Chapter I: Real Variables: Rational Numbers: Examples I

Examples 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 of course meaningless).


Part i): Given r and s are rational numbers. Let r=a/b, s=c/d, where a, b, c and d are integers, and b and d are not zero; where a and b do not have any common factors, where c and d do not have any common factors, and c and d are positive integers.

Then, r+s=a/b+c/d=(ad+bc)/bd, which is clearly rational as both the numerator and denominator are new integers (closure in addition and multiplication).

Part ii) Similar to part (i).

Part iii) By closure in multiplication.

Part iv) By definition of division in fractions, and closure in multiplication.

2) If \lambda , m, n are positive rational numbers, and m > n, then prove that \lambda(m^{2}-n^{2}), 2\lambda mn, \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.


This follows from problem 1 where we proved that the addition, subtraction and multiplication of rational numbers is rational.

Also, Pythagoras’ theorem holds in the following manner:

\lambda^{2}(m^{2}-n^{2})^{2}+(2\lambda m n)^{2}=\lambda^{2}(m^{2}+n^{2})^{2}

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.

Proof Part 1:

This is obvious since the divisors other than 2 or 5, namely, 3,6,7,9, and other prime numbers do not divide 1 into a terminated decimal.

Proof Part 2:

Since the process of division produces a unique quotient.

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 [\frac{p}{q}(p+q-1)(p+q-2)+q]th term of the series.


Suggested idea. Try by mathematical induction.

More later,

Nalin Pithwa



A Tribute to John Nash, Jr. Abel Laureate, Nobel Laureate: A Beautiful Mind

The following is a recollection of John Nash’s seminal contribution to Geometry. It includes some descriptions of his interactions with other mathematicians. I have picked it up  from his famous biography “A Beautiful Mind” by Sylvia Nasar.

There are two kinds of mathematical contributions: work that is important to the history of mathematics and work that is simply a triumph of the human spirit. — Paul J. Cohen, 1996.

In the spring of 1953, Paul Halmos, a mathematician at the University of Chicago, received the following letter from his old friend Warren Ambrose, a colleague of Nash’s:

There’s no significant news from here, as always. Martin is appointing John Nash to an Assistant Professorship (not the Nash at Illinois, the one out of Princeton by Steenrod) and I am pretty annoyed at that. Nash is a childish bright guy who wants to be “basically original,” which I suppose is fine for those who have some basic originality in them. He also makes a damned fool of himself in various ways contrary to this philosophy. He recently heard of the unsolved problem about imbedding a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem were sufficiently worthwhile to justify his efforts;; so he proceeded to write to everyone in the math society to check on that, was told that it probably was, and proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium. Meanwhile, he went to Levinson to inquire about a differential equation that and Levinson says it is a system of partial differential equations and if he could only get to the essentially simpler analog of a single ordinary differential equation it would be a damned good paper — and Nash had only the vaguest notions about the whole thing. So it is generally conceded he is getting nowhere and making an even bigger ass of himself than he has previously been supposed by those with less insight than myself. But we have got him and saved ourselves the possibility of having a gotten a real mathematician. He’s a bright guy but conceited as Hell, childish as Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y.

Ambrose had every reason to be both skeptical and annoyed.

Ambrose was a moody, intense, somewhat frustrated mathematician in his late thirties, full, as his letter indicates of dark humour. He was a radical and nonconformist. He married three times. He gave a lecture on “Why I am an atheist.” He once tried to defend some left-wing demonstrators against police in Argentina — and got himself beaten up and jailed for his efforts. He was also a jazz fanatic, a personal friend of Charlie Parker, and a fine trumpet player. Handsome, solidly built, with a boxer’s broken nose — the consequence of an accident in an elevator — he was one of the most popular members of the department. He and Nash clashed from the start.

Ambrose’s manner was calculated to give an impression of stupidity. “I am a simple man, I can’t understand this.” Robert Aumann recalled. “Ambrose came to class one day with one shoelace tied and the other untied. “Did you know your right shoelace is untied?” we asked. “Oh, my God,” he said, “I tied the left one and thought that the other must be tied by considerations of symmetry.”

The older faculty in the department mostly ignored Nash’s putdowns and jibes. Ambrose did not. Soon a tit-for-tat rivalry was under way. Ambrose, was famous, among other things, for detail. His blackboard notes were so dense that rather attempt the impossible task of copying them, one of his assistants used to photograph them. Nash, who disliked laborious, step-by-step expositions, found much to mock. When Ambrose wrote what Nash considered as an ugly argument on the blackboard during a seminar, Nash would mutter, “Hack, Hack” from the back of the room.

Nash made Ambrose the target of several pranks. “Seminar on the REAL mathematics!” read a sign that Nash posted one day. “The seminar will meet weekly Thursdays at 2PM in the Common Room.” Thursday at 2PM was the hour that Ambrose taught his graduate course in analysis. On another occasion, after Ambrose delivered a lecture at the Harvard mathematics colloquium, Nash, arranged to have a large bouquet of roses delivered to the podium as if Ambrose were a ballerina taking her bows.

