Monthly Archives: February 2015

Bitter Truth of Love

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I love mathematics (click here for WHY?), and many other creations of Nature. But every time you add one of the creations of nature to your “LOVE LIST” , some unwanted (bitter) things creep in. I will talk what so far I have experience in case of (pure) Mathematics by quoting some big shots who feel same as me.

Doing research in mathematics is frustrating and if being frustrated is something you cannot get used to, then mathematics may not be an ideal occupation for you. Most of the time one is stuck, and if this is not the case for you, then either you are exceptionally talented or you are tackling problems that you knew how to solve before you started. There is room for some work of the latter kind, and it can be of a high quality, but most of the big breakthroughs are earned the hard way, with many false steps and long periods of little progress, or even negative progress. There are ways to make this aspect of research less unpleasant. Many people these days work jointly, which, besides the obvious advantage of bringing different expertise to bear on a problem, allows one to share the frustration. For most people this is a big positive (and in mathematics the corresponding sharing of the joy and credit on making a breakthrough has not, so far at least, led to many big fights in the way that it has in some other areas of science). I often advise students to try to have a range of problems at hand at any given moment. The least challenging should still be difficult enough that solving it will give you satisfaction (for without that, what is the point?) and with luck it will be of interest to others. Then you should have a range of more challenging problems, with the most difficult ones being central unsolved problems. One should attack these on and off over time, looking at them from different points of view. It is important to keep exposing oneself to the possibility of solving very difficult problems and perhaps benefiting from a bit of luck.

Mathematicians usually have a hard time explaining to their partner that the times when they work with most intensity are when they are lying down in the dark on a sofa. Unfortunately, with e-mail and the invasion of computer screens in all mathematical institutions, the opportunity to isolate oneself and concentrate is becoming rarer, and all the more valuable

Second, the mathematician must risk frustration. Most of the time, in fact, he finds himself, after weeks or months of ceaseless searching, with exactly nothing: no results, no ideas, no energy. Since some of this time, at least, has been spent in total involvement, the resulting frustration is very nearly total. Certainly it seriously affects his attitude toward all other affairs. This factor is a more important hindrance than any other, I believe; to risk total frustration, and to be almost certain
to lose, is a psychological problem of the first rank.

We are mathematicians by choice. We chose the profession because we love the subject. Reading and assimilating deep results of masters and then solving some of our own small problems brings us pleasure to which nothing else compares much. Yet we live in a world populated mostly be non-mathematicians. We must survive and thrive in their midst. This brings
forth its own challenges and frustrations.

But what I & many other mathematicians feel as a compensation of this Bitterness of Loving Mathematics  is:

  • Love Teaching (it has it’s own bitterness) : Share, propagate and preach the beauty that you can see by teaching others.  Enjoy teaching if it is a course of our choice and the class consists of a few eager, motivated, well-behaved students. That is only a dream. Often we must teach large classes of uninterested students who are there only for completing the requirements. But in spite of all this we must strive to teach, giving it our best and at the same time maintaining the standard of our subject. Compromises have no place here. We believe in teaching in a certain way and it can be fine-tuned depending upon the reactions of the students. But it should not prevent us from communicating the basic spirit of mathematics, especially the importance of logical enquiry. All aspects of mathematics, including history, biographies, motivation, definitions, lemmas, theorems, corollaries, proofs, examples, counterexamples, conjectures, construction, computation and applications can and should find a place in the classroom.
  • Love Travelling: Attend as any conferences, workshops etc. and keep moving.  As Paul Erdős said :

Another roof, another proof

Resting with ‘Nested Radicals’

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One night after hectic schedule at college, I was standing in front of black board with a chalk in my hand (to relax my mind!).

