Notes on Rational and Real Numbers

The notion of a number is as old as mathematics itself, and their developments have been inseparable. Usually a new set of numbers included the old set, or, as we often say, extended it. In this way, the set {1, 2, 3, ...} of natural numbers, has been extended by adding zero to it, negative integers, rational numbers (fractions), irrational numbers, complex numbers. The actual extensions happened not in the order suggested by the previous sentence, and the history of the process is fascinating. The main motivation for this extension came from mathematics itself: having a greater set of numbers allowed mathematicians to express themselves with better precision and fewer words, i.e., with greater ease. Numbers form one of the most important part of mathematical “language”, and in this regard, their development is very similar to the development of live languages, where the vocabulary increases mostly for convenience, rather than of necessity. It is very hard to part with conveniences after getting used to them. Imagine the world without electricity, or even worse – the mathematics without real numbers. Then objects like √ 5, or sin 10◦, or log10 7 would cease to exist, as they do not exist among rational numbers. The use of quadratic equations, or Trigonometry, or Calculus would terminate ... Well, enough of this nightmare. It is often hard to define basic mathematical notions. The rigor of such definition depends on the time they are made, and of the depth of the subject where they are used. For example, a better definition of a function became important with the development of Calculus, and of abstract algebra. Often the development of mathematical techniques and accumulation of mathematical facts far preceded the thorough discussion of the objects being studied. As examples, one can mention integers, functions, limits, rational and complex numbers. It took many centuries between their appearance and use in mathematics, and the time when they were defined at the level meeting now days standards. Of course, mathematics is not special in this regard. For thousands of years comedies were played in theaters, but definitions of humor, or of the notion of ‘funny’ appeared very recently. One may wonder whether those were needed at all, but very few mathematicians will doubt that in order to discuss rational and irrational numbers it is important to first define them. These notes are motivated by the desire to clarify for myself what I wish to say about the rationals and reals in the courses I teach. There are many accounts of these topics in the literature, but I have difficulties of using them in my courses. Either the exposition is too long, or too short, or at the kindergarten level. Or it requires greater mathematical maturity from the students, or it does not mention things I find necessary to be mentioned, or I disagree with the emphases, or with