1、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，
2、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 l
3、inear 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
4、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。 Mathematics comes from mans social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branche
5、s could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。Before 17th century, man confined hi
6、mself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值
7、。 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，可以用它来解决许多实际应用问题。Equations are of very great use. We can use equations in many mathematical problems. （8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。 To solve the equation means to mov
8、e 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. 英译汉1.Algebra has evolved from the operations and rules of arithmetic.The study of arithmetic begins with addition,multiplication,subtraction,and division o
9、f numbers:4+7,37682,49-22,408.In algebra we introduce symbols or letterssuch as a,b,c,d,x,y,zto denote arbitrary numbers and，instead of special cases，we often consider general statements：a+b,cd,x-y,x a.代数是从算术的运算和规则当中逐渐发展起来的，算术的研究是从数的加减乘除开始的。例如 4+7， 37682，49-22，408。在代数学里,我们采用符号或字母。例如 a,b,c,d,x,y,z 来表
10、示任意的数字，而不考虑那些特殊情况。我们经常考虑的是一般的表达式，例如 a+b,cd,x-y,xa。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 many specific statements.代数的语言有两个作用。第一个是使用它作为一种速记法去缩减和
11、减化那些又长又复杂的表达。第二，它被证明是一种概括许多具体的表达方式的便捷途径。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 the expression 2(3+4),we do the addition first and then
12、 we do the multiplication:2(3+4)=2(7)=14.许多数学表达式包含两个或更多的运算。分组符号告诉我们哪一个运算先做。常见的分组符号是圆括号，方括号和分数线。例如，在数学表达公式 2（3+4）里。我们先做加法再做乘法 2（3+4）=2 （7）=142.2 几何与三角词组翻译1.学会 institution，建筑师 architect, 机械师 machinist, 制图员 draftsman, 测量者surveyor, 木匠 carpenter2.点 point, 端点 endpoint, 线 line, 直线 straight line, 线段 line se
13、gment, 曲线 curved line, 折线 broken line, 射线 ray , 平面 plane,曲面 curved surface3.立体 solid, 柱体 cylinder, 立方体 cube,球 sphere, 棱锥 pyramid,圆锥 cone ,4.圆 circle,圆心 center, 直径 diameter, 半径 radius, 半圆 semicircle, 弦 chord, 弧 arc, 优弧 major arc, 劣弧 minor arc5.角 angle, 边 side, 三角形 triangle, 直角三角形 right triangle,斜边 hy
14、potenuse, 直角边 right-angle side6.长度 length,宽度 breadth/width,厚度 thickness, 位置 position7.几何的 geometrical,立体的 three-dimensional , 弯曲的 curved,等距离的 equidistant ,无限的 infinite8.培养创造力 train originality,必须的毅力 necessary perseverance , 提高鉴赏力 raise/improve the appreciation ability9.消失了的边界 vanishing boundaries/la
15、ndmarks,有序性和优美感 orderliness and sense of beauty, 几何图形大量存在 geometric forms abound in , 定理成立的先决条件 a prerequisite to a theorem 汉译英（1）许多专家都认为数学是学习其他科学技术的必备基础和先决条件。 Many experts recognize that mathematics is the necessary foundation and prerequisite of studying other science technology. （2）西方国家的专家认为几何起源于
16、巴比伦和埃及人的土地测量技术，其实中国古代的数学 家对几何做了许多出色的研究。 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 geometry. （3）几何的学习使学生在思考问题时更周密和审慎，他们将不会盲目接受任何结论。 In studying g
17、eometry, the student is taught to think clearly and critically and he is led away from the practice of blind acceptance of any conclusions. （4）数学培养学生的分析问题的能力，使他们能应用毅力、创造性和逻辑推理来解决问题。 Studying mathematics can develop the students ability to analyze problems and utilizing perseverance, originality, and
