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数学专业英语－The Real Number System
The Real-Number System
The real-number system is collection of mathematical objects, called real number, which acquire mathematical life by virtue fundamental principles, or rules, that we adopt. The situation is somewhat similar to a game, like chess, for example. The chess system, or game, is a collection of objects, called chess pieces, which acquire life by virtue of the rules of the game, that is, the principles that are adopted to define allowable moves for the pieces and the way in which they may interact.
Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system. However, to establish a common ground of understanding and avoid certain errors that have become very common, we shall explicitly state and illustrate many of these principles.
The real-number system includes such numbers as –27,-2,2/3,… It is worthy of note that positive numbers, 1/2, 1, for examples, are sometimes expressed as +(1/2), +1. The plus sign, “+”, used here does not express the operation of addition, but is rather part of the symbolism for the numbers themselves. Similarly, the minus sign, “-“, used in expressing such numbers as -(1/2), -1, is part of the symbolism for these numbers.
Within the real number system, numbers of various kinds are identified and named. The numbers 1, 2, 3, 4,… which are used in the counting process, are called natural numbers. The natural numbers, together with–1,-2,-3,-4,…and zero, are called integers. Since 1,2,3,4,…are greater than 0, they are also called positive integers; -1,-2,-3,-4,…are less than 0, and for this reason are called negative integers. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero. The integers are included among the rational numbers since any integer can be expressed as the ratio of the integer itself and one. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.
One of the basic properties of the real-number system is that any two real numbers can be compared for size. If a and b are real numbers, we write a<b to signify that a is less than b. Another way of saying the same thing is to write b>a, which is read “b is greater than a “.
Geometrically, real numbers are identified with points on a straight line. We choose a straight line, and an initial point f reference called the origin. To the origin we assign the number zero. By marking off the unit of length in both directions from the origin, we assign positive integers to marked-off points in one direction (by convention, to the right of the origin ) and negative integers to marked-off point in the other direction. By following through in terms of the chosen unit of length, a real number is attached to one point on the number line, and each point on the number line has attached to it one number.
Geometrically, in terms of our number line, to say that a<b is to say that a is to the left of b; b>a means that b is to the right of a.
Properties of Addition and Multiplication
Addition and multiplication are primary operations on real numbers. Most, if not all, of the basic properties of these operations are familiar to us from experience.
(a) Closure property of addition and multiplication.
Whenever two real numbers are added or multiplied, we obtain a real number as the result. That is, performing the operations of addition and multiplication leaves us within the real-number system.
(b) Commutative property of addition and multiplication.
The order in which two real numbers are added or multiplied does not affect the result obtained. That is, if a and b are any two real numbers, then we have (i) a+ b=b+ a and (ii) ab = ba. Such a property is called a commutative property. Thus, addition and multiplication of real numbers are commutative operations.
(c) Associative property of addition and multiplication.
Parentheses, brackets, and the like, we recall, are used in algebra to group together whatever terms are within them. Thus 2+(3+4) means that 2 is to be added to the sum of 3 and 4 yielding 2+7 =9 whereas (2+3)+4 means the sum of 2 and 3 is to be added to 4 yielding also 9. Similarly, 2•(3•4) yields 2•(12)=24 whereas (2•3) •4 yields the same end result by the route 6•4=24 . That such is the case in general is the content of the associative property of addition and multiplication of real numbers.
(d) Distributive property of multiplication over addition.
We know that 2•(3•4)=2•7=14 and that 2•3+ 2•4=14 ,thus 2•(3+4)=2•3+ 2•4. That such is the case in general for all real numbers is the content of the distributive property of multiplication over addition, more simply called the distributive property.
Substraction and Division
The numbers zero and one. The following are the basic properties of the numbers zero and one.
(a) There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.
There is a unique real number, different from zero, called one and denoted by 1, with the property that a•1=1•a=a, where a is any real number.
(b) If a is any real number, then there is a unique real number x, called the additive inverse of a , or negative of a, with the property that a+ x = x+ a .If a is any nonzero real number, then there is a unique real number y, called the multiplicative inverse of a, or reciprocal of a, with the property that ay = ya = 1
The concept of the negative of a number should not be confused with the concept of a negative number; they are not the same. ”Negative of“ means additive inverse of “. On the other hand, a “negative number” is a number that is less than zero.
The multiplicative inverse of a is often represented by the symbol 1/a or a-1. Note that since the product of any number y and 0 is 0, 0 cannot have a multiplicative inverse. Thus 1/0 does not exist.
Now substraction is defined in terms of addition in the following way.
If a and b are any two real numbers, then the difference a-b is defined by a- b= c where c is such that b+ c=a or c= a+(-b). That is, to substract b from a means to add the negative of b (additive inverse of b) to a.
Division is defined in terms of multiplication in the following way.
