers 4 can be expressed as the sum of two primes.According to Hardy, "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and Faber offered a $1000000 prize to anyone who proved Goldbach's conjecture between March 20, 2000 and March 20, 2002, but the prize went unclaimed and the conjecture remains open.Schnirelman (1939) proved that every even number can be written as the sum of not more than 300000 primes, which seems a rather far cry from a proof for two primes! Pogorzelski (1977) claimed to have proven the Goldbach conjecture, but his proof is not generally accepted. The following table summarizes bounds n such that the strong Goldbach conjecture has been shown to be true for numbers <n.bound referenceDesboves 1885第21 页Pipping 1938Stein and Stein 1965abGranville et al. 1989Sinisalo 1993Deshouillers et al. 1998Richstein 1999, 2001Oliveira e Silva (Mar. 24, 2003)Oliveira e Silva (Oct. 3, 2003)Oliveira e Silva (Feb. 5, 2005)The conjecture that all odd numbers 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. Vinogradov (1937, 1954) proved that every sufficiently large odd number is the sum of three primes, and Estermann (1938) proved that almost all even numbers are the sums of two primes. Vinogradov's original "sufficiently large" N(e e)16573 3.25X10 was subsequently reduced to第22 页(e e)1150343000 by Chen and Wang (1989). Chen (1973, 1978) also showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes.第23 页Problem 1 Given any two operations, symbolized by and , if a(b c)= (a b)(a c) for all possible choices of a, b and c. This fact is called ____ law.A commutativeB associativeC transitiveD distributiveProblem 2 Twin primes are pairs of primes of the form (p, p+2). In the following four pairs of numbers, which is not a twin primesA (3, 5)B (11, 13)C (17, 19)D (19, 21)Problem 3 A semiprime is a composite number that is the product of two (possibly equal) primes. In the following four numbers, which is not a semiprimeA 9B 14C 15D 20第24 页。
数学专业英语【第二版】（吴炯圻）
数学专业英语【第二版】1- A 什么是数学数学来自于人的社会实践，例如，工业和农业生产、商业活动、军事行动和科研工作。
与数学反过来，为实践服务和所有字段中的伟大作用。
没有现代的科学和技术分支机构可以定期制定中的数学，应用无早有需要的人来了数字和形式的概念。
然后，开发出的几何土地和三角测量的问题来自测量的问题。
若要对付一些更复杂的实际问题，男子成立，然后解决方程未知号码，因此代数发生。
17 世纪前, 男子向自己限于小学数学，即几何、三角和代数，只有常量被认为在其中。
17 世纪产业的快速发展促进了经济和技术的进展和所需变量的数量、处理从常量到带来两个分支的数学-解析几何和微积分，属于高等数学，现在有很多分支机构，其中有数学分析、高等代数、微分方程的高等数学中的可变数量的飞跃函数理论等。
数学家研究理念和主张。
所有命题公理、假设、定义和定理都。
符号是一种特殊和功能强大的数学工具，用于表示很多时候的理念和主张。
公式、数字和图表是阿拉伯数字1,2,3,4,5,6,7,8,9,0 与另外的符号"+"、减法"-"，乘"*"，除"\"和平等"="。
数学中的结论得到主要由逻辑推理和计算。
长期的数学史上，以中心地点的数学方法被占领逻辑扣除。
现在，由于电子计算机是迅速发展和广泛应用，计算的作用变得越来越多重要。
在我们这个时代计算不只用于处理大量的信息和数据，而且还进行一些只是可以做的工作较早前的逻辑推理，例如，大部分的几何定理的证明。
1--B 方程方程是平等的语句的两个相等的数字或数字符号之间。
因此(a-5）= 一5a 和x 3 = 5 是方程。
方程的两种——身份和方程的条件。
方程的算术或代数的身份。
这种方程中两名成员是相似的或成为相似的指示操作的性能。
因此12-2=2+8,(m+n)(m-n) = m n 是身份。
1—c 比与测量今天的思想沟通往往根据编号和数量的比较。
数学专业英语
数学专业英语数学专业英语课后答案2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
数学专业英语(只是部分,不是很完整)
Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition “+”, subtraction “-” , multiplication “×”, division “÷” and equality “=”. 数学家研究的是概念和命题，公理，公设，定义和定理都 是命题。符号是数学中一个特殊而有用的工具，常用于表 达概念和命题。 公式，图形和图表都是不同的符号……..