Ambrose needled back. He wrote “F*** Myself” on the To Do list that Nash kept hanging over his desk on a clipboard. It was he who nicknamed Nash “Gnash” for constantly making belittling remarks about other mathematicians. And, during a discussion in the common room, after one of Nash’s diatribes about hacks and drones, Ambrose said disgustedly, “If you are so good, why don’t you solve the embedding problem for manifolds?” — a notoriously difficult problem that had been around since it was posed by Riemann.

So Nash did.

Two years later at the University of Chicago, Nash began a lecture describing his first really big theorem by saying, “I did this because of a bet.” Nash’s opening statement spoke volumes about who he was. He was a mathematician who viewed mathematics not as a grand scheme, but as a collection of challenging problems. In the taxonomy of mathematicians, there are problem solvers and theoreticians, and by temperament, Nash belonged to the first group. He was not a game theorist, analyst, algebraist, geometer, topologist, or mathematical physicist. But he zeroed in on  areas in these fields where essentially nobody had achieved anything. The thing was to find an interesting question that he could say something about.

Before taking on Ambrose’s challenge, Nash wanted to be certain that solving the problem would cover him with glory. He not only quizzed various experts on the problem’s importance, but according to Felix Browder, another Moore Instructor, claimed to have proved the result long before he actually had. When a mathematician at Harvard confronted Nash, recalled Browder: “Nash explained that he wanted to find out whether it was worth working on.”

“The discussion of manifolds was everywhere,” said Joseph Kohn in 1995, gesturing to the air around him. “The precise question that Ambrose asked Nash in the common room one day was the following: Is it possible to embed any Riemannian manifold in a Euclidean space?”

It’s a “deep philosophical question” concerning the foundations of geometry that virtually every mathematician — from Riemann and Hilbert to Elie-Joseph Cartan and Hermann Weyl — working in the field of differential geometry for the past century had asked himself. The question, first posed explicitly by Ludwig Schlaffi in the 1870s, had evolved naturally from a progression of other questions that had been posed and partly answered beginning in the mid-nineteenth century. First mathematicians studied ordinary curves, then surfaces, and finally, thanks to Riemann, a sickly German genius and one of the great figures of nineteenth century mathematics, geometric objects in higher dimensions. Riemann discovered examples of manifolds inside Euclidean spaces. But, in the early 1950s interest shifted to manifolds partly because of the large role that distorted space and time relationships had in Einstein’s theory of relativity.

Nash’s own description of the embedding problem in his 1995 Nobel autobiography hints at the reason he wished to make sure that solving the problem would be worth the effort: “This problem, although classical, was not much talked about is an outstanding problem. It was not like, for example, the four-colour conjecture.”

Embedding involves portraying a geometric object as — or, a bit more precisely, making it a subset of — some space in some dimension. Take the surface of a balloon. You can’t put it on a blackboard, which is a two-dimensional space. But you can make it a subset of spaces of three or more dimensions. Now take a slightly more complicated object, say a Klein bottle. A Klein bottle looks like a tin can whose lid and bottom have been removed and whose top has been stretched around and reconnected through the side to the bottom. If you think about it, it’s obvious that if you try that in three-dimensional space, the thing intersects itself. That’s bad from a mathematical point of view because the neighbourhood in the immediate vicinity of the intersection looks weird and irregular, and attempts to calculate various attributes like distance or rates of change in that part of the object tend to blow up. But, put the same Klein bottle into a space of 4 dimensions and the thing no longer intersects itself. Like a ball embedded in three space, a Klein bottle in four space becomes a perfectly well-behaved manifold.

Nash’s theorem stated that any kind of surface that embodied a special notion of smoothness can actually be embedded in Euclidean space. He showed that you could fold the manifold like a silk  handkerchief without distorting it. Nobody would have expected Nash’s theorem to be true. In fact, everyone would have expected it to be false. “It showed incredible originality,” said Mikhail Gromov, the geometer whose book Partial Differential Relations builds on Nash’s work. He went on:

“Many of us have the power to develop existing ideas. We follow paths prepared by others. But most of us could never produce anything comparable to what Nash produced. It’s like lightning striking. Psychologically the barrier he broke is absolutely fantastic. He has completely changed the perspective of partial differential equations. There has been some tendency in recent decades to move from harmony to chaos. Nash says chaos is just round the corner.”

John Conway, the Princeton mathematician who discovered surreal numbers and invented the game of Life, called Nash’s result “one of the most important pieces of mathematical analysis in this century.”

It was also, one must add, a deliberate jab, at then-fashionable approaches to Riemannian manifolds, just as Nash’s approach to theory of games was a direct challenge to von Neumann’s. Ambrose, for example, was himself involved in a highly abstract and conceptual description of such manifolds at the time. As Jurgen Moser, a young German mathematician who came to know Nash well in the mid-1950’s, put it, “Nash didn’t like that style of mathematics at all. He was out to show that this, to his mind, exotic approach was completely unnecessary since any such manifold was simply a submanifold of a high dimensional Euclidean space.”

Nash’s important achievement may have been the powerful technique he invented to obtain his result. In order to prove his theorem, Nash had to confront a seemingly insurmountable obstacle, solving a certain set of partial differential equations that were impossible to solve with existing methods.

