Opinions Welcome : Should we dream?

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We are living organisms and our biological objective of life is to grow and reproduce. Now if we add the social prospective to this basic concept of living organisms, we must ensure healthy life of our offspring (love, education, health, money…). So, a life without dream would be to work hard enough to be a able to use your capabilities to earn enough money to support your family.

Now, that was very idealistic view of life. Most parents “dream” that their children should achieve much more than them in their lives (like the objective of evolution). But surely “dreams” like to become richest person, greatest scientist etc. are not necessary for living. Moreover many times we have to pay huge price for chasing our dreams like less time for family (which is actually the reason for our existence), more stress, less friends, etc. I think that “dream chasing” is like drug addiction, it becomes inseparable part of your life and leads to various negative effects. So, I would like to ask:

Should we dream?

Maybe, working hard enough to fully utilise our capabilities is what dreaming all about.

Similar posts in past: 

Why I want to be a Mathematician?

Inquisitive Mathematical Thinking

Bitter Truth of Love

Solution to the decimal problem

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In this post I will discuss the solution of the problem I posted a week ago. Firstly I would thank Prof. Purusottam Rath for pointing out that this problem has already been solved. In 1961, Kurt Mahler published the solution in Lectures on Diophantine Approximations. In fact, M=0.12345678.... is called Mahler’s number since Mahler  showed it to be transcendental. For complete proof refer, Section 1.6 of “Making transcendence transparent: an intuitive approach to classical transcendental number theory“, by Edward Burger and Robert Tubbs, Springer-Verlag (2004). But I can give an outline of proof here.

The most basic type of transcendental numbers are the Liouville’s Numbers, these numbers satisfy following theorem (proved here on pp. 19):

Liouville’s Theorem: Let \alpha be a real number . Suppose there exists an infinite sequence of rational numbers p_n/q_n satisfying the inequality \displaystyle{\left|\alpha - \frac{p_n}{q_n}\right|<\frac{1}{q_n^n}}. Then \alpha is transcendental.

To be able to apply this theorem we use truncation procedure i.e. obtain the approximations of the number \alpha  by truncating the decimal expansion of \alpha immediately before each long run of zeros, and using this to get the desired inequality.

For Mahler number, Liouville’s theorem alone is not sufficient. Since, if we attempt truncation procedure, we will see that the number of decimal digits before each run of zeros far exceeds the length of the run. For example, a run of 2 zeros occurs after 189 digits. But, using Liouville’s theorem we can prove a partial result:

(1) There exists an infinite sequence of rational numbers p_n/q_n satisfying the inequality \displaystyle{\left|M - \frac{p_n}{q_n}\right|<\frac{1}{q_n^{4.5}}}. Hence, Mahler number M is either a transcendental number or an algebraic number of degree at least 5.

Since we need a stronger inequality, we will use following theorem (proved here on pp. 54), which states that:

Thue-Seigel-Roth Theorem: Let \alpha be an irrational algebraic number. Then for any \varepsilon > 0 there exists a constant c(\alpha, \varepsilon) depending on \alpha  such that \displaystyle{\frac{c(\alpha,\varepsilon)}{q^{2+\varepsilon}}<\left|\alpha - \frac{p}{q}\right|}

From this theorem we conclude that

(2) If M is algebraic of degree d\geq 2 (I showed in previous post that it is irrational) then for \varepsilon =0.5 there exists a constant c such that for all p_n/q_n, \displaystyle{\frac{c}{q_n^{2.5}}<\left|M-\frac{p_n}{q_n}\right|}

Using (1) and (2) we conclude that for all n,

\displaystyle{0<c<\frac{1}{q_n^2}}

But as q_n\rightarrow \infty, this inequality cannot hold. Hence M is transcendental.

This number, 0.123456789101112... is also known as Champernowne’s constant. I shall discuss more about it in future posts.

