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Introduction to Mathematical Thinking
The Formation of Concepts in Modern Mathematics
By Friedrich Waismann, Theodore J. Benac
Dover Publications, Inc.Copyright © 2003 Dover Publications, Inc.
All rights reserved.
The Various Types of Numbers
The numbers presented to us at the first stage of development are the natural or cardinal numbers 1, 2, 3, 4 ...; they are used for counting purposes. Numbers are usually represented geometrically as points on a straight line. We will frequently employ this technique to make the following investigations clearer. For this purpose we choose an arbitrary point on a line as the starting point, also an arbitrary interval as the unit of length, and then successively mark off this interval in one direction. The numbers 0, 1, 2, 3 ... are assigned to the points thereby generated.
These points are now the "images" of the numbers, and it is advantageous for many purposes to tie our concepts to this scale of points. Henceforth we will speak of the number series also as a point series.
What properties belong to the system of natural numbers?
1. It is an ordered system. This means that if two distinct natural numbers are given, one must precede the other; in other words: the relations a > b, a = b, a < b (a greater than b, a equal to b, a smaller than b) form a complete disjunction.
2. Consequently the concept of "betweenness" can be applied to natural numbers; that is, to say that the number c lies between a and b, implies a > c > b or a < c < b. On examining the natural numbers with respect to this concept, we encounter a rather characteristic property: every number lies between two others, its immediate predecessor and its immediate consequent. Between two numbers immediately following one another no further number can be inserted.
3. There is only one exception. The number 0 does not have a predecessor. On the other hand, there is no number which does not have a consequent. We will express these facts as follows: the number series has a first but not a last element; or: it is infinite. on one side.
The possibility of mapping numbers on the points of a line rests on the fact that the above properties can be ascribed to the point series. Thus it is ordered as soon as we run through the points, let us say, from left to right and think of those points which lie further to the left as antecedent. The other quoted properties also apply. Hence the structure of the number system can be carried over to that of the point system.
We obtain two further properties of the system of natural numbers as soon as we take the arithmetical operations into consideration. Which of the four basic rules of arithmetic (addition, subtraction, multiplication, division) can be performed unlimitedly in this domain, so that the result is always a natural number? Obviously only two—addition and multiplication. In contrast, the subtraction a—b can be carried out only if the minuend a is greater than the subtrahend b; and the division without a remainder only if the dividend is a multiple of the divisor.
If we combine the numbers of the domain arbitrarily by addition and multiplication, we never leave the domain. The natural numbers appear in this respect as a totality, a closed system. We will state these facts as follows: the domain of the natural numbers is closed under addition and multiplication, not closed under the two other arithmetical operations. This latter fact was actually the reason for extending the system of natural numbers in two directions. First we have to introduce the negative and secondly the fractional numbers in order to settle the closure of the number domain. Let us now take a closer look at these number creations.
The negative numbers can be thought of as generated by reversing the formation rule of the number series. Thus, instead of successively adding the number 1, we descend from the number 3 to the number 2, from 2 to 1, from 1 to 0, and then to numbers which we designate sequentially by—1,—2,—3, etc. These are represented by points as indicated below:
The system of positive and negative numbers is called the system of integers.
Let us now compare the integers with the natural numbers. What properties remain intact under this extension? The integers are also an ordered system; therefore the concept "betweenness" has a meaning; however, there is now no longer a number which precedes all others; the system has neither a first nor a last element; it is infinite on both sides. Moreover, three operations can be performed unlimitedly—not only addition and multiplication but also subtraction. However, in general, the quotient of two integers is not an integer.
The fractional numbers must be introduced if division is to be performed unlimitedly. The system of integers and fractional numbers is called the system of rational numbers. This system is closed under all four arithmetical operations. Hence we always stay within the system whenever we combine the individual elements by the four operations. A domain with this property is also called a "field" (the word is used here in a technical sense, say as in "battlefield" or in "field of force"). The system of rational numbers is ordered; for if two unequal fractions are given, one must be greater or smaller than the other. Moreover, we can think of the integers as written formally as fractions. If we try to represent the rational numbers according to the earlier model, as points on a line, we are confronted with a characteristic difficulty. Obviously the fractions lie between the integers, and consequently we must insert further points in the space between the equidistant points of Fig. 2. But how are these points spread out? If we start out, say, from the number ½, is there still an immediate predecessor or an immediate consequent? By no means! For if we choose a fraction which lies as close to ½ as we wish, it is a simple matter to obtain another fraction which lies still closer to ½. (The reader may prove this by showing that the number [a + c]/[b + d] always lies between the two rational numbers a/b and c/d.)
