Basic Concepts in Modern Mathematics

John Edward Hafstrom

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486316277
• Topic: Matematica
• Subtopic: Riferimenti matematici

An in-depth survey of some of the most readily applicable essentials of modern mathematics, this concise volume is geared toward undergraduates of all backgrounds as well as future math majors. By focusing on relatively few fundamental concepts, the text delves deeply enough into each subject to challenge students and to offer practical applications.
The opening chapter introduces the program of study and discusses how numbers developed. Subsequent chapters explore the natural numbers; sets, variables, and statement forms; mappings and operations; groups; relations and partitions; integers; and rational and real numbers. Prerequisites include high school courses in elementary algebra and plane geometry.

CHAPTER 1

INTRODUCTION

1–1 Our program of study. In studying this text, it will be our purpose to become acquainted with certain modern mathematical concepts that will help us better to understand and use mathematics. In a text such as this, which attempts to give only an elementary treatment of these concepts, we shall be able to consider only a few of their applications; in particular, we shall use these concepts to gain a stronger understanding of the structure of some familiar number systems and how these systems are related. Among the concepts that the reader will encounter for perhaps the first time are those of set, mapping, relation, group, and isomorphism. It is hoped that application of these concepts to the study of number systems will result in a new appreciation for the logic of mathematical arguments and an added respect for the powers of the human mind.

1–2 How numbers developed. From our studies in elementary grades and in high school, we are all familiar with a variety of such numbers as whole numbers, fractions, negative numbers, and so on. It is interesting to note that our experience in learning about numbers, i.e., progressing at a rate that maturity allows, roughly parallels that of the human race, except that our learning experience is compressed into a short span of a few years!

Leopold Kronecker, a celebrated German mathematician of the nineteenth century, is supposed to have said, “God created the natural numbers; all else is the work of man.” By “natural numbers” Kronecker meant the numbers 0, 1, 2, 3, . . . , which we often refer to as the “whole” numbers, or the counting numbers. We may ask whether we should take Kronecker’s statement literally, i.e., whether we should believe God handed down to man the symbols ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . together with instructions for using them. Of course not; Kronecker simply meant that God gave man a mind and an instinctive consciousness of natural numbers. It almost seems that man is born to count; we know, for example, that even a very small child is conscious of the number of persons in his immediate family and, if he owns three identical teddy bears, is conscious of the fact when one is missing. Man, himself, devised the symbols which he uses to represent the numbers of which he is conscious. We have an intuitive notion of the meaning of “number” and “number system,” but we shall make no attempt at this time to formulate definitions of these concepts; after studying sets and mappings we shall be able to state a technical definition of “number” (Chapter 4).

We have implied that the symbols ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . are not the numbers of man’s consciousness; strictly speaking, they are symbols (or names) which represent or “stand for” these numbers. However, by habit, we call these symbols the natural numbers and we shall continue to do so. Over the centuries, man has used many other sets of symbols, and the familiar symbols ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . are of fairly recent origin in western culture. The Romans used the symbols ‘I’ ‘II,’ ‘III,’ ‘IV,’ ‘V,’ ‘VI,’ . . . (the Roman numerals), and earlier civilizations used other sets of symbols for the natural numbers.¹ The early Romans and other Mediterranean peoples used no symbol for the natural number we call ‘zero,’ perhaps because they were not yet fully aware that zero actually is a natural number. Note that we are able to write any natural number we wish by using no more than ten different symbols: ‘0,’ ‘1,’ ‘2,’ ‘3,’ ‘4,’ ‘5,’ ‘6,’ ‘7,’ ‘8,’ ‘9.’ The Roman system of notation had no such feature; the Romans wrote ‘V’ for ‘5,’ ‘L’ for ‘50,’ ‘D’ for ‘500,’ and so on. In short, in order to write larger and larger natural numbers, the Romans were forced to invent more and more symbols to represent them. Computations we consider routine were very awkward for the Romans and they relied heavily on the abacus,² mankind’s first digital completing machine. After the Crusades, the Hindu-Arabic symbols ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . were introduced into western Europe, and the relatively simple rules (algorithms) for computing with them made computing a common and widely practiced art.

Today we use numbers other than only the natural numbers (numbers which, according to Kronecker, are “the work of man”) and we shall now briefly discuss how such numbers as fractions, negative numbers, etc., came into use and how they were blended with the natural numbers to form what we call the real number system.

Fractions were invented because, when man began to build temples and survey land, for example, he found that whatever standard unit for measuring length he adopted, whether it was the length of his foot, the length of his forearm, or the length of the king’s beard, he encountered lengths which did not contain the standard unit a natural number of times. Thus, he began to divide the standard units on his measuring devices into halves, thirds, quarters, and so on, and fractions were born. Records tell us that the Babylonians and Egyptians were using fractions as early as 3000 B.C.

The story of negative numbers is different. The early algebraists, mainly the Arabs, found that certain equations had no solution, if by “solution” one meant a natural number or a fraction. For example, if the only numbers familiar to him were the natural numbers and fractions, an algebraist confronted with the task of solving the equation 3x + 12 = 0 would be forced to say, “This equation has no solution; the statement 3x + 12 = 0 can never be true if x is to be a natural number or a fraction. ” To remedy this state of affairs, some bold and imaginative algebraist invented negative numbers and described the rules that must govern their use if they were to blend with the existing numbers and form, with them, a larger system of numbers which we know today as the rational number system. Today, we ordinarily call the numbers

. . ., –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

the integers and define the rational numbers to be the collection of all numbers of the form p/q, where p and q are integers and q ≠ 0 (≠ means “is not equal to”). An integer is itself a very special kind of rational number, i.e., one that may be written in the form p/1, where p is an integer. Today we take for granted the ease with which, by means of elementary algebra, we are able to find a solution for any equation of the form ax + b = c whenever b and c are rational numbers and a is a rational number other than 0; few of us are aware of the debt we owe to the anonymous inventor of negative numbers. Because the need for negative numbers was not, at first, a practical one, such as the need for fractions, for centuries negative numbers were looked upon as freak numbers that algebraists used as playthings. Later, however, as man’s knowledge of the physical world. increased, negative numbers became necessary for describing physical relationships and, by the time of the Renaissance, they had become accepted members in the society of numbers.