1－B Equation
An equation is a statement of the equality between two equal numbers or number symbols. Equations are of two kinds---- identities and equations of condition. An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation. 等式是关于两个数或者数的符号相等的一种描述。 等式有两种－恒等式和条件等式。算术或者代数恒等式都是 等式。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。
数学专业英语(Doc版).Word2
数学专业英语－(a) How to define a mathematical term?数学术语的定义和数学定理的叙述，其基本格式可归纳为似“if …then …”的格式，其他的格式一般地说可视为这一格式的延伸或变形。
如果一定语短语或定语从句，以界定被定义的词，所得定义表面上看虽不是“If ……then ……”的句型，而实际上是用“定语部分”代替了“If ”句，因此我们可以把“定语部分”写成If 句，从而又回到“If ……then ……”的句型。
至于下面将要叙述的“Let …if …then ”，“Let and assume …, If …then …”等句型，其实质也是基本句型“If ……then ……”的延伸。
有时，在定义或定理中，需要附加说明某些成份，我们还可在“if …then …”句中插入如“where …”等的句子，加以延伸（见后面例子）。
总之，绝大部分（如果不是全部的话）数学术语的定义和定理的叙述均可采用本附录中各种格式之。
（a ）How to define a mathematical term?Something something The union of A and B is defined as the set of those elements which are in A, inBor in both.The mapping , ad-bc 0, is called a Mobius transformation.Something something(or adjective) The difference A-B is defined tobe the set of all elements of A which are notin B.A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.Real numbers which are greater than zero are said to be positive.3. We something to be something.We define the intersection of A and B to be the set of those elements common to both A and B. We call real numbers that are less than zero (to be) negative numbers.4. 如果在定义某一术语之前，需要事先交代某些东西（前提），可用如下形式：, then…) be an n-tuple of real numbers. Then the set of all such n-tuples is definthe Euclidean n-space R.Let d(x,y) denote the distance between two points x and y of a set A. Then the numberD=is called the diameter of A.5．如果被定义术语，需要满足某些条件，则可用如下形式：If…, then…If the number of rows of a matrix A equals the number of its columns, thenis called a square matrix.If a function f is differentiable at every point of a domain D, then it is said to be analytic in D.6.如果需要说明被定义术语应在什么前提下，满足什么条件，则可用下面形式：is calledis said to beLetSuppose…. If…then……Let f(z) be an analytic function defined on a domain D (前提条件). If for every pair or points , and in D with , we have f( ) f( ) (直接条件)，then f(z) is called a schlicht function or is said to be schlicht i n D.7. 如果被定义术语需要满足几个条件（大前提，小前提，直接条件）则可用如下形式：supposeassumeLet…and …. If…then…is called…Let D be a domain and suppose that f(z) is analytic in D. If for every pair of points and in D with , we have f( ) f( ), then f(z) is called a schlicht function.Notes:(a) 一种形式往往可写成另一种形式。
数学专业英语第二版吴炯圻
数学专业英语第二版吴炯圻Revised by BETTY on December 25,2020数学专业英语【第二版】1-A 什么是数学数学来自于人的社会实践，例如，工业和农业生产、商业活动、军事行动和科研工作。
与数学反过来，为实践服务和所有字段中的伟大作用。
没有现代的科学和技术分支机构可以定期制定中的数学，应用无早有需要的人来了数字和形式的概念。
然后，开发出的几何土地和三角测量的问题来自测量的问题。
若要对付一些更复杂的实际问题，男子成立，然后解决方程未知号码，因此代数发生。
17 世纪前, 男子向自己限于小学数学，即几何、三角和代数，只有常量被认为在其中。
17 世纪产业的快速发展促进了经济和技术的进展和所需变量的数量、处理从常量到带来两个分支的数学-解析几何和微积分，属于高等数学，现在有很多分支机构，其中有数学分析、高等代数、微分方程的高等数学中的可变数量的飞跃函数理论等。
数学家研究理念和主张。
所有命题公理、假设、定义和定理都。
符号是一种特殊和功能强大的数学工具，用于表示很多时候的理念和主张。
公式、数字和图表是阿拉伯数字 1,2,3,4,5,6,7,8,9,0 与另外的符号"+"、减法"-"，乘"*"，除"\"和平等"="。
数学中的结论得到主要由逻辑推理和计算。
长期的数学史上，以中心地点的数学方法被占领逻辑扣除。
现在，由于电子计算机是迅速发展和广泛应用，计算的作用变得越来越多重要。
在我们这个时代计算不只用于处理大量的信息和数据，而且还进行一些只是可以做的工作较早前的逻辑推理，例如，大部分的几何定理的证明。
1--B 方程方程是平等的语句的两个相等的数字或数字符号之间。
因此 (a-5） = 一 5a 和 x 3 = 5 是方程。
方程的两种——身份和方程的条件。
方程的算术或代数的身份。
这种方程中两名成员是相似的或成为相似的指示操作的性能。
数学专业英语第一讲(修改)
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e.g . Continuous Variables. The quantities treated of in the differential Calculus are regarded as capable of continuous variation in magnitude.