 

A Decimal Problem

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I would like to share a jotting from my diary (dated: 21-April-2016) which is bothering me:

Is the (decimal) number generated the concatenating all natural numbers like 0.1234567891011…, a transcendental number?

We can see that this number is irrational. Since, if it is a rational number then there exists a natural number n such that 0.\overline{123...n}=0.123...n..., which is clearly impossible.

Now, to prove a number is transcendental or not is a difficult question to answer (some open problems). So, either produce a polynomial equation which has this number as a solution or prove that no such polynomial equation exists.

Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines.

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Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point).

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Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics.

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P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“.

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Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics.

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We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.

Mathemagic?

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In my opinion, today is a magical date: 4/8/16 (dd-mm-yy; as I write on my notebook). So let me tell you what I think about “mathematics” and “magic”. I believe that magic is an art of concealing facts leading to astonishing results. Magic trick is interesting from perspective of both observer and performer. Performer gets satisfaction of being able to fool observer (by making him/her believe that he/she can’t do it), and on the other hand observer gets satisfaction of being able of witness an act which he/she can’t perform (a quality of appreciation is expected).

Now, as many of you have observed, mathematics (or nature in general) is very much magical in the same sense. Only experts (algebraist/number theorists…) can “understand” the rules (called theorems) behind the actions (called computations) they ask you to perform (which amaze you). So, in general, whenever you are using a result (for example, an integer has unique factorization into prime numbers) without “knowing” the proof, you are performing “mathemagic” for yourself. When you take magic out of mathematics, you get what mathematicians called rigour.

I believe that the most important rule for performing a magic trick is to never reveal the secret rule (though the audience is free to conjecture and prove the possible secret rule). This is very much different from first rule of cryptanalysis, since while doing cryptanalysis you “must” know the algorithm/rule used to encipher the message and task is to find the key to decipher the cipher. Trying to find the secret rules for a magic trick is much more interesting that trying to decipher a cipher. So, I will leave you with a classic “mathemagic” trick and you as observer of this trick, try to find the secret rule governing it (but never reveal it to others!!!):

Write down the year you were born and under that the year of some great event of your life (like year of graduation, the time you saved somebody…). Now write the only even prime number. Write down your age by the end of this year (i.e. 2016) and the number of years ago the great event (quoted above) took place in your life. Now add all these. I know what the total will be! Today it’s 4034.

Mathematics Today…

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When we talk about doing mathematics, what comes to  our mind is Blackboard-chalk, notebook-pen and books. No doubt that these are and will remain one one of the most important instruments leading to elegant mathematical discoveries.  But, the evolution of technology we use has also affected the way we do, learn and share mathematics. In ancient time, mathematics was shared in form of books and letters. Then in 17th Century people started publishing academic journals periodically, which has today become one the most profitable business (like pharmaceuticals).  In 1960s computer algebra systems were invented (called MATHLAB). Then in 1970s books were digitized and today we have dedicated ebook readers.  Another major challenge  of publishing mathematical knowledge was to be able to typeset weird symbols, and this problem was fully solved using computers in 1978 by Donald Knuth. (to know more about this transition read this discussion).

Now it’s 21st century and the shape of sharing mathematical knowledge has changed significantly in past decade.  To begin with, in 2003 Poincaré conjecture’s solution was not published in any journal but was rather posted on arXiv. Today we have people on social networking sites like Facebook, Twitter, Google+, Tumblr, Weblogs...  who let you know the results just as they are being cooked up. For example, Live-tweet of Babai’s first Graph Isomorphism talk, in this talk one of the most interesting theorem of 2015 was proved. Many big shots announce their big results directly on their Weblogs, for example Terence Tao announced his proof of Erdős Discrepancy Problem on his blog. Today we can have interactive textbooks (like this), articles (like this) and assignments (like this) with advent of MathJax, SageMath

So far I have been concerned about “print” mathematics, but with advent of cheap internet, whole new methods of mathematical ideas sharing have come into picture. Today almost every reputed research organization maintain video lecture archives (IAS, CIRM,  IHÉSIHPInstitut Fourier, MatScience). Apart from mainstream mathematics, popularization of mathematics has become much more interesting today. We have lots of interesting mathematics popularization channels on YouTube like ViHartNumberphile, Mathologer, 3Blue1Brown, The Global Math Project,… and SoundCloud like BBC Radio 4: More or Less, ACMEScience ,… For a big-list of online mathematics videos see this and for big-list of mathematics podcasts see this.

