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

Analysis — Chapter I — Real Variables — Part 5 — Irrational numbers continued

We have thus divided the positive rational numbers into two classes, L and R, such that (i) every member of R is greater than every member of L, and (ii) we can find a member of L and a member of R, whose difference is as small as we please, (iii) L has no greatest and R has not least member. Our common-sense notion of the attributes of a straight line, the requirements of our elementary geometry and our elementary algebra, alike demand the existence of a number x greater than all the members of L and less than all the members of R, and of a corresponding point P on \Lambda such that P divides the points which correspond to members of L from those which correspond to members of R.

Let us suppose for a moment that there is such a number x and that it may be operated upon in accordance with laws of algebra, so that, for example, x^{2} has a definite meaning. Then x^{2} cannot either be less than or greater than 2. For suppose, for example, that x^{2} is less than 2. Then, it follows from what precedes that we can find a positive rational number \xi such that \xi^{2} lies between x^{2} and 2. That is to say, we can find a member of L greater than x; and this contradicts the supposition that x divides the members of L from those of R. Thus, x^{2} cannot be less than 2, and similarly, it cannot be greater than 2. We are therefore driven to the conclusion that x^{2}=2, and that x is the number which in algebra  we denote by \sqrt{2}. And, of course, this number \sqrt{2} is not rational, for no rational number has its square equal to 2. It is the simplest example of what is called an irrational number.

But the preceding argument may be applied to equations other than x^{2}=2, almost word for word; for example, to

x^{2}=N, where N is an integer which is not a perfect square, or to

$latex $x^{3}=3$ and x^{2}=7 and x^{4}=23,

or, as we shall see later on. to x^{3}=3x+8. We are thus led to believe for the existence of irrational numbers x and points P on \Lambda such that x satisfies equations such as these, even when these lengths cannot (as \sqrt{2} can) be constructed by means of elementary geometric methods.

The reader may now follow one or other of two alternative courses. He may, if he pleases, be content to assume that “irrational numbers” such as \sqrt{2} and \sqrt[5]{3} exist and are amenable to usual algebraic laws. If he does this, he will be able to avoid the more abstract discussions of the next few blogs.

If, on the other hand, he is not disposed to adopt so naive an attitude, he will be well advised to pay careful attention to the blogs which follow, in which these questions receive further consideration.

More later,

Nalin Pithwa


What are worthwhile problems as per Richard Feynman, American, Physics Nobel Laureate

What are worthwhile problems as per Richard Feynman

The letter below is from Perfectly Reasonabe Deviations From The Beaten Track, a book of letters of Richard Feynman. It is one of the most moving letters that I have read. Tomonaga mentioned below shared the 1965 Nobel prize for physics along with Feynman and Schwinger.

A former student, who was also once a student of Tomonaga’s, wrote to extend his congratulations. Feynman responded, asking Mr. Mano what he was now doing. The response: “studying the Coherence theory with some applications to the propagation of electromagnetic waves through turbulent atmosphere… a humble and down-to-earth type of problem.”

Dear Koichi,

I was very happy to hear from you, and that you have such a position in the Research Laboratories. Unfortunately your letter made me unhappy for you seem to be truly sad. It seems that the influence of your teacher has been to give you a false idea of what are worthwhile problems. The worthwhile problems are the ones you can really solve or help solve, the ones you can really contribute something to. A problem is grand in science if it lies before us unsolved and we see some way for us to make some headway into it. I would advise you to take even simpler, or as you say, humbler, problems until you find some you can really solve easily, no matter how trivial. You will get the pleasure of success, and of helping your fellow man, even if it is only to answer a question in the mind of a colleague less able than you. You must not take away from yourself these pleasures because you have some erroneous idea of what is worthwhile.

You met me at the peak of my career when I seemed to you to be concerned with problems close to the gods. But at the same time I had another Ph.D. Student (Albert Hibbs) was on how it is that the winds build up waves blowing over water in the sea. I accepted him as a student because he came to me with the problem he wanted to solve. With you I made a mistake, I gave you the problem instead of letting you find your own; and left you with a wrong idea of what is interesting or pleasant or important to work on (namely those problems you see you may do something about). I am sorry, excuse me. I hope by this letter to correct it a little.

