Are real numbers real?
by Roman SIKORSKI (Delta 1/1974)
No doubt, natural numbers are natural. Equally undoubtedly, integers deserve the name "integer". Rational numbers might well be called "measuring" or "measure" numbers, for they are the numbers we use in all our practical measurements - in fact, not only in measurements, since all things we compute in practice on some given numbers fall into the realm of rationals. Why then should we think of the broader, but much harder notion of real numbers, if the rational ones seem to be sufficient for all our computations? The definition of real numbers always tends to be the source of some difficulties, so it is often camouflaged in school texts rather than formulated in a precise way.
The notion of a natural number is quite easy to understand. Indeed, it is so easy and so familiar, that they seem to be taken straight from the material world around us. We usually forget that without much effort we can write down, even on a small piece of paper, the name of a natural number n that cannot be found in the material world, i.e., such that it is impossible to provide an example of an n-element set of real-life objects. The number n = is one of this sort. You can easily imagine initial, small natural numbers, although the same cannot be said about large ones. Nevertheless, the set N of all natural numbers is easily definable.
Obviously, it should have among its elements the number 1. If some number n is in the set, then the same should be with n+1. And that is all! In other words, N is the smallest set with these two properties. This definition of the set of all natural numbers reminds me of the joke about packing paper handkerchiefs into an empty suitcase. Clearly, there is room enough to pack one handkerchief. Moreover, experience proves that if you can pack n handkerchiefs into the suitcase, then you can also pack the n+1st one without much effort. Therefore you can fill the suitcase with as many handkerchiefs as there are natural numbers, i.e., with an infinity of them!
You can now think of a mathematician as a man with an abstract suitcase N, containing all the natural numbers. He also has another suitcase with "mirror reflections in zero" of those numbers and these "mirror reflections" are called negative integers. He has one decision to make: where to pack the zero? There is a difference of opinion: some mathematicians prefer to put it together with the natural numbers, others do not like this solution. Actually, the argument is not very important. Why argue about the zero, the mathematical "nothing"? It's not worth the time.
Besides those two suitcases, the mathematician also has a device for neatly cutting the numbers. More seriously, a mathematician knows how to use integers to easily make rational numbers, i.e., "fractions" m/n, where m is an integer and n is a natural number (but not zero!).Some of these fractions should be considered equal. Then in a natural way you can define the basic arithmetical operations of addition and multiplication and the inverse operations of subtraction and division (not by zero!). The rational numbers can be ordered, too: you can introduce the minority relation x < y. It is not hard to "imagine" a rational number. Indeed, you can easily think of the nth part of something, i.e., of the number 1/n; now you can imagine m such parts, i.e., the number m/n, with a change in sign, if necessary, i.e., with its "mirror reflection in zero".
Good old rationals! How useful they are! All commercial and financial transactions are based on them, as well as all our measurements and technical computations. From a practical point of view, there are too many of them, for they are an infinity. In practice only finitely many such numbers are used in computations (though it might be quite difficult to estimate how many). But so it is with mathematicians. Whenever they create something, they do it in full generality, and thus with some excess, usually much more than practice requires.
Alas, the set of all rational numbers has its drawbacks. It is dense, i.e., has the property that in between any two different rational numbers there is always a third one, and yet it has holes. What is worse, the holes are also densely distributed among the rationals: there is a hole between any two different rational numbers. It is not the old age of the set of rational numbers that should be blamed for the holes, nor did its frequent use wear the material it is made of. Simply, such is the nature of this set. The set of all rational numbers has the structure of a very dense one-dimensional sieve.
It is time to explain what is meant by a hole in the set of all rational numbers (professional mathematicians call it a "gap" rather than a "hole"). A hole is a partition of the set Q of all rational numbers into two nonempty subsets Q1 and Q2 such that each number in Q1 is less than any number in Q2, and there is neither greatest element in Q1 nor a least one in Q2. We say that a hole lies between the rational numbers q1 and q2 (with q1 < q2) if q1 is in Q1 and q2 is in Q2. For instance, let all the negative rational numbers be in Q1 together with all those nonnegative ones for which the square is less than 2, and assign to Q2 all the positive rational numbers with squares greater than 2. Then the partition of the set Q into Q1 and Q2 is a hole (gap) in the set of all rational numbers. You can easily check that the hole lies between 1 and 2; with little effort you can determine the position of the hole with more precision. Indeed, simple computation proves that it lies between 1.41 and 1.42. It should be clear that much better precision could be attained.