18、 logical reasoning in solving the problem. （5）几何主要不是研究数，而是形，例如三角形，平行四边形和圆，虽然它也与数有关。 Geometry mainly studies hot numbers but figures such as triangles, parallelograms and circles, though it is related with numbers. （6）一个立体（图形）有长、宽和高；面（曲面或平面）有长和宽，但没有厚度；线（直线 或曲线）有长度，但既没有宽度，也没有厚度；点只有位置，却没有大小。 A solid (f
19、igure) 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, but no width and thickness. A point has position, but no dimension. （7）射线从某个点出发无限延伸；两条从同一点出发的射线构成了角。这两条射线称为这个 角的两边，当这两边位于同一直线上且方向
20、相反时，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）平面上的闭曲线当其中每一点到一个固定点的距离均相等时叫做圆。这个固定点称为圆 心，经过圆心且其两个端点在圆周上的线段称
21、为这个圆的直径，直径的一半叫做半径，这条 曲线的长度叫做周长。 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 endpoints on the circle. Half of the diameter is ca
22、lled radius. The length of the circle is called circumference.英译汉 1.In geometry an angle is defined as the set of points determined by two rays l and l having the same endpoint O. 在几何学里从同一点 O 出发引出的两12条射线 l 和 l 所组成的点的集合叫做角。 2.In trigonometry we often interpret angles as rotations of rays.To obtain an
23、 angle we may start with a fixed ray l having endpoint O,and rotate it about O,in a plane,to a 1position specified by ray l .We call l the initial side, l the terminal side,and O the 212vertex of angle. 在三角学里，我们经常解释角就是射线的旋转。在平面上，我们许会从端点是 O 的射线 l 开始让它绕着端点 O 旋转，转到一个位置，由射1线 l 标注。我们把 l 叫做角的始边，l 叫做角的终边，O
24、 叫做角的顶点。 2 23.A right angle is a 90 angle . An angle is acute if 0 0，函数 g(x)表示不超过 x 的素数的个数。 For a given real number x0, 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
25、 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. 2.7 序列及其极限 序列及其极限 （1）序列各项对 n 的相关性常利用下标来表示，写成如下形式： a n , x n 等。 The dependence of every team of sequence on n is denoted by using su
26、bscript, and we write a n , x n and so on. （2）以正整数集为定义域的函数称为序列。 A function whose domain is the set of all positive integers is called an infinite sequence. （3）一个复值序列收敛当且仅当它的实部和虚部分别收敛。 A complex-valued sequence converges if and only if both the real part and the imaginary part converge separately. （4
27、） 一个序列 a n 若满足： 对任意正数 ， 存在另一个正数 N （N 可能与 有关） 使得 a n - L 对所有 n N 成立，就称 a n 收敛于 L。 A sequence a n is said to have a limit L if, for every positive number , there is another positive number N (which may depend on ) such that In this case, we say the sequence a n converges to L. an ? L for all n N. （5）
28、重要的是， 该集的每一个成员都用一个正整数标上记号。 这样一来， 就可以谈论第一项、 第二项和一般项，即第 n 项。 The important thing is that each member of the set has been labeled with an integer so that we may speak of the first term, the second term and in general, the nth term. （6）若无另加申明，本章研究的序列都假定具有实的项或复的项。 Unless otherwise specified, all sequence
29、s in this chapter are assumed to have real or complex terms. （7）作为日常用语，sequence 和 series 是同义词；但作为数学术语，它们表示不同的概念。 In everyday usage of the English language, the words “sequence” and “series” are synonyms, but in mathematics these words have special technical meanings. （8）术语“收敛序列 ”指的是具有有限极限的序列，因此，极限为无
30、限的序列不是收敛的， 而是发散的。 The phrase “convergent sequence” is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge not convergence. 2.8 函数的导数和它的几何意义 （1）差商表示函数 f 在连接 x 与 x+h 的区间上的平均变化率。 The different quotient is referred to as the average rate of the change of