I have worked on innumerable problems that you would call humble, but which I enjoyed and felt very good about because I sometimes could partially succeed. For example, experiments on the coefficient of friction on highly polished surfaces, to try to learn something about how friction worked (failure). Or, how elastic properties of crystals depends on the forces between the atoms in them, or how to make electroplated metal stick to plastic objects (like radio knobs). Or, how neutrons diffuse out of Uranium. Or, the reflection of electromagnetic waves from films coating glass. The development of shock waves in explosions. The design of a neutron counter. Why some elements capture electrons from the L-orbits, but not the K-orbits. General theory of how to fold paper to make a certain type of child’s toy (called flexagons). The energy levels in the light nuclei. The theory of turbulence (I have spent several years on it without success). Plus all the “grander” problems of quantum theory.

No problem is too small or too trivial if we can really do something about it.

You say you are a nameless man. You are not to your wife and to your child. You will not long remain so to your immediate colleagues if you can answer their simple questions when they come into your office. You are not nameless to me. Do not remain nameless to yourself – it is too sad a way to be. now your place in the world and evaluate yourself fairly, not in terms of your naïve ideals of your own youth, nor in terms of what you erroneously imagine your teacher’s ideals are.

Best of luck and happiness.
Richard P. Feynman.
An accomplished father giving heartfelt advice to a son struggling to find his way, a teacher who immediately feels from a few gestures what a pupil is going through and reaches out due to his love for his student and due to his own humility, a man who recognizes his greatness and his defects in equal measure

Analysis — Chapter 1 — Real Variables — Part 4 Irrational numbers continued

Part 4. Irrational numbers (continued).

The result of our geometrical interpretation of the rational numbers is therefore to suggest the desirability of enlarging our conception of “number” by the introduction of further numbers of a new kind.

The same conclusion might have been reached without the use of geometrical language. One of the central problems of algebra is that of the solution of equations, such as

x^{2}=1, x^{2}=2.

The first equation has the two rational roots 1 and -1. But, if our conception of number is to be limited to the rational numbers, we can only say that the second equation has no roots; and the same is the case with such equations as x^{3}=2, x^{4}=7. These facts are plainly sufficient to make some generalization of our idea of number desirable, if it should prove to be possible.

Let us consider more closely the equation x^{2}=2.

We have already seen that there is no rational number x which satisfies this equation. The square of any rational number is either less than or greater than 2. We can therefore divide the rational numbers into two classes, one containing the numbers whose squares are less than 2, and the other those whose squares are greater than 2. We shall confine our attention to the positive rational numbers, and we shall call these two classes the class L, or the lower class, or the left-hand class, and the class R, or the upper class, or the right hand class. It is obvious that every member of R is greater than all the members of class R. Moreover, it is easy to convince ourselves that we can find a member of the class L whose square, though less than 2, differs from 2 by as little as possible, and a member of R whose square, though greater than 2, also differs from 2 by as little as we please. In fact, it we carry out the ordinary arithmetical process for the extraction of the square root of 2, we obtain a series of rational numbers, viz.,

1,1.4, 1.41, 1.414, 1.4142, \ldots

whose squares

1, 1.96, 1.9881, 1.999396, 1.99996164, \ldots

are all less than 2, but approach nearer and nearer to it, and by taking a sufficient number of the figures given by the process we can obtain as close an approximation as we want. And if we increase the last figure, in each of the approximations given above, by unity, we obtain a series of rational numbers

2, 1.5, 1.42, 1.415,1.413, \ldots

whose squares

4, 2.25, 2.0164, 2.002225, 2.00024449, \ldots

are all greater than 2, but approximate to 2 as closely as we please.

It follows also that there can be no largest member of L or smallest member of R. For if x is any member of L, then

x^{2} < 2. Suppose that x^{2}=2-\delta. Then we can find a member x, of L such that {x_{1}}^{2} differs from 2 by less than \delta, and {x_{1}}^{2}>x^{2} or x_{1}>x. Thus there are larger members of L than x; and, as x is any member of L, it follows that no member of L can be larger than all the rest. Hence, L has no largest member, and similarly, it has no smallest.

Note: A rigorous analysis of the above can be easily carried out. If you need help, please let me know and I will post it in the next blog.

More later,

Nalin Pithwa

Analysis — Real Variables — Chapter 1 — Examples II

Examples II.

1) Show that no rational number can have its cube equal to 2.

Proof #1.

Proof by contradiction.