3、半专业性术语穿插频繁、词义多变
Eg: function, power, set,right
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数学词汇有自己的特点:
“古代用词” （包括拉丁文）占有一定的比例.新创词汇不 断涌现,合成词、派生词、以数学家名字命名的定理、公式等， 数量可观.
拉丁文：algebra 代数; geometry 几何；
e.g. Assume f(n) is an ascending function and let L denote the least upper bound of the set of function values.
翻译:
假设f(n)是一个增函数,用L表示函数值集合的最小上界.
7.5 表示原因句型常用since , as , owing to 等引起的从 句,少用because(非常强烈的因果关系才用!).
Half open interval Left limit Independent variable Remainder term Power series Linearly dependence
半开区间 左极限 自变量 余项 幂级数 线性相关
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(7)表示条件推理的句型相对固定.
B(a,r), we have h(x)d(x)h(a). B
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为什么要学习数学专业英语？
现在国际数学教育与研究交流日益频繁，很多 最新的研究成果是以英文形式呈现的。专业人员需 要有阅读翻译英语，甚至用英文写作论文的能力。 对于学习数学的学生和准备从事数学研究的人员， 在掌握了公共英语的基本知识的基础上，如果希望 较快地掌握阅读英文版数学教程和科研资料的基本 方法，进一步学习数学专业英语是必须的。 由于数学内容的英语表达有其特殊性，阅读时 不仅要靠个人在公共英语上的基本功，而且还要靠 数学基本知识的掌握及其运用的能力。
本课程分四部分讲解：
精选课文：1、理解数学专业文章和一般英语文章写作的不同；
2、掌握数学一般的主要不同专业文章中的基本术语和普遍常用 词汇，英文习作的一般习惯。
专业文选：1、掌握数学英语专业文献和普通英语文章的不同；
2、了解专业文献方面知识。
英文数学论文写作基础：掌握英文论文的基本写作格式、
用词和符号等方面的规范。
翻译： 所有企业现在必须根据市场来作出决策.
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3. 主动语态句型也多数用于强调事实，而 不是强调行为发出者及其情感
Eg1:Given 0 ，there exists a n N .
Eg2: Since h(x) is harmonic on neighborhood of B(a,r), we have h( x)d ( x) h(a ) .
开设本课程目的
1、为数学专业本科大学生阅读英文数学文章 和发表数学英文文章做培训，同时扩大学生英 语知识面，为相关大学生的研究生考试做英语 保温和一定强化。 2、掌握数学专业词汇，培养快速阅读数学专业 英语的能力。了解数学的学科分类，英文学术 论文摘要的检索，数学文章的英译汉和汉译英 的技巧。
3、能用英语书写文章摘要、学术会议通知、 学术交流信件等。同时培养简单的英语会 话能力。 4、为部分优秀学生攻读研究生奠定数学专 业英语的基础，同时让大部分同学了解数 学专业英语与生活英语的区别，为今后走 上工作岗位，特别是服务于IT业或外资企 业有独当一面的能力。
英语每个简单句中，只能用一个谓语动词，如果读到几个动作，就必须 选出主要动作当谓语，而将其余动作用非谓语动词形式，才能符合英语 语法要求. 非谓语动词有三种：动名词、分词（包括现在分词和过去分词）和不定 式. 例如： 要成为一个名符其实的内行，需要学到老. 这句中，有“成为”、“需要”和“学”三个表示动作的词，译成英语 后为： To be a true professional requires lifelong learning. 可以看出，选好“需要”（require）作为谓语，其余两个动作：“成为 ”用不定式形式 to be，而“学”用动名词形式learning，这样才能符合 15 英语语法要求.
