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.

cube   立方

induction   归纳法