Let, if possible, p/q, q \neq 0, and p and q do not  have any common factor and are integers. Then, if (p/q)^{3}=2, we have

p^{3}=2q^{3}. So, p contains a factor of 2. So, let p=2k. So, q contains a factor of 2. Hence, both p and q have a common factor have a common factor 2, contradictory to out assumption. Hence, the proof.

2) Prove generally that a rational function p/q in its lowest terms cannot be the cube of a rational number unless p and q are both perfect cubes.

Proof #2.

Let, if possible, p/q = (m/n)^{3} where m,n,p,q are integers, with n and q non-zero and p and q are in lowest terms. This implies that m and n have no common factor.  Hence, p=m^{3}, q=n^{3}.

3) A more general proposition, which is due to Gauss and includes those which precede as particular cases, is the following: an algebraical equation

z^{n}+p_{1}z^{n-1}+p_{2}z^{n-2}+ \ldots + p_{n}=0 with integral coefficients, cannot have rational, but non-integral root.

Proof #3.

For suppose that, the equation has a root a/b, where a and b are integers without a common factor, and b is positive. Writing

a/b for z, and multiplying by b^{n-1}, we obtain

-(a^{n}/b)=p_{1}a^{n-1}+p_{2}a^{n-2}b+ \ldots + p_{n}b^{n-1},

a function in its lowest terms equal to an integer, which is absurd. Thus, b=1, and the root is a. It is evident that a must be a divisor of p_{n}.

4) Show that if p_{n}=1 and neither of

1+p_{1}+p_{2}+p_{3}+\ldots and 1-p_{1}+p_{2}-p_{3}+\ldots

is zero, then the equation cannot have a rational root.

Proof #4. Please try this and send me a solution.. I do not have a solution yet 🙂

5) Find the rational roots, if any, of x^{4}-4x^{3}-8x^{2}+13x+10=0.

Solution #5.

Use problem #3.

The roots can only be integral, and so find the roots by trial and error. It is clear that we can in this way determine the rational roots of any such equation.

More later,

Nalin Pithwa

Analysis — Chapter I — part 3 — Real Variables — Irrational numbers

Part 3. Irrational numbers.

If the reader will mark off on the line all the points corresponding to the rational numbers whose denominators are 1,2,3, …in succession, he will readily  convince himself that he can cover the line with rational points, as closely as he likes. We can state this more precisely as follows: If we take any segment BC on A, we can find as many rational points on it as we please on BC.

Suppose, for example, that BC falls within the segment A_{1}A_{2}. it is evident that if we choose a positive integer k such that

k.BC>1 Equation I

(The assumption that this is possible is equivalent to the assumption of what is known as the Axiom of Archimedes.)

and divide A_{1}A_{2} into k equal parts, then at least one of the points of division (say P) must fall inside BC, without coinciding with either B or C. For if this were not so, BC would be entirely included in one of the k parts into which A_{1}A_{2} has been divided, which contradicts the supposition I. But P obviously corresponds to a rational number whose denominator is k. Thus at least one rational point P lies between B and C. But, then we can find another such point Q between B and P, another between B and Q, and so on indefinitely; that is, as we asserted above, we can find as many as we please. We may express this by saying that BC includes infinitely many 

rational points. (We will investigate the meaning of infinite more closely later).

From these considerations, the reader might be tempted to infer that an adequate view of the nature of the line could be obtained by imagining it to be formed simply by the rational points which lie on it. And, it is certainly the case that if we imagine the line to be made up of  solely of the rational points, and all other points (if there are any such) to be eliminated, the figure would possess most of the properties which common sense attributes to the straight line, and would, to put the matter roughly, look and behave very much like a line.

A little further consideration, however, shows that this view would involve us in serious difficulties.

Let us look at the matter for a moment with the eye of common sense, and consider some of the properties which we may reasonably expect a straight line to possess if it is to satisfy the idea which we have formed of it in elementary geometry.

The straight line must be composed of points, and any segment of it by all the points which lie between its end points.  With any such segment must be associated a certain entity called its length, which must be a quantity capable of numerical measurement in terms of any standard or unit length, and these lengths must be capable of combination with another, according to the ordinary rules of algebra, by means of addition or multiplication. Again, it must be possible to construct a line whose length is the sum or product of any two given lengths. If the length PQ along a given line is a, and the length QR, along the same straight line, is b, the length PR must be a+b.

