What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

INTRODUCTION

We must greatly extend the original concept of number as natural number in order to create an instrument powerful enough for the needs of practice and theory. In a long and hesitant evolution zero, negative integers, and fractions were gradually accepted on the same footing as the positive integers, and today the rules of operation with these numbers are mastered by the average school child. But to gain complete freedom in algebraic operations we must go further by including irrational and complex quantities in the number concept. Although these extensions of the concept of natural number have been in use for centuries and are at the basis of all modern mathematics it is only in recent times that they have been put on a logically sound basis. In the present chapter we shall give an account of this development.

§1. THE RATIONAL NUMBERS

1. Rational Numbers as a Device for Measuring

The integers are abstractions from the process of counting finite collections of objects. But in daily life we need not only to count individual objects, but also to measure quantities such as length, area, weight, and time. If we want to operate freely with the measures of these quantities, which are capable of arbitrarily fine subdivision, it is necessary to extend the realm of arithmetic beyond the integers. The first step is to reduce the problem of measuring to the problem of counting. First we select, quite arbitrarily, a unit of measurement— foot, yard, inch, pound, gram, or second as the case may be—to which we assign the measure 1. Then we count the number of these units which together make up the quantity to be measured. A given mass of lead may weigh exactly 54 pounds. In general, however, the process of counting units will not “come out even,” and the given quantity will not be exactly measurable in terms of integral multiples of the chosen unit. The most we can say is that it lies between two successive multiples of this unit, say between 53 and 54 pounds. When this occurs, we take a further step by introducing new sub-units, obtained by subdividing the original unit into a number n of equal parts. In ordinary language, these new sub-units may have special names; for example, the foot is divided into 12 inches, the meter into 100 centimeters, the pound into 16 ounces, the hour into 60 minutes, the minute into 60 seconds, etc. In the symbolism of mathematics, however, a sub-unit obtained by dividing the original unit 1 into n equal parts is denoted by the symbol 1/n; and if a given quantity contains exactly m of these sub-units, its measure is denoted by the symbol m/n. This symbol is called a fraction or ratio(sometimes written m:n). The next and decisive step was consciously taken only after centuries of groping effort: the symbol m/n was divested of its concrete reference to the process of measuring and the quantities measured, and instead considered as a pure number, an entity in itself, on the same footing with the natural numbers. When m and n are natural numbers, the symbol m/n is called a rational number.

The use of the word number (originally meaning natural number only) for these new symbols is justified by the fact that addition and multiplication of these symbols obey the same laws that govern the operations with natural numbers. To show this, addition, multiplication, and equality of rational numbers must first be defined. As everyone knows, these definitions are:

(1) 

for any integers a, b, c, d. For example:

Precisely these definitions are forced upon us if we wish to use the rational numbers as measures for lengths, areas, etc. But strictly speaking, these rules for the addition, multiplication, and equality of our symbols are established by our own definition and are not imposed upon us by any prior necessity other than that of consistency and usefulness for applications. On the basis of the definitions (1) we can show that the fundamental laws of the arithmetic of natural numbers continue to hold in the domain of rational numbers:

For example, the proof of the commutative law of addition for fractions is exhibited by the equations

of which the first and last equality signs correspond to the definition (1) of addition, while the middle one is a consequence of the commutative laws of addition and multiplication of natural numbers. The reader may verify the other four laws in the same way.

For a real understanding of these facts it must be emphasized once more that the rational numbers are our own creations, and that the rules (1) are imposed at our volition. We might whimsically decree some other rule for addition, such as , which in particular would yield , an absurd result from the point of view of measuring. Rules of this type, though logically permissible, would make the arithmetic of our symbols a meaningless game. The free play of the intellect is guided here by the necessity of creating a suitable instrument for handling measurements.

2. Intrinsic Need for the Rational Numbers. Principle of Generalization

Aside from the “practical” reason for the introduction of rational numbers, there is a more intrinsic and in some ways an even more compelling one, which we shall now discuss quite independently of the preceding argument. It is of an entirely arithmetical character, and is typical of a dominant tendency in mathematical procedure.

In the ordinary arithmetic of natural numbers we can always carry out the two fundamental operations, addition and multiplication. But the “inverse operations” of subtraction and division are not always possible. The difference b – a of two integers a, b is the integer c such that a + c = b, i.e. it is the solution of the equation a + x = b. But in the domain of natural numbers the symbol b – a has a meaning only under the restriction b > a, for only then does the equation a + x = b have a natural number x as a solution. It was a very great step towards removing this restriction when the symbol 0 was introduced by setting a – a = 0. It was of even greater importance when, through the introduction of the symbols –1, –2, –3, · · ·, together with the definition

b – a = – (a – b)

for the case b < a, it was assured that subtraction could be performed without restriction in the domain of positive and negative integers. To include the new symbols –1, –2, –3, · · · in an enlarged arithmetic which embraces both positive and negative integers we must, of course, define operations with them in such a way that the original rules of arithmetical operations are preserved. For example, the rule

(3)  (–1)(–1) = 1,

which we set up to govern the multiplication of negative integers, is a consequence of our desire to preserve the distributive law a(b + c) = ab + ac. For if we had ruled that (–1)(–1) = – 1, then, on setting a = – 1, b = 1, c = – 1, we should have had –1(1 – 1) = –1 – 1 = – 2, while on the other hand we actually have –1(1 – 1) = –1 · 0 = 0. It took a long time for mathematicians to realize that the “rule of signs” (3), together with all the other definitions governing negative integers and fractions cannot be “proved.” They are created by us in order to attain freedom of operation while preserving the fundamental laws of arithmetic. What can–and must–be proved is only that on the basis of these definitions the commutative, associative, and distributive laws of arithmetic are preserved. Even the great Euler resorted to a thoroughly unconvincing argument to show that (–1)(–1) “must” be equal to +1. For, as he reasoned, it must either be +1 or –1, and cannot be –1, since –1 = (+1)(–1).

