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    The Number System
    The Number System
    The Number System
    Ebook197 pages10 hoursDover Books on Mathematics

    The Number System

    By H. A. Thurston

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    About this ebook

    The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's underlying concepts and their logical development.
    This volume's great merit lies in its wealth of explanatory material, designed to promote an informal and intuitive understanding of the rigorous logical approach to the number system. The first part explains and comments on axioms and definitions, making their subsequent treatment more coherent. The second part presents a detailed, systematic construction of the number systems of rational, real, and complex numbers. It covers whole numbers, hemigroups and groups, integers, ordered fields, the order relation for rationals, exponentiation, and real and complex numbers. Every step is justified by a reference to the appropriate theorem or lemma. Exercises following each chapter in Part II help readers test their progress and provide practice in using the relevant concepts.
    LanguageEnglish
    PublisherDover Publications
    Release dateOct 23, 2012
    ISBN9780486154947
    The Number System

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      Book preview

      The Number System - H. A. Thurston

      Part I

      EXPLANATORY TREATMENT

      CHAPTER I

      Counting

      1. Numbers are used for counting; that is to say, for comparing the many-fold-ness of groups of objects. If we want to know whether there are more states in the United States than counties in England, the obvious thing to do is to count first the states and then the English counties. But the most obvious process is not in fact the simplest. We can see at a glance that there are more dots in [……..] than in […]. Some animals can distinguish between groups of different sizes (e.g. between a group of three men going into a hut, and a group of two of them coming out again), but there is no evidence that they can count.

      The difference between counting and comparison without counting is brought out clearly if we imagine a primitive tribe whose language contains no numbers above 20. The tribesmen can then distinguish between groups of 17 and of 18 by counting, but not between groups of 25 and of 75. By the comparison method, on the other hand, the 25 and the 75 are far more easily distinguished than are the 17 and the 18. Comparison does not, however, depend on merely looking at the two groups. The crude method can be refined.

      2. Let us imagine that a master of ceremonies wants to know whether there are more men or more women at a dance, and that he uses the comparison method, not just by glancing at the dancers but by asking everyone to take a partner. If there are men left over when all the women are partnered, then there are more men than women, and vice versa; if there is no one left over, then the sexes are present in equal strength. This process, simple though it is, is important in modern mathematics. I shall use the word matching to describe the pairing of the members of two sets,* and I shall call two sets matchable if they can be paired against each other with no members left over. One set is of smaller size than another if and only if it is matchable with a set consisting of some members of the second set, but is not matchable with the second set itself.

      3. A peasant farmer will perhaps be able to tell infallibly when one cow is more valuable than another. A more sophisticated merchant will not be content with this; he will want their values in terms of money. In other words, he wants a standard scale—the scale of money-values—instead of mere comparisons. We can construct a scale for our sizes-of-sets.

      Consider [.], [..], […], [….], and so on. If a set is matchable with the set of dots in [.], we say that the number of members of the set is one. If it is matchable with the set of dots in [..], the number is two. Similarly for three, four, and higher numbers. We have now practically reached the stage of counting: counting is simply replacing the rather clumsy standard set by signs. The trick is to use a set of signs arranged in a fixed order. Let us suppose that we use the familiar signs 1, 2, 3, 4, etc. Then for our standard sets we take, not sets of dots, but sets of these signs, in the fixed order

      When we match a set of objects against one of the standard sets, we do not need to remember the whole standard set: it is enough to know which is the last number in it. Now at last we are really counting. We have used the fact that the set of standard sets is matchable with the set of number-signs, and have replaced each standard set by the number-sign with which it is paired. The matching can be illustrated thus:

      Usually, matchings are written in the form

      or by a general formula such as [1, 2, …, n] ↔ n.

      1 and 2 are the mates of [1] and of [1, 2] respectively.

      4. I want to emphasize that all we need for counting is a set of signs in a fixed order going on for ever. (Clearly the set must not ever come to an end: if it did we should not be able to count sets larger than the set of available signs.) The signs may be words:

      That in our familiar system—the decimal system—we have only a small number of simple signs (1 to 9 and 0), using compound signs (10, 11, …, 100, …) for bigger numbers, is only a matter of convenience of writing. Calculation with these signs—the process taught at school as arithmetic—is really the study of the decimal notation rather than the study of numbers.

