Introspection

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I have written many philosophical blog posts motivated by the idea of existence of reason for everything.  So far I have written (and revised) my point of view regarding topics like Being Alive, Loving MathematicsBecoming Mathematician and Dreaming Big.

A more fundamental and more disturbing question is:

Since each and everything is made up of atoms, what is the borderline between a living organism and non-living object.

Clearly, mathematics will fail to answer this question since there are no absolute quantifiers involved. But still we can try deducing an answer from logical arguments. Following video by Kurzgesagt – In a Nutshell illustrates my question:

We generally describe living organism as something capable of reproduction, growth and consciousness. So, can we convert anything into living organism by somehow adding artificial intelligence to it? Is internet itself a living organism?

Once you call yourself a living organism, the immediate question is about the purpose of your existence (since we believe that there is reason for everything). So we can use this as a quantifier to classify something as living and non-living. Many people have tried (and failed) to answer this question. I came across a possible answer for this question in the film Kubo and the two strings:

We live to write a story and then become immortal in memories of others in form of our stories.

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Kubo and the two strings (© 2016, Focus Features)

I really liked this point of view. Being alive is all about being able to create memories. But this view point is very much human centred since we don’t know how other organisms (like other animals, plants, cells, organelles…) interact. Moreover, non-living objects also have stories associated with them (like monuments, paintings,…). So this view point also fails to capture the central idea for classification of something as living or non-living.

I will be happy to know your viewpoint of being able to classify something as living or non-living.

Not all numbers are computable

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When we hear the word number, symbols like 1,\sqrt{2},¼, π (area enclosed by a unit circle), ι (symbol for \sqrt{-1}), ε (infinitesimal),  ω (ordinal infinity), ℵ (cardinal infinity), …. appear in our mind.  But not all numbers are  computable:

A computable number [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape].

In other words, a real number is called computable if there is an algorithm which, given n, returns the first n digits of the number. This is equivalent to the existence of a program that enumerates the digits of the real number. For example, π is a computable number (why? see here).

Using Cantor’s diagonal argument on a list of all computable numbers, we get a non-computable number (here is the discussion). For example, a sum of series of real numbers called Chaitin’s constant, denoted by Ω, is a non-computable number (why? see here).

Fun fact: We don’t know whether π is a normal number or not (though we want it to be a normal number), but Ω is known to be a normal number (just like the Mahler’s Number discussed here).

Celebrity Mathematicians

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In my opinion, currency notes are one of the biggest motivation to learn arithmetical operations (like addition, multiplication,…). In fact, most of our elementary school problems are about buying a particular quantity of something.

Historically, there had been currencies notes featuring great mathematicians like Carl Friedrich Gauss, Leonard Euler and Rene Descartes. But, today there are no currencies featuring mathematicians. The database of currency notes featuring mathematicians is available here: http://web.olivet.edu/~hathaway/math_money.html

Since honouring people by featuring them on currency notes is politically challenging, government rather issues special postage stamps. The database of stamps featuring mathematicians is available here:  http://jeff560.tripod.com/stamps.html

Apart from illustrating various mathematical concepts (like graphs, metric system, binomial theorem… ) on stamps, India Post has issued stamps to honour mathematicians like Damodar Dharmananda Kosambi , Srinivasa Ramanujan Iyengar and Bertrand Russell.

 

Real Numbers

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Few days ago I found something very interesting on 9gag:

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When I first read this post, I was amazed because I never thought about this obvious fact!!

There are lots of interesting comments, but here is a proof from the comments:

…. Infinite x zero (as a limit) is indefinite. But infinite x zero (as a number) is zero. So lim( 0 x exp (x²) ) = 0 while lim ( f(X) x exp(X) ) with f(X)->0 is indefinite …

Though the statement made in the post is very vague and can lead to different opinions, like what about doing the product with surreal numbers, but we can safely avoid this by considering the product of real numbers only.

Now an immediate question should be (since every positive real number has a negative counterpart):

Is the sum of all real numbers zero?

In my opinion the answer should be “no”. As of now I don’t have a concrete proof but the intuition is:

Sum of a convergent series is the limit of partial sums, and for real numbers due to lack of starting point we can’t define a partial  sum. Hence we can’t compute the limit of this sum and the sum of series of real numbers doesn’t exist.

Moreover, since the sum of all “positive” real numbers is not a finite value (i.e. the series of positive real numbers is divergent) we conclude that we can’t rearrange the terms in series of “all” real numbers (Riemann Rearrangement Theorem). Thus the sum of real numbers can only be conditionally convergent. So, my above argument should work. Please let me know if you find a flaw in these reasonings.

Also I found following interesting answer on Quora:

The real numbers are uncountably infinite, and the standard notions of summation are only defined for countably many terms.

Note: Since we are dealing with infinite product and sum, we can’t argue using algebra of real numbers (like commutativity etc.).

Packing Problems

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Easy to state and hard to solve problems make mathematics interesting. Packing problems are one such type. In fact there is a very nice Wikipedia article on this topic:

 Packing problems involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible.

I came across this problem a couple of years ago, while watching the following TED Talk by Eduardo Sáenz de Cabezón:

In this talk, the object of attraction is the Weaire-Phelan structure made from six 14-hedrons and two dodechedrons. This object is believed to be the solution of Kelvin’s problem:

How you can chop 3d space into cells of equal volume with the minimum surface area per cell?

The packing problem for higher dimensions was in news last year, since a problem about “Densest Packing Problem in Dimensions 8 and 24” was solved by a young mathematician (Maryna Viazovska). The mathematics involved in the solution is very advanced but we can start gaining knowledge from this book:

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A classic reference in this field by two well known geniuses.