Before this internet era, there were similar mathematics popularization attempts. Like my favourite: “Donald Duck in Mathmagic Land” (1959)

But I’m not aware of existence of mathematical radio programs back then. So, if you know about such radio programs,o please let me know about them as comments below.

What is Algebra?

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About 8 months ago I wrote about Analysis:

Thus, algebraic approximations produced the algebra of inequalities. The application of Algebra of inequalities lead to concept of Approximations in Calculus.

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You may have seen/heard this quote several times…

Now the time has come to understand the term “Algebra” itself, which has very rich history and dynamic present. I will use following classification (influenced by Shreeram Abhyankar) of algebra in 3 levels:

  1. High School Algebra (HSA)
  2. College Algebra (CA)
  3. University Algebra (UA)

HSA (8th Century – 16th Century) is all about learning tricks and manipulations to solve mensuration problems which involve solving linear, quadratic and “special” cubic equations for real (or rational) numbers. This level was developed by Muḥammad ibn Mūsā al-KhwārizmīThābit ibn QurraOmar KhayyámLeonardo Pisano (Fibonacci)Maestro Dardi of PisaScipione del FerroNiccolò Fontana (Tartaglia)Gerolamo CardanoLodovico Ferrari and Rafael Bombelli.

CA (18th Century – 19th Century) is commonly known as abstract algebra. Its development was motivated by the failure of HSA to solve the general equations of degree higher than the fourth and later on the study of symmetry of equations, geometric objects, etc. became one of the central topics of interest. In this we study properties of various algebraic structures like fields, linear spaces, groups, rings and modules. This level was developed by Joseph-Louis LagrangePaolo RuffiniPietro Abbati MarescottiNiels AbelÉvariste GaloisAugustin-Louis Cauchy Arthur CayleyLudwig SylowCamille JordanOtto HölderCarl Friedrich GaussLeonhard EulerWilliam Rowan Hamilton, Hermann GrassmannHeinrich Weber Emmy Noether and Abraham Fraenkel .

UA (19th Century – present) has derived its motivations from many diverse subjects of study in mathematics like Number Theory, Geometry and Analysis.  In this level of study, the term “algebra” itself has a different meaning

An algebra over a field is a vector space (a module over a field) equipped with a bilinear product.

and topics are named like Commutative Algebra, Lie  Algebra and so on. This level was initially developed by Benjamin Peirce,  Georg FrobeniusRichard DedekindKarl WeierstrassÉlie CartanTheodor MolienSophus LieJoseph WedderburnMax NoetherLeopold Kronecker,  David HilbertFrancis Macaulay,  Emanuel LaskerJames Joseph SylvesterPaul Gordan, Emil ArtinKurt HenselErnst SteinitzOtto Schreier ….

Since algebra happens to be a fast developing research area, the above classification is valid only for this moment. Also note that, though Emmy Noether was daughter of Max Noether I have included the contributions of Emmy in CA and those of Max in UA. The list of contributors is not exhaustive.

References:

[1] van der Waerden, B. L.  A history of algebra. Berlin and Heidelberg: Springer-Verlag, 1985. doi: 10.1007/978-3-642-51599-6

[2] Kleiner, I.  A History of Abstract Algebra. Boston : Birkhäuser, 2007. doi: 10.1007/978-0-8176-4685-1

[3] Burns, J. E. “The Foundation Period in the History of Group Theory.” American Mathematical Monthly 20, (1913), 141-148.  doi: 10.2307/2972411