That obstacle cropped up  in many mathematical and physical problems. It was the difficulty that Levinson, according to Ambrose’s letter, pointed out to Nash, and it is a difficulty that crops up in many many problems — in particular, nonlinear problems. Typically, in solving an equation, the thing that is given is some function, and one finds estimates of derivatives of a solution in terms of derivatives of the given function. Nash’s solution was remarkable in that the a priori estimates lost derivatives. Nobody knew how  to deal with such equations. Nash invented a novel iterative method — a procedure for making a series of educated guesses — for finding roots of equations, and combined it with a technique for smoothing to counteract the loss of derivatives.

Newman described Nash as a ‘very poetic, different kind of thinker.” In this instance, Nash used differential calculus, not geometric pictures or algebraic manipulations, methods that were classical outgrowths of nineteenth-century calculus. The technique is now referred to as Nash-Moser theorem, although there is no dispute that Nash was its originator. Jurgen Moser was to show how Nash’s technique could be modified and applied to celestial mechanics — the movement of planets — especially, for establishing the stability of periodic orbits.

Nash solved the problem in two steps. He discovered that one could embed a Riemannian manifold in a three-dimensional space if one ignored smoothness. One had, so to speak, to crumple it up. It was a remarkable result, a strange and interesting result, but a mathematical curiosity, or so it seemed. Mathematicians were interested in embedding without wrinkles, embedding in which the smoothness of the manifold could be preserved. 

In his  autobiographical essay, Nash wrote:

So, as it happened, as soon as I heard in conversation at MIT about the question of embeddability being open I begann to study it. The first break led to a curious result about the embeddability being realizable in surprisingly low-dimensional ambient spaces provided that one would accept that the embedding would have only limited smoothness. And later, with “heavy analysis”, the problem was solved in terms of embedding with a more proper degree of smoothness.”

Nash presented his initial “curious” result at a seminar in Princeton, most likely in the spring of 1953, at around the same time that Ambrose wrote his scathing letter to Halmos. Emil Artin was in the audience. He made no secret of his doubts.

“Well, that’s all well and good, but what about the embedding theorem?” said Artin. “You’ll never get it.”

“I’ll get it next week,” Nash shot back.

One night, possibly en route to this very talk, Nash was hurtling down the Merritt Parkway. Poldy Flatto was riding with him as far as the Bronx. Flatto, like all the other graduate students, knew that Nash was working on the embedding problem. Most likely to get Nash’s goat and have the pleasure of watching his reaction, he mentioned that Jacob Schwartz, a brilliant young mathematician at Yale whom Nash knew slightly, was also working on the problem.

Nash became quite agitated. He gripped the steering wheel and almost shouted at Flatto, asking whether he had meant to say that Schwartz had solved the problem. “I didn’t say that,” Flatto corrected. “I said I heard he was working on it.”

“Working on it?” Nash replied, his whole body now the picture of relaxation. “Well, then there’s nothing to worry about. He doesn’t have the insights I have.”

Schwartz was indeed working on the same problem. Later, after Nash had produced his solution, Schwartz wrote a book on the subject of implicit-function theorems. He recalled in 1996:

“I got half the idea independently, but I couldn’t get the other half. It’s easy to see an approximate statement to the effect that not every surface can be exactly embedded, but that you can come arbitrarily close. I  got that idea and I was able to produce the proof of the easy half in a day. But then I realized that there was a technical problem. I worked on it for a month and couldn’t see any way to make headway. I ran into an absolute stone wall. I didn’t know what to do. Nash worked on that problem for two years with a sort of ferocious, fantastic tenacity until he broke through it.”

Week after week, Nash would turn up  in Levinson’s office, much as he had in Spencer’s, at Princeton. He would describe to Levinson what he had done and Levinson would show him why it didn’t work. Isadore Singer, a fellow Moore Instructor, recalled:

“He’d show the solutions to Levinson. The first few times he was dead wrong. But, he didn’t give up. As he saw the problem get harder and harder, he applied himself more, and more and more. He was motivated just to show everybody how good he was, sure, but on the other hand he didn’t give up even when the problem turned out to much harder than expected. He put more and more of himself into it.”

There is no way of knowing what enables one man to crack a problem while another man, also brilliant, fails. Some geniuses have been sprinters who have solved problems quickly. Nash was a long-distance runner. If Nash defied von Neumann in his approach to the theory of games, he now took on the received wisdom of nearly a century. He went into a classical domain where everybody understood what was possible and what was not possible. “It took enormous courage to attack these problems,” said Paul Cohen, a mathematician at Stanford University and a Fields medalist. His tolerance for solitude, great confidence in his own intuition, indifference to criticism — all detectable at a young age but now prominent and impermeable features of his personality — served him well. He was a hard worker by habit. He worked mostly at night in the MIT office — from ten in the evening until 3.00AM — and on weekends as well, with, as one observer said, “no references, but his own mind and his supreme self-confidence.” Schwartz called it “the ability to continue punching the wall until the stone breaks.”

The most eloquent description of Nash’s single-minded attack on the problem comes from Moser:

“The difficulty that Levinson had pointed out, to anyone in his right mind, would have stopped them cold and caused them to abandon the problem. But Nash was different. If he had a hunch, conventional criticism didn’t stop him. He had no background knowledge. It was totally uncanny. Nobody could understand how somebody like that could do it. He was the only person I ever saw with that kind of power, just brute mental power.”