Recently, while reading Matt Parker’s book, I discovered a wonderful website called Packomania by Eckard Specht (Otto-von-Guericke-Universität Magdeburg) containing data about packing problems in 2D and 3D. (also checkout his Math4u.de website, it’s a good reference for elementary triangle geometry and inequalities problems.)

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Screen-shot of Dr. Eckard Specht’s homepage, his online problem collection (with solutions; each problem in GIF, PS and PDF formats) and Packomania.

Computers also have a role to play in solving such optimization problems For example, in 2015 Thomas Hales formally proved Kepler’s Conjecture about 3D packing:

No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.

using HOL Light proof assistant.

The proof was a huge collaboration and was called Flyspeck Project. The details about this proof are available in this book:

aas

NOTE: You can construct your own Truncated octahedron and Weaire-Phelan polyhedra by printing the nets available at Matt Parker’s website.

Clocks

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Clocks are amazing. They tell us time. In this post I want to talk about working of analog clocks. In case you haven’t seen an analog clock, this is how it looks like:

 

 

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A wall clock, it’s working part in the back and inside of the working part.

But, what clocks have to do with mathematics? As I have mentioned several times, one major part of mathematics is about counting and clocks “count”! Clocks are amazing counting device, they perform mod 12 and mod 60 calculations (that’s why modular arithmetic is also called clock arithmetic).

Unfortunately, the ideal cases exist only in our abstract world of mathematics. In real world, whatever we build has some error percentage and our motive to minimize this error. A mathematical construction, called Stern-Brocot tree, was created to help build timepieces and understand number theory.

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By Aaron Rotenberg (Own work) [GFDL or CC-BY-SA-3.0], via Wikimedia Commons

This “tree” gives an exceptionally elegant way to enumerate the positive rational numbers and is a surprisingly useful tool for constructing clocks.  For more information about this construction read this feature column article by David Austin.

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Just like continued fractions, this tree gives us good rational approximations of a given real number. Clocks typically have a source of energy–such as a spring, a suspended weight, or a battery–that using gears turns a shaft at a fixed rate. We can increase the precision by using more number of gears of different teeth count in appropriate combination.

I will end this post with an example from Austin’s article:

Suppose we place a pinion on a shaft that rotates once every hour and ask to drive a wheel that rotates once in a mean tropical year, which is 365 days, 5 hours, 49 minutes. Converting both periods to minutes, we see that we need the ratio 720 / 525,949. The problem here is that the denominator 525,949 is prime so we cannot factor it. To obtain this ratio exactly, we cannot use gears with a smaller number of teeth. It is likewise impossible to find a multi-stage gear train to obtain this ratio. But, as we slide down the “tree” toward 720 / 525,949, the rationals we meet along the way will give good approximations with relatively small numerators and denominators. As we descend the Stern-Brocot tree towards 720 / 525,949, we find the fraction 196 / 143,175, which may be factored into four rational factors, 2/3, 2/25, 7/23 and 7/83. We can therefore construct a four-stage gear train and can get a pretty accurate clock.

I hope I have been able to convince you that clocks are much more interesting than they would appear and you should read the article by David Austin for further references.

Arithmetic Operations

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There are only 4 binary operations which we call “arithmetic operations”. These are:

  • Addition (+)
  • Subtractions (-)
  • Multiplication (×)
  • Division (÷)

Reading this fact, an obvious question is:

Why only four out of the infinitely many possible binary operations are said to be arithmetical?

Before presenting my attempt to answer this question, I would like to remind you that these are the operations you were taught when you learnt about numbers i.e. arithmetic.

In high school when \sqrt{2} is introduced, we are told that real numbers are of two types: “rational” and “irrational”. Then in college when \sqrt{-1} is introduced, we should be told that complex numbers are of two types: “algebraic” and “transcendental“.

As I have commented before, there are various number systems. And for each number system we have some valid arithmetical operations leading to a valid algebraic structure. So, only these 4 operations are entitled to be arithmetic operations because only these operations lead to valid algebraic numbers when operated on algebraic numbers.

Now this leads to another obvious question:

Why so much concerned about algebraic numbers?

To answer this question, we will have to look into the motivation for construction of various number systems like integers, rational, irrationals, complex numbers… The construction of these number systems has been motivated by our need to be able to solve polynomials of various degree (linear, quadratic, cubic…). And the Fundamental Theorem of Algebra says:

Every polynomial with rational coefficients and of degree n in variable x has n solutions in  complex number system.

But, here is a catch. The number of complex numbers which can’t satisfy any polynomial (called transcendental numbers) is much more than the number of complex numbers which can satisfy a polynomial equation (called algebraic numbers). And we wish to find solutions of a polynomial equation (ie.e algebraic numbers) in terms of sum, difference, product, division or m^{th} root of rational numbers (since coefficients were rational numbers). Therefore, sum, difference, product and division are only 4 possible arithmetic operations.

My previous statement may lead to a doubt that:

Why taking m^{th} root isn’t an arithmetic operation?

This is because it isn’t a binary operation to start with, since we have fixed m. Also, taking m^{th} root is allowed because of the multiplication property.

CAUTION: The reverse of m^{th} root is multiplying a number with itself m times and it is obviously allowed. But, this doesn’t make the binary operation of taking exponents, \alpha^{\beta} where \alpha and \beta are algebraic numbers, an arithmetic operation. For example, 2^{\sqrt{2}} is transcendental (called Gelfond–Schneider constant or Hilbert number) even though 2 and \sqrt{2} are algebraic.