Moreover, if the lengths OP and OQ, along one straight line, are 1 and a, and the length OR along another straight line is b, and if we determine the length OS by Euclid’s construction for a fourth proportional to the lines OP, OQ, OR, this length must be ab, the algebraic fourth proportional to 1, a and b. And, it is hardly necessary to remark that the sums and products thus defined must obey the ordinary laws of algebra; viz.,






The lengths of our lines must also obey a number of obvious laws concerning inequalities as well as equalities: thus, if A, B, C are three points lying along A from left to right, we must have AB<AC, and so on. Moreover, it might be possible, on our fundamental line \Lambda to find a point P such that A_{0}P is equal to any segment whatever taken along \Lambda or along any other straight line. All these properties of a line, and more, are involved in the presuppositions of our elementary geometry.

Now, it is very easy to see that the idea of a straight line as composed of a series of points, each corresponding to a rational number, cannot possibly satisfy all these requirements. There are various elementary geometrical constructions, for example, which purport to construct a length x such that x^{2}=2. For instance, we may construct an isosceles right angled triangle ABC such that AB=AC=1.. Then, if BC=x, x^{2}=2. Or we may determine the length x by means of Euclid’s construction for a mean proportional to a and 2, as indicated in the figure. Our requirements therefore involve the existence of a length measured by a number x, and a point P on \Lambda such that A_{0}P=x, x^{2}=2.

But, it is easy to see that there is no rational number such that its square is 2. In fact, we may go further and say that there is no rational number whose square is m/n, where m/n is say positive fraction in its lowest terms, unless m and n are both perfect squares.

For suppose, if possible, that \frac {p^{2}}{q^{2}}=m/n.

p having no common factor with q, and m no common factor with n. Thus, np^{2}=mq^{2}. Every factor of q^{2} must divide np^{2}, and as p and q have no common factor, every factor of q^{2} must divide n. Hence,

n={\lambda}q^{2}, where \lambda is an integer. But, this involves m={\lambda}p^{2}: and as m and n have common factor, \lambda must be unity. Thus, m=p^{2} and n=q^{2}, as was to be proved. In particular, it follows by taking n=1, that an integer cannot be the square of a rational number, unless that rational number is itself integral.

it appears that our requirements involve the existence of a number x and a point P, not one of the rational points already constructed, such that A_{0}P=x and x^{2}=2; and, (as the reader will remember from elementary algebra) we write x = \sqrt {2}.

Alternate proof. 

The following alternate proof that no rational number can have its square equal to 2 is interesting.

Suppose, if possible, that p/q is a positive fraction, in its lowest terms such that (p/q)^{2}=2. It is easy to see that this involves (2q-p)^{2}=2(p-q)^{2}, and so \frac {2q-p}{p-q} is also another fraction having the same property. But, clearly,

q<p<2q and so p-q<q. Hence, there is another fraction equal to p/q and having a smaller denomination, which contradicts the assumption that p/q is in its lowest terms.

In the next blog, we shall look at examples,

More later,

Nalin Pithwa



Career Advice by Prof Terence Tao, Mozart of Mathematics

Career advice by Terence Tao

Advice is what we ask for when we already know the answer but wish we did not. (Erica Jong).

Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages).

Disclaimer: The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues. You will of course need to evaluate many factors, contexts, and needs specific to your own situation, as well as employing a healthy dose of common sense, before making any important career decisions. I would in particular recommenddiscussing such decisions with your advisor if you have one, as he or she will be familiar with your situation and will likely be able to provide pertinent advice.  Also, it should be clear that most of this advice is targeted towards academic careers in mathematics; of course, there are many other career options available besides this, but I have no particularly informed advice to offer for such alternatives.

Talk to your advisor

It is the province of knowledge to speak and it is the privilege of  wisdom to listen — Oliver Wendell Holmes — The Poet at the Breakfast Table.

Your advisor is one of the best sources of guidance you have; not only in directly assisting you with your research topic, but in directing you (both explicitly and implicitly) to relevant researchers, conferences, publications, open problems, folklore, or other pieces of good mathematics. Your advisor also knows your situation well and can give career advice which is tailored to your specific strengths and weaknesses (unlike the generic advice in these pages).

If things get to the point that you are actively avoiding your advisor (or vice versa), that is a very bad sign. In particular, you should be aware of your advisor’s schedule, and conversely your advisor should be aware of when you will be available in the department, and what you are currently working on.