A thought of nested roots came to my mind and I wrote:
\sqrt{2 +\sqrt{2 +\sqrt{2 + \ldots}}} = 2
As I had `learnt’ from some olympiad book in class 8, the trick to solve such problems.
Let, \sqrt{2 +\sqrt{2 +\sqrt{2+ \ldots}}} = x
As we have infinite number of terms on left hand side, if we keep one of the two’s aside, we can rewrite it as:\\
\sqrt{2 + x} = x
\Rightarrow 2 + x = x^2
Solving this quadratic we will get x = 2, -1
Clearly x\geq 0, thus x = 2.
Now thrilled by this I proceeded to evaluate nested roots for other arithmetic operations in similar fashion:
\sqrt{2 -\sqrt{2 -\sqrt{2 - \ldots}}} = 1
\sqrt{2\sqrt{2\sqrt{2\ldots}}} = 2
\sqrt{\frac{2}{\sqrt{\frac{2}{\sqrt{\frac{2}{\vdots}}}}}} = \sqrt[3]{2}
Now I knew that no one knows the answer for:
1-1+1-1+1-1\ldots = ??
so, inspired by this I wrote:
\sqrt{1-\sqrt{1 +\sqrt{1-\ldots}}} = ??
But then one of my friend suggested:

Since we are dealing with infinite terms we can say

\sqrt{1-\sqrt{1 +x}} = x
\Rightarrow 1-\sqrt{1 +x} = x^2
\Rightarrow x^4 - 2x^2 - x = 0
\Rightarrow x(x^3 - 2x - 1) = 0
\Rightarrow x(x+1)(x^2-x-1) = 0
\Rightarrow x = 0, -1, \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}
But I can’t say which one out of  0\quad \& \quad \frac{1+\sqrt{5}}{2} is solution.
I know that flaw in this argument is similar to that in saying:
1-1+1-1+1-1\ldots = 0, by pairing +1\quad \& \quad -1.
But I found the idea itself flawless and tried to apply it to my starting expression:
\sqrt{2 +\sqrt{2 + x}} = x
\Rightarrow 2 + \sqrt{2 + x} = x^2
\Rightarrow x^4 - 4x^2 - x +2 = 0
\Rightarrow (x-2)(x+1)(x^2 + x -1) = 0
\Rightarrow x = 2, - 1, \frac{-1-\sqrt{5}}{2}, \frac{-1+\sqrt{5}}{2}
Unlike last case I can decide between 2\quad \& \quad \frac{-1+\sqrt{5}}{2}  [Thanks to Sagar Shrivastava]

Since, \sqrt{2 +\sqrt{2 +\sqrt{2+ \ldots}}} > \sqrt{2} > \frac{-1+\sqrt{5}}{2}

Thus, 2 is only answer.

Thus if I take out more and more 2’s out of square root, I will get more and more values of x and among them we will get only one appropriate value of x.

But still I can’t figure out the correct value of\sqrt{1-\sqrt{1 +\sqrt{1-\ldots}}} since I don’t have much experience with ‘series’ at this stage, but will surely investigate this question in detail by end of this semester.
If you have some ideas regarding this please do share.

Also these nested roots remind me of Ramanujam’s famous problem:
Find the value of : \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\ldots}}}}  (See this article by B. Sury)

Two Gems

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I would like to share following two mathematical problems which were brought to my knowledge by different people.

  • Boy or Girl Paradox by Guru boy-girl-paradox

Statement of the problem is:

X has two children.  Find the probability that:

  •  Both children are girls, given that the older child is a girl.

  • Both children are boys, given that at least one of them is a boy.

Some historical reference can be found at Wikipedia: http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

Also you can refer to this interesting paper published on this problem by Peter Lynch : http://mathsci.ucd.ie/~plynch/Publications/BIMS-TwoChildParadox.pdf 

  • The twelve-coin problem by Sagar Shrivastava

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Statement of the problem is:

There is a pile of twelve coins, all of equal size. Eleven are of equal weight. One is of a different weight. What are minimum number of times one need to weigh to find the faulty coin and determine if it is heavier or lighter?

Some historical reference can be found at this wikipedia page: http://en.wikipedia.org/wiki/Balance_puzzle

Also, a discussion on this problem is available at: http://mathforum.org/library/drmath/view/55618.html