If a and b are any real numbers, where b≠0, then a+ b is defined by a +b= a•(1/b) =a•b-1. That is, to divide a by b means to multiply a by the multiplicative inverse ( reciprocal)of b. The quotient a +b is also expressed by the fraction symbol a/b.
Vocabulary
real number 实数
negative 负的
the real number system 实数系
rational number 有理数
collection 集体,总体
ratio 比,比率
object 对象,目的
denominator 分母
principle 原理,规则
numerator 分子
adopt 采用
irrational number 无理数
define 定义(动词)
signify 表示
definition 定义(名词)
geometrical 几何的
establish 建立
straight line 直线
explicit 清晰的,明显的
initial point 初始点
illustrate 说明
point of reference 参考点
positive 正的
origin 原点
express 表达
assign 指定
plus 加
unit 单位
sign 记号,符号,正负号
property 性质
operation 运算,操作
closure property 封闭性质
addition 加法
commutative 交换的
multiplication 乘法
associative 结合的
substraction 减法
parentheses 圆括号
division 除法
brackets 括号
sum 和,总数
algebra 代数
procuct 乘积
yield 产生
difference 差,差分
term 术语,项
quotient 商
distributive 分配的
symbolism 符号系统
unique 唯一的
minus 减
additive inverse 加法逆运算
identify 使同一
multiplicative inverse 乘法逆运算
count 计数
reciprocal 倒数,互逆
natural number 自然数
concept 概念
zero 零
fraction 分数
integer 整数
arithmetic 算术的
greater than 大于
solution 解,解法
less than 小于
even 偶的
be equal to 等于
odd 奇的
arbitrary 任意的
square 平方
absolute value 绝对值
square root 平方根
cube 立方
induction 归纳法
Note
1. Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system.
意思是:我们对数的实际工作经验使我们大家对支配着实数系的各原则早已有了某些熟悉,这里working作”实际工作的”解,govern作”支配”解.
2.The plus sign,”+”, used here not express the operation of addition, but is rather part of the symbolism for the numbers themselves.
意思是:这里的正符号”+”不是表示加法运算,而是数本身的符号系统的一部分.
3. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero.
这是定义数学术语的一种形式.下面是另一种定义数学术语的形式.
A matrix is called a square matrix if the number of its rows equals the number of its columns.
这里is called与is said to be 可以互用,注意is called后面一般不加to be而is said后面一般要加.
4. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.
与注3比较,这里用定语从句界定术语.
5. There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.
意思是:存在唯一的一个实数,叫做零并记为0,具有性质a+0=0+a,这里(其中)a是任一实数.
1) 这里called和denoted都是过去分词,与后面的词组成分词短语,修饰number.
2) with the property是前置短语,修饰number.
3) 注意本句和注3.中where的用法,一般当需要附加说明句子中某一对象时可用此结构.
Exercise
I. Turn the following arithmetic expressions into English:
i) 3+(-2)=1 ii) 2+3(-4)=-10
iii) = -5 iv) =3
v)2/5-1/6=7/30
II. Fill in each blank the missing mathematical term to mark the following sentences complete.
i) The of two real numbers of unlike signs is negative.
ii) An integer n is called if n=2m for some integer m.
iii) An solution to the equation xn=c is called the n is of c.
iv) If x is a real number, then the of x is a nonnegative real number denoted by |x| and defined as follows
x, if x 0
|x|= -x, if x <0
III. Translate the following exercises into Chinese:
i) If x is an arbitrary real number, prove that there is exactly one integer n such that x<n<x+1.
ii) Prove that there is no rational number whose square in 2.
iii) Given positive real numbers a1,a2,a3,…such that an<can-1 for all n>2, where c is a fixed positive number, use induction to prove that an<cn-1a1, for all n>1.
iv) Determine all positive integers n for which 2n<n!
Ⅳ Translate the following passage into Chinese:
There are many ways to introduce the real number system. One popular method is to begin with the positive integers 1,2,3,…and use them as building blocks to construct a more comprehensive system having the properties desired. Briefly, the idea of this method is to take the positive integers as undefined concepts, state some axioms concerning them, and them use the positive integers to build a larger system consisting of the positive rational numbers. The positive irrational numbers, in turn, may then be used as basis for constructing the positive irrational numbers. The final step is the introduction of the negative numbers and zero. The most difficult part of the whole process is the transition from the rational numbers to the irrational num
Ⅴ. Translate the following theorems into English:
1. 定理A: 每一非负数有唯一一个非负平方根.
2. 定理B: 若x>0, y是任意一实数,则存在一正整数n使得nx > y.
Ⅵ. 1. Try to show the structure of the set of real numbers graphically.
2. List and state the laws that operations of addition and multiplication of real numbers obey.
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发表于 2006-1-10 19:18
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