The editors of the Annals of Mathematics hardly knew what to make of Nash’s manuscript when it landed on their desks at the end of October 1954. It hardly had the look of a mathematics paper. It was as thick as a book, printed by hand rather than typed and chaotic. It made use of concepts and terminology more familiar to engineers than to mathematicians. So, they sent it to a mathematician at Brown University, Herbert Federer, and Austrian born refugee from Nazism and a pioneer in surface area theory, who, although only thirty-four, already had a reputation for high standards, superb taste, and an unusual willingness to tackle difficult manuscripts.

Mathematics is often described, quite rightly, as the most solitary of endeavours. But when a serious mathematician announces that he had found the solution to an important problem, at least one other serious mathematician, and sometimes several, as a matter of longstanding tradition that goes back hundreds of years, will set aside his own work for weeks and months at a time, as one former collaborator of Federer put it, “to make a go of it, and to straighten everything out.” Nash’s manuscript presented Federer with a sensationally complicated puzzle and he attacked the task with relish.

The collaboration between the author and referee took months. A large correspondence, many telephone conversations, and numerous drafts ensued. Nash did not submit the revised version of the paper until nearly the end of the following summer. His acknowledgement to Federer was, by Nash’s standards effusive. “I am profoundly indebted to H. Federer, to whom may be traced most of the improvement over the first chaotic formulation of this work.”

Armand Borel, who was a visiting professor at Chicago when Nash gave a lecture on his embedding theorem, remembers the audience’s shocked reaction. “Nobody believed his proof at first,” he recalled in 1995. “People were very skeptical. It looked like a beguiling idea. But when there’s no technique you are skeptical. You dream about a vision. Usually you are missing something. People did not challenge him publicly, but they talked privately.” (Characterically, Nash’s report to his parents merely said, ‘talks went well.’)

Gian-Carlo Rota, professor of mathematics and philosophy at MIT confirmed Borel’s account. “One of the great experts on the subject told me that if one of his graduate students had proposed such an outlandish idea he’d have thrown him out of his office.

The result was so unexpected and Nash’s methods so novel, that even the experts had tremendous difficulty understanding what he had done. Nash used to have drafts lying around the MIT common room. A former MIT graduate student recalls a long and confused discussion between Ambrose, Singer and Masatake Kuranishi, (a mathematician at Columbia University who later applied Nash’s result), in which each one tried to explain Nash’s result to the other without much success.

Jack Schwartz recalled:

“Nash’s solution was not just novel, but very mysterious, a mysterious set of weird inequalities that all came together. In my explication of it I sort of looked at what happened and could generalize and give an abstract form and realize it was applicable to situations other the specific one he treated. But, I didn’t quite get to the bottom of it either.”

Later, Heinz Hopf, professor of mathematics in Zurich and a past president of the International Mathematical Union, “a great man with a small build, friendly, radiating a warm glow, who knew everything about differential geometry,” gave a talk on Nash’s embedding theorem in New York. Usually, Hopf’s lectures were models of crystalline clarity. Moser, who was in the audience recalled. “So we thought NOW we will understand what Nash did. He was naturally skeptical. He would have been an important validator of Nash’s work. But, as the lecture went on, my God, Hopf was befuddled himself. He couldn’t convey a complete picture. He was completely overwhelmed.”

Several years later, Jurgen Moser tried to get Nash to explain how he had overcome the difficulties that Levinson had originally pointed out:”I did not learn so much from him. When he talked, he was vague, hand waving. ‘You have to control this. You have to watch out for that.’ You couldn’t follow him. But, his written paper was complete and correct.” Federer not only edited Nash’s paper to make it more accessible, but also was the first to convince the mathematical community that Nash’s theorem was indeed correct.

Martin’s surprise proposal, in the early part of 1953, to offer Nash a permanent faculty position set off a storm of controversy among the eighteen-member mathematics faculty. Levinson and Wiener were among Nash’s strongest supporters. But, others like Warren Ambrose and George Whitehead, the distinguished topologist, were opposed. Moore Instructorships weren’t meant to lead to tenure-track positions. More to the point, Nash had plenty of enemies and few friends in his first year and a half. His disdainful manner towards his colleagues and his poor record as a teacher rubbed many the wrong way.

Mostly, however, Nash’s opponents were of the opinion that he hadn’t proved he could produce. Whitehead recalled, “He talked big. Some of us were not sure he could live up to his claims.” Ambrose, not surprisingly, felt similarly. Even Nash’s champions could not have been completely certain. Flatto remembered one occasion on which Nash came to Levinson’s office to ask Levinson whether he’d a draft of his embedding paper. Levinson said, “To tell you the truth I don’t have enough background in this area to pass judgement.”

When Nash finally succeeded, Ambrose did what a fine mathematician and sterling human being would do. His applause was as loud as or louder than anyone else’s. The bantering became friendlier and among other things, Ambrose took to telling his musical friends that Nash’s whistling was the purest, most beautiful tone he had ever heard.


