数学专业英语第2章课后答案上课讲义

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2.1

2.比:ratio 比例:proportion 利率:interest rate 速率:speed 除:pide 除法:pision 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:percentage

3.(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.

2.2

2.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:adjacent side 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 as

3.(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 are

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known, 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.

2.3

2.素数:prime 合数:composite 质因数:prime factor/prime pisor 公倍数:common multiple 正素因子: positive prime pisor 除法算式:pision equation 最大公因数:greatest common pisor(G.C.D) 最小公倍数: lowest common multiple(L.C.M) 整除:pide by 整除性:pisibility 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 to

3.(1)We need to study by which integers an integer is pisible, 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 pisible by 1 and itself, can also be pisible 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.

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(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.

2.4

2.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 axiom

3.(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.

2.5

2.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

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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:quadratic equation 判别式:discriminant 行列式:determinant

3.(1)In the planar rectangular coordinate system, one can set up a

one-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 hyperbola

or 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）。一个方程组的图像给了我们关于解的信息。而代数解给了我们关于图像的信息。一个方程组的代数和它的解析几何之间的

相互作用引人注目并且有用。

（3）考虑一个非退化的以Q为圆心的实圆，设P不等于Q是在经过Q且垂直于原所在平面的任意的一个实值定点。在连接P和圆上的点得到直线簇的点的

轨迹称为正圆锥。

2.6

2．向量：vector 单位向量：unit vector 法向量：normal vector 位置向量：position vector 基点：base point 向量的尖端：tip of a vector 向量的分量：component of a vector 点乘：dot product参数方程：parametric equation 位移：displacement 在集合上：geometrical 平行线：parallel lines 平行于：parallel to 平行四边形：parallelogram 不平行的：nonparallel 垂直的：perpendicular指向：point 点：point 加：add 乘以：multiply by

3.(1)In space any two parallel vectors are collinear; two vectors can determine a plane if and only if they are nonparallel.

(3)The product a A of a real number a and a vector A is still a

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vector which has length |a||A| and the same direction as A if a is a positive number and the opposite direction if a is a negative number.

(5)By giving a point on a line and a nonzero vector parallel to the line, the parametric equation of the line can be obtained. A line can be expressed as a system of equations consisting of the equations for two plane.

4.(1)众所周知，笛卡尔用一个数对来定位平面上的一个点，用一个数的三元数组来定位空间的一个点。当这个思想被推广为n元实数组u（a1，a2，……an），对于任何整数N>=1,这个n元数组称为一个n维点或者一个n维向量，各个数

a1，a2，……an称为这个向量的坐标或者分量。

(3)向量x1,x2,……xn称为线性相关的，如果存在不全为0的标量使得

a1x1+a2x2+……anxn=0.向量x1，x2……xn称为线性无关的，如果x1，

x2……xn不是线性相关的。

2.7

2.向量空间：vector space 行向量：row vector 列向量：column vector 线

性相关：linearly dependent 线性无关:linearly independent 线性组

合:linear combination 数量级:scalar

product 矩阵:matrix 方阵:square matrix 行列式:determinant 逆矩

阵:inverse matrix 单位矩阵:identity matrix 零矩阵:zero matrix 变

换:transformation 到上的:onto 同

构:isomorphism 同构的:isomorphic 应用微分方程:applied differential equations 数理经济:mathematical economics 量子力学:quantum mechanics 相容的:consistent 最终的:ultimately

3.(1)Linear combination, linear dependence and linear independence

are all important concepts of linear spaces.

(3)Not only matrix can be used to solve a system of linear equations, but also can be used to judge whether the system of equations have solutions and whether the solution is unique.

(5)This conclusion is contradictory to the hypothesis of the problem, so the proposition to be proved is true.

(7)let V be an n-dimensional vector space over the field F and W be an m-dimensional vector space over the field F. Let B and B' be

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