For similar reasons, you should give your advisor some advance warning if you want to take a long period of time away from your studies.

If your advisor is unavailable, you should regularly discuss mathematical issues with at least one other mathematician instead, preferably an experienced one.  [Also, it is not uncommon for a student to have both a formal advisor, who handles all the official paperwork, and an informal advisor, with which you discuss research and career issues.]

Of course, you should not rely purely on your advisor; you also need to take the initiative when it comes to your mathematical career.

Primary school level

  • Advice on gifted education
  • If you can give your son or daughter only one gift let it be the gift of enthusiasm. — Bruce Barton.

Education is a complex, multifaceted, and painstaking process, and being gifted does not make this less so. I would caution against any single “silver bullet” to educating a gifted child, whether it be a special school, private tutoring, home schooling, grade acceleration, or anything else; these are all options with advantages and disadvantages, and need to be weighed against the various requirements and preferences (both academic and non-academic) of the child, the parents, and the school. Since this varies so much from child to child, I cannot give any specific advice on a given child’s situation. [In particular, due to many existing time commitments and high volume of requests, I am unable to personally respond to any queries regarding gifted education.]

I can give a few general pieces of advice, though. Firstly, one should not focus overly much on a specific artificial benchmark, such as obtaining degree X fromprestigious institution Y in only Z years, or on scoring A on test B at age C. In the long term, these feats will not be the most important or decisive moments in the child’s career; also, any short-term advantage one might gain in working excessively towards such benchmarks may be outweighed by the time and energy that such a goal takes away from other aspects of a child’s social, emotional, academic, physical, or intellectual development. Of course, one should still work hard, and participate in competitions if one wishes; but competitions and academic achievements should not be viewed as ends in themselves, but rather a way to develop one’s talents, experience, knowledge, and enjoyment of the subject.

Secondly, I feel that it is important to enjoy one’s work; this is what sustains and drives a person throughout the duration of his or her career, and holds burnout at bay. It would be a tragedy if a well-meaning parent, by pushing too hard (or too little) for the development of their child’s gifts in a subject, ended up accidentally extinguishing the child’s love for that subject. The pace of the child’s education should be driven more by the eagerness of the child than the eagerness of the parent.

Thirdly, one should praise one’s children for their efforts and achievements (which they can control), and not for their innate talents (which they cannot). This article by Po Bronson describes this point excellently. See also the Scientific American article “The secret to raising smart kids” for a similar viewpoint.

Finally, one should be flexible in one’s goals. A child may be initially gifted in field X, but decides that field Y is more enjoyable or is a better fit. This may be a better choice, even if Y is “less prestigious” than X; sometimes it is better to work in a less well known field that one feels competent and comfortable in, than in a “hot” but competitive field that one feels unsuitable for. (See alsoRicardo’s law of comparative advantage.)

My own education is discussed in the following articles. While I am very happy with the way things turned out for me, I would again caution that each child’s situation, strengths, and weaknesses are different, and that my experience might not necessarily be the ideal template to follow for others.

For professional advice on gifted education, I can recommend the Center for Talented Youth. See also my page on career advice.

  • High School Education
  • Advice on mathematics competitions
  • Sports serve society by providing vivid examples of excellence. — George Will.
  • I greatly enjoyed my experiences with high school mathematics competitions (all the way back in the 1980s!). Like any other school sporting event, there is a certain level of excitement in participating with peers with similar interests and talents in a competitive activity. At the olympiad levels, there is also the opportunity to travel nationally and internationally, which is an experience I strongly recommend for all high-school students.
  • Mathematics competitions also demonstrate thatmathematics is not just about grades and exams. But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does.(While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)
  • Also, the “classical” type of mathematics you learn while doing Olympiad problems (e.g. Euclidean geometry, elementary number theory, etc.) can seem dramatically different from the “modern” mathematics you learn in undergraduate and graduate school, though if you dig a little deeper you will see that the classical is still hidden within the foundation of the modern. For instance, classical theorems in Euclidean geometry provide excellent examples to inform modern algebraic or differential geometry, while classical number theory similarly informs modern algebra and number theory, and so forth. So be prepared for a significant change in mathematical perspective when one studies the modern aspects of the subject. (One exception to this is perhaps the field of combinatorics, which still has large areas which closely resemble its classical roots, though this is changing also.)
  • In summary: enjoy these competitions, but don’t neglect the more “boring” aspects of your mathematical education, as those turn out to be ultimately more useful.
  • For advice on how tosolve mathematical problems, you can try my book on the subject.
  • Some collected quotes on mathematics competitions can befound here.