Until the next blog,

Good bye

Nalin Pithwa




Abel Laureates 2015 John Nash, Jr. and Louis Nirenberg

The leading lights at Courant were very much at the forefront of rapid progress, stimulated by World War II, in certain kinds of differential equations that serve as mathematical models for an immense variety of physical phenomena involving some sort of change. By the mid-fifties, as Fortune noted, mathematicians knew relatively simple routines for solving ordinary differential equations using computers. But there were no straightforward methods for solving most nonlinear partial differential equations that crop up when large or abrupt changes occur — such as equations that describe the aerodynamic shock waves produced when a jet accelerates past the speed of sound. In his 1958 obituary of von Neumann, who did important work in this field in the thirties, Stanislaw Ulam called such systems of equations “baffling analytically” saying that they “defy even qualitative insights by present methods.” As Nash was to write that same year, “The open problems in the area of non-linear partial differential equations are very relevant to applied mathematics and science as a whole, perhaps, more so than the open problems in any other area of mathematics, and this field seems poised for rapid development. It seems clear however that fresh methods must be employed.”

Nash, partly because of his contact with Norbert Wiener and perhaps his earlier interaction with Weinstein at Carnegie, was already interested in the problem of turbulence. Turbulence refers to the flow of gas or liquid over any uneven surface, like water rushing into a bay, heat or electrical charges travelling through metal, oil escaping from an underground pool, or clouds skimming over an air mass. It should be possible to model such motion mathematically. But, it turns out to be extremely difficult. As Nash wrote:

Little is known about the existence, uniqueness and smoothness of solutions of the general equations of flow for a viscous, compressible, and heat conducting fluid. These are a non-linear parabolic system of equations. An interest in these questions led us to undertake this work. It became clear that nothing could be done about the continuum description of general fluid flow without the ability to handle non-linear parabolic equations and this in turn required an a priori estimate of continuity.

It was Louis Nirenberg, a short, myopic, and sweet-natured young protege of Courant’s, who had handed Nash a major unsolved problem in the then fairly new field of nonlinear theoty. Nirenberg, also in his twenties then, and already a formidable analyst, found Nash a bit strange. “He’d often seemed to have an internal smile, as if he was thinking of a private joke, as he was laughing at a private joke that he had never told anyone about.” But he was extremely impressed with the technique Nash had invented for solving his embedding theorem and sensed that Nash might be the man to crack an extremely difficult outstanding problem that had been open since the late 1950s:

He (Nirenberg) had recalled:

I worked in partial differential equations. I also worked in geometry. The problem had to do with certain kinds of inequalities associated with elliptic partial differential equations. The problem had been around in the field for some time and a number of people had worked on it. Someone had obtained such estimates much earlier in the 1930s in two dimensions. But the problem was open for almost thirty years in higher dimensions.

Nash had begun working on the problem almost as soon as Nirenberg suggested it, although he knocked on doors until he had been satisfied that the problem was as important as Nirenberg had claimed. Peter Lax, who was one of these he had consulted, had commented some time back: In physics, everybody knows the most important problems. They are well-defined. Not so in mathematics. People are more introspective. For Nash, though, it had to be important in the opinion of others.

Nash had started visiting Nirenberg’s office to discuss his progress. But, it was weeks before Nirenberg got any real sense that Nash was getting anywhere. “We would meet often. Nash would say, “I seem to need such and such an inequality. I think it’s true that…” Very often, Nash’s speculations were far off the mark. He was sort of groping. He gave that impression. I wasn’t very confident he was going to get through.

Nitenberg had then sent Nash around to talk to Lars Hormander, a tall, steely Swede who was then already one of the top scholars in the field. Precise, careful, and immensely knowledgeable, Hormander knew Nash, by reputation but had reacted even more skeptically than Nirenberg. “Nash had learned from Nirenberg the importance of extending the Holder estimates known for second-order elliptic equations with two variables and irregular coefficients to higher dimensions,” Hormander had recalled in 1997. “He came to see me several times. ‘What did I think of such and such an inequality?’ At first, his conjectures were obviously false. They were easy to disprove by known facts on constant coefficient operators. He was rather inexperienced in these matters. Nash did things from scratch without using standard techniques. He was always trying to extract problems…(from conversations with others). He had not the patience to study them.”

Nash had continued to grope, but with more success. “After a couple more times,” said Hormander, “he would come up with things that were not so obviously wrong.”

By  the spring, Nash was able to obtain basic existence, uniqueness, and continuity theorems once again using novel methods of his own invention. He had a theory that difficult problems couldn’t be attacked frontally. He had approached the problem in an ingeniously roundabout manner, first transforming the nonllnear equations into linear equations and then attacking these by nonlinear means. “It was a stroke of genius,” said Peter Lax, who had followed the progress of Nash’s research closely. “I have never seen that done. I always kept it in my mind, thinking may be, it will work in another circumstance.”

(Note: Peter Lax is an earlier Abel Laureate).

Nash’s new result had gotten far more immediate attention than his embedding theorem. It had convinced Nirenberg, too, that Nash was a genius. Hormander’s mentor of the University of Lund, Lars Garding, a world class specialist in partial differential equations, had immediately declared, “You have to be a genius to do that.”


More later,

Nalin Pithwa

Analysis — Chapter 1: continued — Real Variables part 9

9. Relations of magnitude between real numbers. 

It is plain, that, now that we have extended our conception of number, we are bound to make corresponding extensions of our conceptions of equality, inequality, addition, multiplication, and so on. We have to show that these ideas can be applied to the  new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of Chapter 1, part 1 blog. To do all this systematically would occupy considerable space/time, and we shall be content to indicate summarily how a more systematic discussion would proceed.