31、f in the interval joining x to x+h. （2）速度等于位置函数的导数。 Velocity is equal to the derivative of positing. （3）由定义导数的过程所提供的几何解释以一种自然的方式导出了关于曲线的切线思想。 The procedure used to define the derivative has a geometric interpretation which leads in a natural way to the idea of a tangent line to a curve. （4）差商表示直线 PQ
32、 与水平线的夹角的正切。 The difference quotient represents the trigonometric tangent of the angle that PQ makes with the horizontal. （5）在直线运动中，速度的一阶导数称为加速度。 For rectilinear motion, the first derivative of velocity is called acceleration. （6）我们约定 f（0）=f ，即函数 f 的零阶导数就等于它本身。 We make the convention that f（0）=f, th
33、at is the zeroth derivate is the function itself. （7）在运动的 9 秒钟内，物体的速度由 v (0) = -144 变成了 v (9) =144，也就是说，速度总共 增加了每秒 288 英尺。 During the 9 seconds of motion the velocity changes from v (0) = -144 to v (9) =144, that is, the total increase in velocity is 288 feet per second. （8）当 从 0 增加到 /2 时，tan 所对应的直线
34、趋于竖直位置。As increases from 0 to /2 , tan approach a vertical position. 2.9 微分方程简介（1）此时，微分方程就有无穷多个解，C 的每个值对应一个解。The differential equation has infinitely many solutions, one for each value of C.（2）微分方程的阶指的是方程中最高阶导数的阶。By the order of an equation is meant the order of the highest derivative which appears.
35、（3）我们可以由已知的粒子运动速度或者加速度计算出粒子的位置。We could try to compute the position of a moving particle from a knowledge of its velocity or acceleration.（4）如果一个微分方程的未知函数是多元函数，则称为偏微分方程。Ordinary and partial, depend on whether the unknown is a function of just one variable or of two or more variables.（5）微分方程的研究直接受到力学
36、、天文学和数学物理的推动。The study of differential equations has been directly inspired by mechanics, astronomy, and mathematical physics.（6）许多应用问题要求我们从方程的解集中选出一个在某个点具有指定值的解。In many problems it is necessary to select from the collection of all solutions one having a prescribed value at some point.（7）确定满足边界条件的解的
37、问题称为边值问题。The problem of determining such a solution that satisfies boundary condition is called a boundary-value problem.（8）人们设计许多高速运行的计算机来对各种积分做出近似估计。Automatic high-speed computing machines are often designed with this kind of problem in mind.2.10 线性空间中的相关与无关集（1）该式的两边同时关于 t 积分，我们就得到一个所需要的结论。Integra
38、ting both sides of this formula with respect to t. we can obtain a conclusion we need.（2）不难看出，这个命题仅仅建立在该空间是线性的这一事实上，与空间的其他性质无关。We clearly find that this proposition is based only on the fact that this space is a linear space and not on any other special property of this space.（3）如果空间不存在有限基，就称该空间是无限维
39、的。A space is called infinite dimensional if it doesnt have a finite basis.（4）假定这个结论对 n-1 个指数函数成立，我们将证明此结论对 n 个指数函数也成立。Assuming the conclusion is true for n-1 exponential functions, we will prove that it is true for n exponential function.（5）这两个定义在逻辑上是互相等价的。These two definitions are logically equival
40、ence.（6）设 X 是线性空间 V 中 k 个元素组成的一个线性无关集合，L（X）是由 X 张成的子空间。那么，L（X）的每一个元素都可以表示成 X 的元素的线性组合。Let X be an independent set consisting of k elements in a linear space V and let L（X ）be the subspace spanned by X, then each element of L(X) can be expressed as a linear combination of element of X.（7）设 V 是一个 n 维线性空间，考虑它的一个基，其元素按给定的次序排列为 ， 。Let V be a linear space of dimension n and consider a basis whose elements , , , are take in a given order.（8）该线性表示的系数构成一个 n 元组，它由向量 x 唯一确定。The coefficients in this linear representation determine an n-tuple of numbers that is uniquely determined by x.