The totality of rational numbers therefore possesses a structure completely different from that of the natural numbers and integers. Between two rational numbers there always exists another rational number. In order to characterize this specific structure the concept "dense" has been coined. We define an ordered system of elements as dense if between any two elements of the system there always lies another element of the system. The system of rational numbers is our first example of a dense system. The natural numbers and integers do not have this property.
The property of denseness makes it especially difficult to obtain an intuitive picture of the distribution of the rational numbers. We can certainly insert further points between those corresponding to the integers; however this process of insertion must be thought of as continued without end, so that in every interval of the number axis, no matter how small, there will lie an infinite number of rational points. In trying to visualize this system as a completed totality we are inevitably confronted with certain oddities. To illustrate, let us consider the totality of proper fractions with the exception of 0 and 1. The class of these points can obviously be placed in the interval of the number axis between 0 and 1; therefore we will think of it as a point set which covers this interval as an infinitely fine dust. This view leads to the following thought. When I run over the line, say, from the left to the right and beginning from a point to the left of 0, I must at some time or other meet a first element of the point set and when I have run through the whole interval, also a last element; the set must possess a point which is furthest to the left and another which is furthest to the right. A moment's reflection shows that this is absolutely impossible. It follows from the structure of the rational numbers that there is no smallest proper fraction (and also no greatest one). Conceptually this situation presents no difficulty whatsoever; the class of proper fractions is clearly and sharply defined; however the attempt to realize this concept as an intuitively clear picture leads to paradoxes. This illustrates the fact that, though we can learn some things about such relations by graphic methods, we will be misled if we entrust ourselves to them alone.
The stepwise extension of the number domain, which we have sketched above, is brought to some kind of a close with the rational numbers. And many will be tempted to assume that the extension can be carried no further, since the rational points fill up the number axis completely and without holes. That this is a mistake, that the rational numbers, even though sown infinitely dense, still do not cover the entire number axis, is the great discovery of Pythagoras. He first recognized that there are numbers which, though completely different from the rational numbers, are still related to them. We will illustrate this extraordinary discovery by constructing a square on an interval of length 1 and drawing a diagonal in this square. Our intuition tells us that this diagonal must have a very definite length. Let us try to compute it. According to the theorem of Pythagoras the square of the diagonal is equal to the sum of the squares of the two legs of the right triangle, therefore 2. Consequently the diagonal has a length [square root of 2]; [square root of 2] is that number whose square is 2. Can it be represented as a fraction? First it is clear that it lies between 1 and 2; therefore we experiment with 1½; the square of this number is 9/4, therefore it is too large. The number under investigation must accordingly be greater than 1 but less than 1½; 4/3 is such a number; the test shows however that it is too small. Let us continue this process of inserting a fraction between those hitherto investigated and testing whether its square is exactly 2. In this way we will find numbers which are either too large or too small, and which we can arrange in the form of two sequences:
Now one could think that if we continue this search on and on and take all the time and effort that is needed, we must eventually arrive at a number whose square is exactly 2. This point will be clarified as soon as we have determined whether the trials attempted so far have failed due to chance or because there is a deeper reason underlying it. If there is a rational number which is exactly [square root of 2], then there is a fraction p/q such that p2/q2 =2. Now a fraction is equal to an integer only if the denominator goes into the numerator without a remainder. Hence p2 is divisible by q2. But this is only possible if p is also divisible by q. For, if p and q are two relatively prime numbers (i.e., two numbers which have no common prime factors), then p2 and q2 are also relatively prime; by the squaring process no prime factors can be generated which do not already exist. Hence if p2/q2 =2, that is, an integer, then p/q must also be an integer. However this is impossible since p/q lies between 1 and 2, an interval in which there are no integers.
Thus the simplest reflection on the divisibility properties of numbers readily shows that the attempt to find a rational number whose square is exactly 2 must be fruitless. On the other hand there can be no doubt that the diagonal of the unit square has a very definite length. If we think of this length as laid on the number axis with one end at 0, we obtain a point which is the geometrical representative of [square root of 2]; the point where [square root of 2] lies can not be a rational point of the number axis. Hence we have the following result: Even though the rational points cover the number axis as an infinitely fine dust, they still do not completely fill it up. They form, as it were, a porous system, in which cracks and crevices leave room for another type of number, the irrationals.