Just as the introduction of the negative integers and zero clears the way for unrestricted subtraction, so the introduction of fractional numbers removes the analogous arithmetical obstacle to division. The quotient x = b/a of two integers a and b, defined by the equation

(4)  ax = b,

exists as an integer only if a is a factor of b. If this is not the case, as for example when a= 2, b = 3, we simply introduce a new symbol b/a, which we call a fraction, subject to the rule that a(b/a) = o, so that b/a is a solution of (4) “by definition.” The invention of the fractions as new number symbols makes division possible without restriction— except for division by zero, which is excluded once for all.

Expressions like 1/0, 3/0, 0/0, etc. will be for us meaningless symbols. For if division by 0 were permitted, we could deduce from the true equation 0.1 = 0.2 the absurd consequence 1 = 2. It is, however, sometimes useful to denote such expressions by the symbol ∞ (read, “infinity”),provided that one does not attempt to operate with the symbol ∞ as though it were subject to the ordinary rules of calculation with numbers.

The purely arithmetical significance of the system of all rational numbers— integers and fractions, positive and negative—is now apparent. For in this extended number domain not only do the formal associative, commutative, and distributive laws hold, but the equations a + x = b and ax= b now have solutions, x = b – a and x = b/a, without restriction, provided in the latter case that a ≠ 0. In other words, in the domain of rational numbers the so-called rational operations—addition, subtraction, multiplication, and division—may be performed without restriction and will never lead out of this domain. Such a closed domain of numbers is called a field. We shall meet with other examples of fields later in this chapter and in Chapter III.

Extending a domain by introducing new symbols in such a way that the laws which hold in the original domain continue to hold in the larger domain is one aspect of the characteristic mathematical process of generalization. The generalization from the natural to the rational numbers satisfies both the theoretical need for removing the restrictions on subtraction and division, and the practical need for numbers to express the results of measurement. It is the fact that the rational numbers fill this two-fold need that gives them their true significance. As we have seen, this extension of the number concept was made possible by the creation of new numbers in the form of abstract symbols like 0, –2, and 3/4. Today, when we deal with such numbers as a matter of course, it is hard to believe that as late as the seventeenth century they were not generally credited with the same legitimacy as the positive integers, and that they were used, when necessary, with a certain amount of doubt and trepidation. The inherent human tendency to cling to the “concrete,” as exemplified by the natural numbers, was responsible for this slowness in taking an inevitable step. Only in the realm of the abstract can a satisfactory system of arithmetic be created.

3. Geometrical Interpretation of Rational Numbers

An illuminating geometrical interpretation of the rational number system is given by the following construction.

On a straight line, the “number axis,” we mark off a segment 0 to 1, as in Fig. 8. This establishes the length of the segment from 0 to 1 as the unit length, which we may choose at will. The positive and negative integers are then represented as a set of equidistant points on the number axis, the positive numbers to the right of the point 0 and the

Fig. 8. The number axis.

negative numbers to the left. To represent fractions with the denominator n, we divide each of the segments of unit length into n equal parts; the points of subdivision then represent the fractions with denominator n. If we do this for every integer n, then all the rational numbers will be represented by points of the number axis. We shall call such points rational points, and we shall use the terms “rational number” and “rational point” interchangeably.

In Chapter I, §1, we defined the relation A < B for natural numbers. This has its analog on the number axis in the fact that if natural number A is less than natural number B, then point A lies to the left of point B. Since the geometrical relation holds between all rational points, we arc led to try to extend the arithmetical relation in such a way as to preserve the relative geometrical order of the corresponding points. This is achieved by the following definition: The rational number A is said to be less than the rational number B (A < B), and B is said to be greater than A (B > A), ifB – A is positive. It then follows that, if A < B, the points (numbers) between A and B arc those which are both > A and < B. Any such pair of distinct points, together with the points between them, is called a segment, or interval, [A, B].

The distance of a point, A, from the origin, considered as positive, is called the absolute value of A and is indicated by the symbol

| A |.

In words, if A ≥ 0, we have | A | = A; if A ≤ 0, we have | A | = – A. It is clear that if A and B have the same sign, the equation | A + B | = | A | + | B | holds, while if A and B have different signs, we have |A + B | < | A | + | B |. Hence, combining these two statements, we have the general inequality

| A + B | ≤ | A | + | B |,

which is valid irrespective of the signs of A and B.

A fact of fundamental importance is expressed in the statement: The rational points are dense on the line. By this we mean that within each interval, no matter how small, there are rational points. We need only take a denominator n large enough so that the interval [0,1/n] is smaller than the interval [A, B] in question; then at least one of the fractions m/n must lie within the interval. Hence there is no interval on the line, however small, which is free from rational points. It follows, moreover, that there must be infinitely many rational points in any interval; for, if there were only a finite number, the interval between any two adjacent rational points would be devoid of rational points, which we have just seen to be impossible.