We denote a real number by a Greek letter such as \alpha, \beta, \gamma\ldots; the rational numbers of its lower and upper classes by the corresponding English letters a, A; b, B; c, C; …We denote the classes themselves by (a), (A),…

If \alpha and \beta are two real numbers, there are three possibilities:

i) every \alpha is a b and every A a B; in this case, (a) is identical with (b) and (A) with (B);

ii) every a in a b, but not all A’s are B’s; in this case (a) is a proper part of (b)^{*}, and (B) a proper part of (A);

iii) every A is a B, but not all a’s are b’s.

(These three cases may be indicated graphically on a number line).

In case (i) we write \alpha=\beta, in case (ii) \alpha=\beta, and in case (iii) \alpha>\beta. It is clear that, when \alpha and \beta are both rational, these definitions agree with the ideas of equality and inequality between rational numbers which we began by taking for granted; and that any positive number is greater than any negative number.

It will be convenient to define at this stage the negative -\alpha of a positive number \alpha. If

(\alpha), (A) are the classes, which consitute \alpha, we can define another section of the rational numbers by putting all numbers -A in the lower class and all numbers -\alpha in  the upper. The real number thus defined, which is clearly negative, we denote by -\alpha. Similarly, we can define

-\alpha when \alpha is negative or zero; if \alpha is negative, -\alpha is positive, It is plain also  that -(-\alpha)=\alpha. Of the two numbers \alpha and -\alpha one is always positive (unless \alpha=0). The one which is positive we denote by |\alpha| and call the modulus of \alpha.

More later,

Nalin Pithwa

Analysis — Chapter 1 — Real Variables — part 8

8. Real numbers. We have confined ourselves so far to certain sections of the positive rational numbers, which we have agreed provisionally to call “positive real numbers.” Before we frame our final definitions, we must alter our point of view a little. We shall consider sections, or divisions into two classes, not merely of the positive rational numbers, but of all rational numbers, including zero. We may then repeat all that we have said about sections of the positive rational numbers in part 6 and 7 merely omitting the word positive occasionally.

Definitions. A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a real number, or simply a number.

A number which does not correspond to a rational number is called an irrational number.

If the real number does correspond to a rational number, we shall use the term “rational” as applying to the real number line.

The term “rational number” will, as a result of our definitions, be ambiguous, it may mean the rational number of part 1, or the, corresponding real number. If we say that 1/2 > 1/3, we may  be asserting either of the two different propositions, one a proposition of elementary arithmetic, the other a proposition concerning sections of the rational numbers. Ambiguities of this kind are common in mathematics, and are perfectly harmless, since the relations between different propositions are exactly the same whichever interpretation is attached to the propositions themselves. From 1/2>1/3 and 1/3>1/4 we can infer 1/2>1/4; the inference is in no way affected by any doubt as to whether 1/2, 1/3 and 1/4 are arithmetic fractions or real numbers. Sometimes, of course, the context in which (example) ‘1/2‘ occurs is sufficient to fix its interpretation. When we say (next blog part 9) that 1/2 < \sqrt{1/3}we must mean by ‘1/2‘ the real number 1/2.

The reader should observe, moreover, that no particular logical importance is to be attached to the precise form of definition of a ‘real number’ that we have adopted. We defined ‘a real number’ as being a section, that is, a pair of classes. We might equally well have defined it to being the lower, or the upper class; indeed it would be easy to define an infinity of classes of entities of each of which would possess the properties of the class of real numbers. What is essential in mathematics is that its symbols should be capable of some interpretation; generally they are capable of many, and then so far as mathematics is concerned, it does not matter which we adopt. Mr. Bertrand Russell has said that “mathematics is the science in which we do not know what we are talking about, and do not care what we say about it is true”, a remark which is expressed in the form of paradox but which in reality embodies a number of important truths. It would take too long to analyze the meaning of Mr Russell’s epigram in detail, but one at any rate of the implications is this, that the symbols of mathematics are capable of varying interpretations, and that we are in general at liberty to adopt whatever we prefer.

There are now three cases to distinguish. It may happen that all negative rational numbers belong to the lower class and zero and all positive rational numbers to the upper. We describe this section as the real number zero. Or, again it may happen that the lower class includes some positive numbers. Such a section we as a positive real number. Finally, it may happen that some negative numbers belong to the upper class. Such a section we describe as a negative real number. 

Note: The difference between our presentation of a positive real number here and that or part 7 of the blogs amounts to the addition to the lower class of zero and all the negative rational numbers. An example of a negative real number is given by taking the property P of part 6 of the blogs to be x+1<0 and Q to be x+1 \geq 0/ This section plainly corresponds to the negative rational number -1. If we took P to be x^{3}<-2 and Q to be x^{3}>-2, we should obtain a negative real number which is not rational.

More later,

Nalin Pithwa



Analysis — Chapter 1 Real Variables — part 7 — continued

Part 7. Irrational numbers (continued).

In the first two cases, we say that the section corresponds to a positive rational number a, which is l in the one case and r in the other. Conversely, it is clear that to any such number a corresponds a section which we shall denote by

\alpha^{*}. For we might take P and Q to be the properties expressed by

x \leq a, x > a

respectively, or by x<a and x \leq a. In the first case, a would be the greatest number of L, and in the second case the least member of R. These are in fact just two sections corresponding to any positive rational number. In order to avoid ambiguity we select one of them; let us select that in which the number itself belongs to the upper class. In other words, let us agree that we will consider only sections in which the lower class L has no greatest number.