What can we say about the distribution of the irrational numbers? That is, are they exceptions—to be found between the rational numbers only here and there? We can answer this question by a very simple argument. Let us think of the entire number axis with the rational points on it as enlarged in the ratio 1: [square root of 2], that is, in such a way that [square root of 2] is used as unit interval. Then every rational number will go over to a number which can be shown, as in the case of [square root of 2], to be irrational; for instance, 1 in [square root of 2], 3/10 in 3/10 [square root of 2], etc. We thereby obtain a second system which is also dense, consists only of irrational numbers, and is somehow squeezed in between the rational numbers. But irrational numbers can be generated in many ways; for instance not only by square roots but also by cube roots, fourth roots, etc. In fact there are infinitely many operations which will produce irrational results when "set loose" on an individual rational number. These remarks lead us to surmise that the irrational numbers make up the principal part of the structure of the number axis instead of being its exceptional points. In the theory of sets it is actually shown that most of the points on the number axis are irrational and that the rational numbers are vanishing exceptions.
The system of rational and irrational numbers is called the system of real numbers.
We can also look at these relations from another point of view by proceeding from the representation of numbers as decimal fractions. A decimal fraction can terminate or continue without end. We will state, first of all, that every terminating decimal fraction can be transformed into a non-terminating one by diminishing the last numeral by unity and permitting only nines to follow afterwards. For instance, we have ½ = 0.5 = 0.4999 ...
If we use this property, we can represent the totality of the real numbers by non-terminating decimal fractions. These fall into two categories: the periodic and non-periodic decimal fractions, the former corresponding to the rational numbers, the latter to the irrationals. (We omit the proof, which is simple.) This shows anew that between two rationals there must always lie irrational numbers; for we can always insert arbitrarily many non-periodic decimal fractions between two periodic ones.
By passing on to the imaginary numbers we take a new direction in the extension of the number concept. They were unknown in antiquity. They appeared for the first time in a work of Cardano (1545), whose name is connected with the solution of cubic equations. However, the mathematicians of that day did not have a clear understanding of the nature of these quantities. On the contrary, the imaginary numbers forced themselves into calculations against the desires and inclinations of mathematicians. This situation resulted from algorithmic requirements. Cardano's formula often represents the solution of a cubic equation, even though it may be real, in a form in which square roots of negative numbers must be extracted. Now, there is no real number whose square is negative; therefore these roots are "impossible." However, in spite of these scruples this new type of expression was manipulated like an ordinary root; and the end justified the means. We encounter here a factor which on the whole played an important role in the history of mathematics. It seems that an independent, onward-driving force is inherent in the manipulation of formulae, in the algorithms; and that in our case it induced the mathematicians to handle imaginary numbers; to the great advantage of mathematics; for the pedantic requirements of rigor would probably have paralyzed the further development. Fortunately the mathematician of that day treated subtle logical concepts with indifference; not, however, to such an extent that he would not retain, when operating with these remarkable entities, a certain uneasiness, a bad conscience, which betrayed itself in names as "impossible" or "fictitious numbers." As evidence of this, Leibniz made the following statement in the year 1702: "The imaginary numbers are a fine and wonderful refuge of the Divine Spirit, almost an amphibian between being and non-being." We note here a reflection of the strange impression which these numbers must have made on the mathematician. Thus we find that Euler was candidly astonished by the remarkable fact that a number as [square root of -1] is neither smaller nor greater than 5, neither positive nor negative, and that it cannot be compared with ordinary numbers at all. And when a student hears about imaginary numbers for the first time, he again experiences this impression of mysteriousness, which disappears later in proportion as he learns to use these numbers. However, the nature of these numbers is not made clearer by usage. We simply have to become accustomed to them and ask nothing further. Under these circumstances it marks an epoch-making development that Gauss should give a geometrical representation of the imaginary numbers. It is found for the first time in his own abstract of a number-theoretical work in the year 1831 and has made an extraordinary deep impression. However, we know from Gauss' diary, which was left among his papers, that he was already in possession of this interpretation by 1797. Through this representation Gauss intended to clarify the "true metaphysics of imaginary numbers" and bestow on them complete franchise in mathematics. Now what is this representation? We already know that the rational and irrational numbers fill up the number axis, so that there remains not even the smallest of gaps. Hence, if we now wish to interpret the imaginary numbers geometrically, we must use a second line. In the Gaussian interpretation the real numbers are represented as points on the x-axis, the imaginaries as points on the y-axis of a rectangular Cartesian system of coordinates, whose intersection point represents the number 0. Hence a rotation through 90° takes the positive real number axis into the positive imaginary number axis. Gauss did not give a basis for this representation; however he derived from it the right to operate with imaginary numbers.
Excerpted from Introduction to Mathematical Thinking by Friedrich Waismann, Theodore J. Benac. Copyright © 2003 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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