Which universities should one apply to?

A college degree is not a sign that a one is a finished product but an indication that a person is prepared for life. (Edward Malloy).

Going to college is a major event in one’s education, but the choice of exactly which college to go to is not as critical as it is sometimes portrayed to be; usually, there will be several good choices that suit your specific strengths and weaknesses, and it is not absolutely necessary to secure the “best” choice for your undergraduate or graduate education. I would recommend a flexibleattitude towards this decision; by focusing too much on one institution, you might overlook others which may in fact be a better fit for you.

It is common to focus on the general prestige of the institution, but actually it is the specific strengths of an institution which should play a more important role in your decisions. Examples of specific strengths include particular research strengths, teaching programs or initiatives, campus resources, academic culture, location, flexibility, affordability, availability of financial assistance, and so forth. On the other hand, given that your interests or situation may change somewhat as you learn more about your chosen field, one should not be too narrowly focused in one’s selection criteria; for instance, if you wish to go to an institution purely because of a single faculty member there, then you might run the risk that that faculty member moves, or is no longer accepting students.

I do however strongly urge that you study at different places; it’s good to move a little bit out of your own “comfort zone” and broaden your education. It’s also good to talk to your advisor about these matters.

My final advice is to have no regrets once one has made one’s choice; get the most out of the place one has chosen, and don’t spend too much time worrying about whether the grass would have been greener elsewhere. In particular, I would not recommend trying to “have the best of both worlds” by somehow trying to study simultaneously at your two top choices; this is very complicated to execute and usually does not work out very well.

I myself earned my undergraduate degree at Flinders University in my hometown of Adelaide, Australia – a small and not widely known institution, but one which was very friendly, close to home, and whose maths department was willing to accommodate my unusual educational experience. My graduate degree was at (the somewhat better known) Princeton University, which turned out to be a good fit for me, as I ended up with an excellent advisor and a challenging, self-driven environment which shook up my complacency about my own mathematical knowledge. My first postdoctoral position was at UCLA, which I liked so much that I have stayed here ever since, even though some of the faculty that I originally came to UCLA to work with have since left. Of course, there are many other good schools, which each have their own strengths and weaknesses. (For example, if the activity of big-city life is important to you, then Princeton does not fare terribly well in this regard.)

Undergraduate level:

Chaque vérité que je trouvois étant une règle qui me servoit après à en trouver d’autres [Each truth that I discovered became a rule which then served to discover other truths]. (René Descartes, “Discours de la Méthode“)Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one. Later in your research career, you will find that problems are mainly solved by knowledge (of your own field and of other fields), experience, patience andhard work; but for the type of problems one sees in school, college or in mathematics competitions one needs a slightly different set of problem solving skills. I do have a book on how to solve mathematical problems at this level; in particular, the first chapter discusses general problem-solving strategies. There are of course several other problem-solving books, such as Polya’s classic “How to solve it“, which I myself learnt from while competing at the Mathematics Olympiads.

Solving homework problems is an essential component of really learning a mathematical subject – it shows that you can “walk the walk” and not just “talk the talk”, and in particular identifies any specific weaknesses you have with the material. It’s worth persisting in trying to understand how to do these problems, and not just for the immediate goal of getting a good grade; if you have a difficulty with the homework which is not resolved, it is likely to cause you further difficulties later in the course, or in subsequent courses.

I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See “ask yourself dumb questions“.

It’s also best to keep in mind that obtaining a solution is only the short-term goal of solving a mathematical problem.  The long-term goal is to increase your understanding of a subject.  A good rule of thumb is that if you cannot adequately explain the solution of a problem to a classmate, then you haven’t really understood the solution yourself, and you may need to think about the problem more (for instance, by covering up the solution and trying it again).  For related reasons, one should value partial progress on a problem as being a stepping stone to a complete solution (and also as an important way to deepen one’s understanding of the subject).

See also Eric Schechter’s “Common errors in undergraduate mathematics“.  I also have a post on problem solving strategies in real analysis.


Prof Terence Tao blogs at

If you like the above, do send a word of appreciation to him…

More later,

Nalin Pithwa