翻译：
因为序列有界，由实数系统的公理10，该序列有 最小上界.
Axiom A 公理A
Axiomatic approach 公理法 Axiomatization 公理化 Axiom of induction 归纳法公理
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7.6 表示推理的根据常用“by 短语”，有时也用 according to. e.g. By lemma 2, we have x=y.
查阅文献基本知识：
1、了解文献查阅是做科研必不可少的能力； 2、掌握关键词索引查阅办法、学科分类查阅等办法。
学习本课程的基本要求：
1、认真自学好教材，并勾画出学习重点。 2、积极参与本课程的讨论。 3、按时独立完成平时作业。
1.1 数学专业英语的基本特点
特点一：注意对客观事实与真理的描述 1. 时态常用一般现在时
Eg1:The Fermat Conjecture has been proved to be true. Eg2: The function idea may be illustrated schematically in many ways.
e.g . To obtain the root (roots) of an equation is called solving an equation 翻译：
翻译： 连续变量. 微分学里所研究的量，一般认为在大小上
是可以连续变化的.
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科技英语，有三分之一以上用被动语态.例如：
(a) No work can be done without energy. 翻译： 没有能量决不能做功.
(b) All business decisions must now be made in the light of the market.
度无关的数.
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e.g . Continuous Variables. The quantities treated of in the differential Calculus are regarded as capable of continuous variation in magnitude.
假设D是开集，闭包在G中.
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7.3 用such that为连接词的从句表示条件或者补充条件.
翻译：
such that 也可以用satisfying 代替. 可用虚拟语气表示条件.
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7.4 用特殊动词(suppose , bet , set, assume )的命令式表 示假设(大前提)等意义. e.g. Assume f(n) is an ascending function and let L denote the least upper bound of the set of function values. 翻译：
派生词：interdependence 互相依存；hypergeometric 超几 何的.
数学家命名：Laplace equation 拉普拉斯方程
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合成词举例：
Half open interval
半开区间
左极限 自变量 余项 幂级数
Left limit
Independent variable Remainder term Power series Linearly dependence
7.7 有时用现在分词表示：经过…………而得到（结 论）. e.g. Integrating the above inequality twice , we see that…… 翻译：
将上一不等式两次积分得到………
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(8)数字特殊符号及其表示形式.
8.1充分必要条件：if and only if ; when and only when; sufficient and necessary condition.
翻译：
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8.2
表示事先任意取定的量.
翻译：
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8.3
表示一个结论成立的范围.
8.4 逻辑符号的使用.
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课堂作业 1
Definition Suppose the function y f ( x) is defined on some neighborhood of x0 , if lim y lim f ( x0 x) f ( x0 ) 0 , then x0 x0 we call that y f ( x) is continuous at the point x0 .
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课堂作业 2
定理 2. 设 f ( x) 在区间 a, b 上有界,且只有有限个间断点, 则 f ( x)在区间 a, b 可积.
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假设f(n)是一个增函数，用L表示函数值集合的最小上界.
7.5
表示原因句型常用since , as , owing to 等引起的从
句，少用because(非常强烈的因果关系才用！).
22
e.g. Since the sequence is bounded, it has a least upper bound by Axion 10 of the real-number system.
9
e.g . A ratio is always an abstract number; i.e., it has no units, a number considered apart from the measured units from which it came.
翻译： 比例是一个抽象的数，即没有单位，它是一个与测
线性相关
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(7)表示条件推理的句型相对固定.
7.1 用if 、where、 as 为连接词的条件从句. e.g. The function f(x) approaches infinity as x tends to zero. 翻译：
当x趋于0时， 函数 f(x)趋于无穷大.
7.2 用with 短语表示条件或补充条件. e.g. Suppose D is an open set with its closure in G. 翻译：
e.g . One and two make three ;
One plus two is three. A right angle is a 90 angle. Equations are of two kinds-identities and equations of condition.
2. 被动语态出现频率高，应用范围广
求方程根的过程称为解方程.
e.g . Two sets A and B are said to be equal if they consist of exactly the same elements in which case we write A=B.
翻译：
如果两个集合A、B所有的元素相同，称其相 等，记为A=B.
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课堂作业 1
定义 设函数 y f ( x) 在点 x0 的某一邻域内有定义 , 如 果 lim y lim f ( x0 x) f ( x0 ) 0 , 那么就称函数 y f ( x) x0 x0 在 x0 连续.