The existence of holes in the set of rational numbers creates many problems. You might say that the mathematical content of many beautiful mathematical theorems leaks away through the holes, which is particularly true for theorems of a rather subtle structure. The set of rational numbers is very suitable for computing on concrete data, but it is quite unsuitable for theoretical purposes as well as for operations more advanced than addition, subtraction, multiplication or division, especially when you want to perform them accurately rather than approximately. Even taking the square root is not feasible in this set. Just try to define so useful a function as is log x (for x > 0) in such a way that both x and log x be rational. In both cases you will find that the set of all rational numbers is too small to allow considering logarithms or roots (with precise values).
The mathematician heavily dislikes situations when some seemingly natural or useful operations cannot be carried out. In many cases he fixes the problem by extending the set of objects, to which the operations are to be applied or which have to be values of the operations. Even the most modern mathematics provides numerous examples, but we shall consider only one: the extension of the set of all rational numbers to the set of real ones.
The extension is made in the following way. The mathematician places a peg at each hole in the set Q (it may be convenient to interpret rational numbers as points on the number axis, and then the peg may be interpreted as a point, too). Then with one single hammer strike he simultaneously drives all the pegs into their respective holes. Let's get it right: he drives the pegs into the holes by a single act of will! You shouldn't think of the process as if the mathematician numbered the holes with consecutive natural numbers and then passed from the nth hole to the n+1st one, hitting one peg after another. This would just be impossible, for it can be proved that natural numbers are not sufficient to number all the holes: there are too many of them.
All the rational numbers can be numbered with natural ones, but not the holes in the set Q! Strange, surprising, but true. Whatever way of numbering the holes with natural numbers you may choose, you will always be left with infinitely many unnumbered holes. Professionals say that a set is countable if all its elements can be numbered with consecutive natural numbers, and it is uncountable when this cannot be done. Uncountable sets are much bigger than the countable ones, much richer in elements. The set all rational numbers is countable, whereas the set of all holes is not. This means that there are by far more holes in the set of rational numbers than numbers! How can you trust such a set? No wonder so many mathematical beauties leak through.
Let's return to the extension of the rational numbers to the set of real ones. The pegs are called irrational numbers and the whole, i.e., both rational and irrational numbers are called real numbers, as is well established. You should easily guess that the peg used to fill the hole used as example to illustrate the notion will be denoted by the symbol .
As with any other mathematical construction, a clear distinction must be made between two different facets of the same problem: first, the intuitive presentation of the purpose and methods of the construction, and second, a precise description of it following the highest strictness requirements of modern mathematics. The story about pegs being driven into the holes was but an intuitive presentation of the construction, an explanation of its purposes. A precise description - that's another story and I would say, quite an unrewarding one. We know two methods of "driving the pegs": the first is due to Dedekind, Cantor has invented the other. Both are very precise and both feature the same drawback: the basic, simple and clear intention of the construction is dimmed by not so essential technical details. Therefore we shall not see any of them here. Let's just have a look at the main difference between them. Clearly, the pegs must be made of something, of some material, by which we obviously mean some abstract mathematical notions. In fact, the two methods differ mainly in the material applied to the construction of pegs. With Dedekind the peg filling the hole is the hole itself! The hole plugs itself! You might say that material costs have been reduced to the minimum, to nothing!
The construction is complete, the pegs are where they should be. It remains to verify whether the job has been done properly, whether no new holes appeared in the process. Fortunately, everything is in the best of orders, the set of real numbers is indeed absolutely leak-proof. The definition of hole could be formulated for the set of reals too, but there is no need to do it: there are no holes in it.
It turns out that the set of all real numbers has even better properties than the set of rational numbers. It can be ordered and basic arithmetic operations (addition, subtraction, multiplication and division) can be generalised to include all real numbers. Moreover, a great many new useful operations can be performed in this set without any restrictions, like taking roots, raising to a power, taking logarithms, etc., very difficult to do in the set of rational numbers, if not impossible.
The notion of a real number proved to be sufficient for the development of the entire mathematical analysis, a huge branch of contemporary mathematics. All its theorems rely on the fundamental fact that the set of reals is "leak-proof" (mathematicians tend to say "complete" instead of "leak-proof"). The theorems of mathematical analysis are no longer true if applied to the set of rational numbers only. That's what I meant when I stated jokingly that theorems leak through the holes in the set of rational numbers. The notion of a real number is essential to the entire theoretical mathematics. It is also essential in the construction of general methods of applied mathematics until the moment when approximate computations are required. Then we return to the more elementary rational numbers.
No doubt the notion of a rational number is simpler than that of a real one. For a non-mathematician the real number as defined by Dedekind or Cantor appears to be quite a mystical object, much less real - in the current sense of the word - than the rational one. For a professional mathematician the real number is a basic work tool, as real as any other mathematical notion. Real numbers are as real as rational ones, since both are correctly defined notions existing in the mathematician's mind. Both have the same degree of reality.