      5. The reasoning in paragraph 3 shows that there is a close connection between counting and comparison without counting—so close that the reader may think that I am making too much fuss about the difference. But there is a reason. The process of pairing standard sets with number-signs arranged in order works only for finite numbers. Not only does the pairing process break down for infinite numbers, but it can be shown that the two ways of looking at numbers give two different systems of arithmetic, known as cardinal and ordinal. (This is not the same as the grammatical usage of the terms cardinal and ordinal. In grammar, the ordinal numbers are first, second, third, ….)

      6. Infinite (and similar words) are sometimes used in ordinary conversation to mean very great, and sometimes as synonyms for unlimited, boundless, etc. In certain other contexts their meaning is vague and mysterious: for example, Of all the arts, dancing is perhaps the one most attuned to the infinite, having its essence in nature itself, Gopal and Dadachanji, Indian Dancing, p. 13. But in mathematics they are used in a perfectly definite sense. Quite simply, a set which has so many elements that the process of counting them one by one would never come to an end is said to be infinite, or to have an infinite number of (or an infinity of, or infinitely many) elements. To avoid confusion with the popular uses of the word, mathematicians often replace the term infinite by the less familiar word transfinite. An account of transfinite numbers will be found in J. E. Little wood’s The Elements of the Theory of Real Functions, or in Paul Halmos’ Naive Set Theory.

      7. We have had to introduce the concept of set, and this concept is in fact a very important one in modem mathematics. It is even more fundamental than number, in spite of the fact that numbers are usually considered to be what mathematics is about.

      The concepts of set and property are interchangeable inasmuch as we could use either concept in place of the other in any treatment of the subject. A given property P, for example, determines a corresponding set: namely the set of things which have the property P. And a set P determines a corresponding property: the property of belonging to P. Finally, if P corresponds to P in this way, then obviously P corresponds to P. However, the mathematician prefers to work with sets, which seem to be a little more definite and amenable to calculation. For example, it is quite clear when two sets are the same: they are the same if they consist of the same elements; in other words, if every member of each belongs to the other. It is not quite so clear when properties are the same. As Bertrand Russell says in his Introduction to Mathematical Philosophy (p. 12), Men may be defined as featherless bipeds, or as rational animals, or (more correctly) by the traits by which Swift delineates the Yahoos. Does this mean that the property of being a featherless biped is the same as that of being a rational animal? Again, does the phrase odd prime number denote the same property as prime number greater than 2? Odd is certainly not the same as greater than 2. We might agree to consider two properties to be equal if and only if they define the same set; but this is precisely equivalent to working with sets instead of properties.

      The concept of set is so important that there is a special notation for it: the expression a ∈ A denotes that a belongs to the set A. Usually small letters will denote elements; large letters, sets. The symbol ∈ will always denote belongs to.

      * We use the word set rather than group, because group has a special meaning in mathematics, whereas set is used quite generally. This is the reverse of ordinary usage, where, for example, a set of cigarette cards is a special group of cigarette cards: it contains just one card of each sort.

      CHAPTER II

      Whole Numbers

      1. If x and y are any two whole numbers, then x + y = y + x. In any given case we are able to verify this. For example, if we work out 5 + 3 we get the answer 8. If we work out 3 + 5 we get the same answer. But we cannot prove the general statement "if x and y are any two numbers, then x + y = y + x like this, because we cannot work out" x + y. Nor can we prove it by testing all possible pairs of whole numbers, because there are an infinite number of them. And most people, when they try to prove the statement, find it extraordinarily difficult; not because it is too complicated but because it is too simple. To prove a complicated fact we use the simpler facts at our disposal. These simpler facts will usually have been proved from still simpler facts. And so on. Eventually we must get down to the simplest facts of all from which everything starts. These basic facts are called axioms.

      The first thing to do, then, is to find a suitable set of axioms for an arithmetic of whole numbers. This simply means that from all the facts at our disposal we choose a few to form a foundation for our system—and all that is required of them is that they should suffice for this, i.e. that the whole system should be deducible from them. It is, of course, a practical advantage if they are fairly simple, and again it is convenient if they are self-evident, but neither of these properties is nowadays regarded as essential, though Euclid required his axioms to be self-evident. Usually there are any number of possible choices. To take a simple example, the laws of arithmetic (p. 12) could be taken as the axioms of a certain arithmetical system. If we were to replace the law (x + y)·z = x·z + y·z by the law x·(y + z) = x·y + x·z, we should get a slightly different but equally good set of axioms for the same system—the same because clearly either law can be deduced from the other, using the commutative law for multiplication.

      In order to know the simplest properties of whole numbers we must know what a whole number is, that is, we must consider not only axioms but also definitions. We shall first define whole number, addition, and multiplication. Subtraction and

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