There being this correspondence between the positive rational numbers and the sections defined by means of them, it would be perfectly legitimate, for mathematical purposes, to replace the numbers by the sections, and to regard the symbols which occur in our formulae as standing for the sections instead of for the numbers. Thus, for example,

\alpha > \alpha^{'} would mean the same as a > a^{'}. If \alpha and \alpha^{'} are

the sections which correspond to a and a^{'}.

But, when we have in this way substituted sections of rational numbers for the rational numbers themselves, we are almost forced to a generalization of our number system. For there are sections (such as that of blog on Chapter 1 — part 4) which do not correspond to any rational number. The aggregate of sections is a larger aggregate than that of the positive rational numbers; it includes sections corresponding to all these numbers, and more besides. It is this fact which we make the basis of our generalization of the idea of a number. We accordingly frame the following definitions, which will however be modified in the next blog, and must therefore be regarded as temporary and provisional.

A section of the positive rational numbers, in which both classes exist and the lower class has no greatest member, is called a positive real number.

A positive real number which does not correspond to a positive rational number is called a positive irrational


More later,

Nalin Pithwa

The Universal Appeal of Mathematics — Geetha S. Rao

I am reproducing an article, “The Universal Appeal of Mathematics — Geetha S. Rao” from “The Mathematics Student” , volume 83, Numbers 1 to 4, (2014), 01-04.

The purpose is just to share this beautiful article with the wider student community and math enthusiasts.

The Universal Appeal of Mathematics: Geetha S. Rao:

Mathematics is the Queen of all Sciences, the King of all Arts and the Master of all that is being surveyed. Such is the immaculate and immense potential of the all-pervasive, fascinating subject, that it transcends all geographical barriers, territorial domains and racial prejudices.

The four pillars that support the growth, development, flowering and fruition of this ever green subject are analytic thinking, logical reasoning, critical reviewing and decision thinking.

Every situation in real life can be modelled and simulated in mathematical language. So much so, every human must be empowered with at least a smattering of mathematical knowledge. Indeed, the field of Artificial Intelligence is one where these concepts are implemented and imparted to the digital computers of today.

From times immemorial, people know how to count and could trade using the barter system. Those who could join primary schools learnt the fundamental arithmetic and algebraic rules. Upon entry into high school and higher secondary classes, the acquaintance with the various branches of this exciting subject commences. It is at this point that effective communication skills of the teacher impact the comprehension and conceptual understanding of the students.

Unfortunately, if the teacher is unsure of the methods and rules involved, then begins a dislike of the subject by the students being taught. To prevent a carcinogenic spread of the dislike, the teacher ought to be suitably oriented and know precisely how to captivate the imagination of the students. If this is the case, the students enjoy learning process and even start loving the subject, making them eagely await Mathematics classes, with bated breath!

Acquiring necessary knowledge of algebraic operations, permutations and combinations, rudiments of probabilistic methods, persuasive ideas from differential and integral calculus and modern set theory will strengthen the bonds of mathematical wisdom.

From that stage, when one enters the portals of university education, general or technical, the opportunity to expand one’s horizon of mathematical initiation is stupendous. Besides, the effective use of Mathematics in Aeronautical, Agricultural, Biological, Chemical, Geographical and Physical Sciences, Engineering, Medicine, Meteorology, Robotics, Social Sciences and other branches of knowledge is indeed mind boggling.

Armed with this mathematical arsenal, the choice of a suitable career becomes very diverse. No two humans need to see eye to eye as far as such a choice is concerned, as the variety is staggering! So, it is crystal clear that studying Mathematics,at every level, is not only meaningful and worthwhile but absolutely essential.

A natural mathematical genius like Srinivasa Ramanujan was and continues to be an enigma and a Swayambhu, who could dream of extraordinary mathematical formulae, without any formal training.

A formally trained mathematician is capable of achieving laudable goals and imminent success in everything that he chooses to learn and if possible, discover for himself, the eternal truths of mathematics, provided he pursues the subject with imagination, passion, vigour and zeal.

Nothing can be so overwhelming as a long standing problem affording a unique solution, bu the creation of new tools, providing immense pleasure, a sense of reward and tremendous excitement in the voyage of discovery.

These flights of imagination and intuition form the core of research activities. With the advent of the computers, numerical algorithms gained in currency and greater precision, enabling the mathematical techniques to grow by leaps and bounds!

Until the enumeration of the Uncertainty Principle by Werner Heisenberg, in 1932, mathematics meant definite rules of certainty. One may venture to say that this is the origin of Fuzziness. Lotfi Zadeh wrote a seminal paper, entitled Fuzzy sets, Information and Control,

8, 1965, 328-353. He must be considered a remarkable pioneer who invented the subject of Fuzzy mathematics, which is the amalgam of mathematical rules and methods of probability put together to define domains of fuzziness.

