Preface to analysis of functions of a single variable: a detailed  (Page 2/4)

There are many many fantastic mathematical truths (facts), and it seems to me that some of them are so beautiful andfundamental to human intellectual development, that a student who wants to be called a mathematician,ought to know how to explain them, or at the very least should have known how toexplain them at some point. Each professor might make up a slightly different list of such truths.Here is mine:

1. The square root of 2 is a real number but is not a rational number.
2. The formula for the area of a circle of radius $r$ is $A=\pi r^2.$
3. The formula for the circumference of a circle of radius $r$ is $C=2\pi r.$
4.   $e^{i\pi }=-1.$
5. The Fundamental Theorem of Calculus, ${\int }_a^{b}f\left(t\right)dt=F\left(b\right)-F\left(a\right).$
6. The Fundamental Theorem of Algebra, every nonconstant polynomial has at least one root in the complex numbers.
7. It is impossible to trisect an arbitrary angle using only a compass and straight edge.

Other mathematical marvels, such as the fact that there are more real numbers than there are rationals, the set of all sets is not a set,an arbitrary fifth degree polynomial equation can not be solved in terms of radicals, a simple closed curve dividesthe plain into exactly two components, there are an infinite number of primes, etc.,are clearly wonderful results, but the seven in the list above are really of a more primary nature to me, an analyst, for they stem from the work of ancient mathematicians and except for number 7,which continues to this day to evoke so-called disproofs, have been accepted as true by most people even in the absence of precise “arguments”for hundreds if not thousands of years. Perhaps one should ruminate on why it took so long for us to formulateprecise definitions of things like numbers and areas?

Only with the advent of calculus in the seventeenth century, together with the contributions of people like Euler, Cauchy, and Weierstrass duringthe next two hundred years, were the first six items above really proved, and only with the contributions ofGalois in the early nineteenth century was the last one truly understood.

This text, while including a traditional treatment of introductory analysis,specifically addresses, as kinds of milestones, the first six of these truths and gives careful derivations of them.The seventh, which looks like an assertion from geometry, turns out to be an algebraic result that is not appropriate for this course in analysis, but in my opinion it should definitely be presented in an undergraduate algebra course.As for the first six, I insist here on developing precise mathematical definitions of all the relevant notions,and moving step by step through their derivations. Specifically, what are the definitions of $\sqrt{2},A,\pi ,r,r^2,C,2,e,i,,$ and $-1?$ My feeling is that mathematicians should understand exactly where these concepts come from in precise mathematical terms, why it took so longto discover these definitions, and why the various relations among them hold.

The numbers $-1,2,$ and $i$ can be disposed of fairly quickly by a discussion of what exactlyis meant by the real and complex number systems. Of course, this is in fact no trivial matter,having had to wait until the end of the nineteenth century for a clear explanation, and in fact I leave the actual proof of the existence of the real numbers to an appendix.However, a complete mathematics education ought to include a study of this proof, and if one finds the time in thisanalysis course, it really should be included here. Having a definition of the real numbers to work with, i.e., havingintroduced the notion of least upper bound, one can relatively easily prove that there is a real number whose square is 2,and that this number can not be a rational number, thereby disposing of the first of our goals.All this is done in [link] . Maintaining the attitude that we should notdistinguish between functions of a real variable and functions of a complex variable,at least at the beginning of the development, [link] concludes with a careful introduction of the basic properties of the field of complex numbers.