Analysis — Chapter I Part I — Real Variables

Real Variables.

1Rational Numbers. A fraction r=p/q, where p and q are positive or negative integers, is called a rational number. We can assume (i) that p and q have no common factors, as if they have a common factor we can divide each of them by it, and (ii) that q is positive, since

p/(-q)=(-p)/q and (-p)/(-q)=p/q.

To the rational numbers thus defined we may add the “rational number 0” obtained by taking p=0.

We assume that you are familiar with the ordinary arithmetic rules for the manipulation of rational numbers. The examples which follow demand no knowledge beyond this.

Example I. 1. If r and s are rational numbers, then r+s, r-s, rs and r/s are rational numbers, unless in the last case s=0 (when r/s is meaningless, of course).

2. If \lambda, m, and n are positive rational numbers, and m>n, then

\lambda (m^{2}-n^{2}), 2\lambda mn, and \lambda (m^{2}+n^{2}) are positive rational numbers. Hence, show how to determine any number of right angled triangles the lengths of all of whose sides are rational.

Proof: Let the hypotenuse be \lambda (m^{2}+n^{2}) and the two arms of the right angled triangle be

2\lambda mn and \lambda (m^{2}-n^{2}). Then, the Pythagoras’s theorem holds. But, the sides and the hypotenuse are all rational.

3. Any terminated decimal represents a rational  number whose denominator contains no  factors other than 2 or 5. Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.

(We will look into this matter a bit deeper, a little later).

4. The positive rational numbers may be arranged in the form of a simple series as follows:

1/1,2/1,1/2, 3/1,2/2, 1/3,4/1,3/2,2/3,1/4, \ldots.

Show that p/q is the \{ (1/2)(p+q-1)(p+q-2) +q \}th term of the series.

(In this series, every rational number is repeated indefinitely. Thus 1 occurs as 1/1, 2/2, 3/3, \ldots We can of course avoid this by omitting every number which has already occurred in a simple form, but then the problem of determining the precise position of p/q becomes more complicated.) Check this for yourself! If you do not get the answer, just write back in the comment section and I will help clarify the matter.

More later…

Nalin Pithwa



Does one have to be a genius in order to be a mathematician

I plan to continue some such articles/discussion about the psychology of learning mathematics or being a great mathematician etc. since we all want to achieve greatness.

I had posed this question(Does one have to be a genius in order to be a mathematician) when I was beginning or many a times finding my graduate program in Math very tough. One of my bright friends, Muthu Muthiah, who had come from Indiana told me: Nalin, it’s very easy. Just become a monk of Mathematics !!!

Given below is the opinion of Prof. Terence Tao, the Mozart of Mathematics (I selected it from his blog):(perhaps, it will give you some hope):

To me, the biggest misconception that non-mathematicians have about how mathematicians think is that there is some mysterious mental faculty that is used to crack a problem all at once. In reality, one can ever think only a few moves ahead, trying out possible attacks from one’s arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas one understands. This is the same way that one solves problems in one’s first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try, perhaps making better guesses about what is likely to yield progress. Sometimes, during this process, a sudden insight comes, but it would not be possible without the painstaking groundwork [… ].


More later,

Nalin Pithwa

How to take lecture notes in maths or physics (Paul Halmos way)

How to take lecture notes (Paul Halmos way)

Assume: Math/Physics esp.

Assume: the lecturer is not dictating notes; but of course, he is writing as well as talking as well as thinking!!!

Lecture notes are a standard way to learn something — one of the worst ways. Too passive, that’s the trouble. Standard recommendation: take notes. Counter-argument: yes, to be sure, taking notes is an activity, and, if you do it, you have something solid to refer back to afterward, but you are likely to miss the delicate details of the presentation as well as the big picture, the Gestalt — you are too busy scribbling to pay attention. Counter counter argument: if you don’t take notes, you won’t remember what happened, in what order it came, and, chances are your attention will flag part of the time, you’ll daydream, and, who knows, you might even nod off.

It’s all true, the arguments both for and against taking notes. My own solution is a compromise: I take very skimpy notes, and then, whenever possible, I transcribe them, in much greater detail, as soon afterward as possible. By very skimpy notes, I mean something like one or two words a minute, plus, possibly a crucial formula or two — just enough to fix the order of events, and, incidentally to keep me awake and on my toes. By transcribe I mean in enough detail to show a friend who wasn’t there, with some hope that he’ll understand what he missed.

Note that if the lecturer draws some pictures (with notations or otherwise), I draw *all* pictures in my notebook.


Hope this is of use to serious students, enthusiastic students, and *amateur* mathematicians (like me!!! :- ))

More later,

Nalin Pithwa