Mathematical English Dr. Xiaomin Zhang
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§2.4 Integers, Rational Numbers and Real numbers
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TEXT A Integers and rational numbers
There exist certain subts of R which are distinguished becau they have special properties not shared by all real numbers. In this ction we shall discuss two such subts, the integers and the rational numbers.
To introduce the positive integers we begin with the number 1, who 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 becau 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 em
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clear, in a careful treatment of the real-number system it is necessary to give a more preci definitio
n of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive t.
培根人物介绍DEFINITION OF AN INDUCTIVE SET A t of real numbers is called an inductive t if it has the following two properties:
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(a) The number 1 is in the t.
佑福(b) For every x in the t, the number x+1 is also in the t.
For example, R is an inductive t. So is the t R+. Now we shall define the positive integers to be tho real numbers which belong to every inductive t.
DEFINITION OF POSITIVE INTEGERS A real number is called a positive integer if it belongs to every inductive t.
Let P denote the t of all positive integers. Then P is itlf an inductive职代会主持词
t becau (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P belong to every inductive t, we refer to P as the smallest inductive t. This property of the t P fo
忽视rms 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 t Z which we call simply the t of integers.
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 b 0) are called rational number. The t长相思汴水流
of rational numbers, denoted by Q, contains Z as a subt. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the t of rational numbers is an ordered field. Real numbers that are not in Q are called irrational.
Notations
Field axioms A field is any t 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 inver, but multiplication is not necessarily commutative.
Order axioms A total order (or "totally ordered t," or "linearly ordered t") is a t plus a relation on the t (called a total order) that satisfies the conditions for a partial order plus an additional condition known as