The need for still another kind of number first arose from the study of geometry. In about 500 B.C. Greek geometers proved that, using a given unit of length, they could construct a line segment whose length did not contain the given unit a rational number of times. For example, if we construct a right triangle whose legs are each one inch in length, the hypotenuse, by the Theorem of Pythagoras, will have a length equal to

(the “square root” of 2) inches, where

is a “number” whose square is 2. The number

is called an irrational number; that is, it is not possible to express this number as the ratio of two integers. In other words, it is not possible to find a number of the form p/q, where p and q are integers and q ≠ 0, whose square is 2. Algebraists were also forced to deal with irrational numbers in solving quadratic equations; an innocent-appearing equation such as x² – 5 = 0 has two solutions,

and –

both of which are irrational numbers. Fortunately, it was found that it was possible to blend irrational numbers with rational numbers so that, together, they formed a larger family of numbers which we today call the real number system. This system now meets most of man’s everyday needs. (The reader interested in knowing more about the historical development of the real number system is urged to consult such books on the history of mathematics as Howard Eves, An Introduction to the History of Mathematics, Rinehart.)

1–3 The mathematician’s view of the development of numbers. The previous discussion of the development of numbers may have given the impression that negative numbers, rational numbers, and irrational numbers have always existed but had remained hidden from view, waiting only to be discovered so that, along with the natural numbers, they could be fitted together like pieces of a jigsaw puzzle to form the real number system. This is not the mathematician’s point of view; remember Kronecker’s quotation: “. . . . all else is the work of man.” Most mathematicians prefer to think of the development of the real number system as a process of orderly and logical growth which progressed from the natural numbers to the integers, from the integers to the rationals, and, finally, from the rationals to the reals; they look at the development of the real numbers as they would like to have planned it. They are like the “Monday morning quarterbacks” who replay last Saturday’s football game; they say, “We would have done things differently.” Almost a hundred years ago mathematicians began to be bothered by the fact that real numbers developed without their supervision, like plants in an untended garden. Fortunately, however, they were able to prove that the end product (the real numbers) of this unsupervised growth is good; in other words, they proved that there is a logical plan which, had it been followed, would have yielded the real number system as we know it today. We shall learn something of this plan in Chapters 7, 8, and 9.

1–4 A word to the reader. If you have had a course in plane geometry, you may recall that your study began with a few such undefined concepts as “point” and “line,” and these, together with basic definitions and assumptions (axioms), were used to prove geometric properties of triangles, circles, polygons, and so on. It was not permissible to make a statement unless it could be defended by quoting appropriate definitions, axioms, or previously proved theorems. In this text we shall follow a somewhat similar procedure; we shall start with certain undefined concepts, a few definitions, and some assumptions about the arithmetical properties (structure) of the natural number system. We shall then use these to construct several other number systems (particularly the integers, the rationals, and the reals) and to study their structures. Above all, we shall stress the importance of making precise statements and being able to defend them.

The importance of the role that definitions will play in our study cannot be overemphasized. Each new concept will usually be introduced by the statement of one or more definitions, and each new definition should be immediately memorized. Of course, mere memorization of a definition does not guarantee that it will be understood or appreciated, but it is a first step toward understanding. After each new definition is presented, several clarifying examples will usually be given to aid in understanding it, and perhaps to indicate how it is to be applied. At this point it will be helpful to try to restate the definition in your own words in as many ways as possible but, in rephrasing, you must be careful to neither add nor subtract meaning from the definition as originally stated. If at first the significance of a new definition is difficult to grasp, you will find it rewarding to return to it again and again. If you thoroughly understand each step as it is presented, your work will be made easier in the long run.

1–5 Numbers and numerals. Doubtless, the reader has observed (Section 1–2) that single quotes were used to make a distinction between the numbers 0, 1, 2, 3, . . . and the symbols (or names) ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . for these numbers. Logically, this distinction is one that should always be made for there is a difference between a thing and its name. We wrote 0, 1, 2, 3, . . . when the reader was to think of the natural numbers and ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . when he was to interpret 0, 1, 2, 3, . . . as symbols or names for these numbers. The symbols ‘0,’ ‘1,’ ‘2,’ ‘3,’ . . . are called numerals.

In the chapters that follow we will not distinguish between things and their names and, in particular, we will not distinguish between numbers and numerals. In other words, we will omit single quotes when writing numerals. In cases where a distinction between number and numeral should be made it will be left to the reader to determine from the context whether or not single quotes logically ought to be present. For example, the reader should be able to correctly interpret 2 as the number 2 or the numeral ‘2’ from the context in which it occurs. For further reading on the distinction between things and the names of things, the excellent book An Introduction to Modern Mathematics by Robert W. Sloan (Prentice-Hall, 1960) is recommended. In particular, refer to Chapter 1.

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¹ J. Houston Banks, Elements of Mathematics, New York, Allyn and Bacon, 1956 (Sec. 2.1–2.6).

² Robert L. Swain, Understanding Arithmetic, New York, Rinehart, 1957 (Ch. 1).

CHAPTER 2

THE NATURAL NUMBERS

2–1 Introduction. When we write the natural numbers (the counting numbers) as 0, 1,...