Fuzzy means frayed, fluffy, blurred or indistinct. On a cold wintry day, haziness is seen around at dawn, and a person or an object at a distance, viewed through the mist, will appear hazy. This is a visual representation of fuzziness. The input variables in a fuzzy control systems are mapped into sets of membership functions known as fuzzy sets. The process of converting a crisp input value to a fuzzy value is called fuzzification.

A control system may also have various types of switches or on-off inputs along with its analog inputs, and such switch inputs will have a truth value equal to either 0 or 1.

Given mappings of input variables into membership functions and truth values, the micro controller makes decisions concerning what action should be taken, based on a set of rules. Fuzzy concepts are those that cannot be expressed as true or false, but rather as partially true!

Fuzzy logic is involved in approximating rather than precisely determining the value. Traditional control systems are based on mathematical models in which one or more differential equations that define the system’s response to the inputs will be used. In many cases, the mathematical model of the control process may not exist, or may be too expensive, in terms of computer processing power and memory, and a system based on empirical rules may be more effective.

Furthermore, fuzzy logic is more suited to low cost implementation based on inexpensive sensors, low resolution analog-to-digital converters and 4-bit or 8 bit microcontroller chips. Such systems can be easily upgraded by adding new rules/novel features to improve performance. In many cases, fuzzy control can be used to enhance the power of existing systems by adding an extra layer of intelligence to the current control system. In practice, there are several different ways to define a rule, but the most simple one employed is the max-min inference method, in which the output membership function is given the truth value generated by the underlying premise. It is important to note that rules involved in hardware are parallel, while in software they are sequential.

In 1985, interest in fuzzy systems was sparked by the Hitachi company in Japan, whose experts demonstrated the superiority of fuzzy control systems for trains. These ideas were quickly adopted and fuzzy systems were used to control accelerating, braking, and stoppage of electric trains, which led to the historic introduction, in 1987, of the bullet train, with a speed of 200 miles per hour, between Tokyo and Sendai.

During an international conference of fuzzy researchers in Tokyo, in 1987, T. Yamakawa explained the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an inverted pendulum experiment. The Japanese soon became infatuated with fuzzy systems and implemented these methods in a wide range of astonishing commercial and industrial applications.

In 1988, the vacuum cleaners of Matsushita used micro controllers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. The Hitachi washing machines used fuzzy controllers to load-weight, fabric-mix and dirt sensors and automatically set the wash cycle for the optimum use of power, water and detergent.

The renowned Canon camera company developed an auto-focusing camera that used a charge coupled device to measure the clarity of the image in six regions in its field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing and controls its speed to prevent overshoot.

Work on fuzzy systems is also being done in USA, Europe, China and India. NASA in USA has studied fuzzy control for automated space docking, as simulation showed that a fuzzy control system can greatly reduce fuel consumption. Firms such as Boeing, General Motors, Allen-Bradley, Chrysler, Eaton and Whirlpool have used fuzzy logic to improve on automotive transmission, energy efficient electric meters, low power refrigerators, etc.

Researchers are concentrating on many applications of fuzzy control systems, have developed fuzzy systems and have integrated fuzzy logic, neural networks and adaptive genetic software systems, with the ultimate goal of building self-learning fuzzy control systems.

This, in my opinion, is sufficient reason to  induce you to start learning mathematics!

Geetha S. Rao,

Ex Professor, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, 

Chepauk, Chennai 600005.



More later,

Nalin Pithwa

Analysis — Chapter 1 — Real Variables: part 6: Irrational numbers continued

6. Irrational numbers (continued).

In Part 4, we discussed a special mode of division of the positive rational numbers x into two classes, such that x^{2}<2 for the numbers of one class and x^{2}>2 for those of the others. Such a mode of division is called a section of the numbers in question. It is plain that we could equally well construct a section in which the numbers of the two classes were characterized by the inequalities

x^{3}<2 and x^{3}>2, or x^{4}>7 and x^{4}<7. Let us now attempt to state the principles of the construction of such “sections” of the positive rational numbers in quite general terms.

Suppose that P and Q stand for two properties which are mutually exclusive and one of which must be possessed by every positive rational number. Further, suppose that every such number which possesses P is less than any such number which possesses Q. Thus, P might be the property “x^{2}<2” and Q the property “x^{2}>2“. Then, we call the numbers which possess P the lower or left-hand class L and those which possess Q the upper or right hand class R. In general, both classes will exist; but, it may happen in special cases that one is non-existent and every number belongs to the other. This would obviously happen, for example, if P (or Q) were the property of being rational, or of being positive. For the present, however, we shall confine ourselves to cases  in which both the classes do exist; and then it follows, as in Part 4, that we can find a member of L and a member of R, whose difference is as small as we please.

In the particular case, which we considered in Part 4, L had no greatest member and R no least. This question of the existence of greatest or least members of the classes is of the utmost importance. We observe first that it is impossible in any case that L should have a greatest member and R least. For, if l were the greatest member of L, and r the least of R, so that l<r, then (1/2)(l+r) would be a positive rational number lying between l and r, and so could neither belong to L nor to R, and this contradicts our assumption that every such number belongs to one class  or to the other.  This being so, there are but three possibilities, which are mutually exclusive. Either

(i) L has a greatest member l, or (ii) R has a least member, r, or (iii) L has no greatest member and R no least.

(In Part 4, there is an example of the last possibility.)

More later,

Nalin Pithwa