Showing posts with label Mathematics. Show all posts
Showing posts with label Mathematics. Show all posts

Tuesday, August 31, 2010

The solution: Binary numbers

So what is the way to convert a binary number, say, 11011 to its decimal equivalent? A basic simple question for anyone who has gone through high school mathematics. Since I am one of them let me solve it here.

1*16 + 1*8 + 0*4 + 1*2 + 1*1 = 27

In words, number 27 is the sum of 1, 2, 8, and 16. More importantly any decimal number can be seen as a sum of numbers that are powers of two. And the binary equivalent of the number decides which of the power of two numbers take part in the summation. Like in the above case it was 1, 2, 8 and 16. Basic binary number concept right?

Lets take a fresh look at the puzzle I shared at this link. And the solution would become obvious to you. So all numbers less than 128 (binary equivalent of 128 is 1111111) can be seen as summation of numbers 1,2,4,8,16,32 and 64.

Off-course I didn't come to the solution through this route. I picked 1 and 2 and then kept including all the numbers that I could not sum up by using the numbers I already had. So 3 I got from 1+2. Because I could not get 4 from 1 and 2, I included 4 and marched on. By the time I reached 8, I noticed the pattern and  tried 16 to verify. And then the above concept of binary numbers came to my mind. And so I was sure I had the answer.

Also I would like to share one interesting solution Rajesh gave. I will quote his response directly here. You can find his answer at this link in the comment section.

The answer is:
ONE each weight of 1,3,9,27 and 81 kg.
...
I want to weigh 2 as 3-1 and 4 kg as 9-(3+1).
I think this sequence can be extended up to 1000. If fact 1,3,9,27 and 81 can weigh up to 121 kgs.

Monday, June 14, 2010

Karl Popper and the Black Swan

I was reading the novel 'The Black Swan' by Nassim Nicholas Taleb during the winter of '08-'09. Since the internet has become just a click away, as a habit I read reviews and make general google research on the book and the author that I am reading. It was during one of these research readings that I came across Karl Popper - who is generally regarded as one of greatest philosophers of science. Being a student of science for most of my life it was surprising, and ignorant of me, to not have heard of the man and his contributions. Nevertheless, I tried to make amends and read a sufficient amount of material on the man. To write a post on a personality, let alone a philosopher is a herculean task and beyond the scope of the post. More importantly I cannot safely say I have read enough of his work to really build an article on the subject. Instead what I will do is introduce Karl Popper, if you are not already introduced to him, with slices of material I read with regard to Black Swan, Nicholas Taleb and Karl Popper.

Writing for the New York Times, Taleb starts his article/book with this paragraph.
Before the discovery of Australia, people in the old world were convinced that all swans were white, an unassailable belief as it seemed completely confirmed by empirical evidence. The sighting of the first black swan might have been an interesting surprise for a few ornithologists (and others extremely concerned with the coloring of birds), but that is not where the significance of the story lies. It illustrates a severe limitation to our learning from observations or experience and the fragility of our knowledge. One single observation can invalidate a general statement derived from millennia of confirmatory sightings of millions of white swans. All you need is one single (and, I am told, quite ugly) black bird. 
He goes on to elaborate on the subject and give a three characteristic of a Black Swan event and more. Also, the following set of paragraphs would allow you to see the link between Taleb and Popper.

Every genuine scientific theory then, in Karl Popper's view, is prohibitive, in the sense that it forbids, by implication, particular events or occurrences. As such it can be tested and falsified, but never logically verified. Thus Popper stresses that it should not be inferred from the fact that a theory has withstood the most rigorous testing, for however long a period of time, that it has been verified; rather we should recognize that such a theory has received a high measure of corroboration. and may be provisionally retained as the best available theory until it is finally falsified (if indeed it is ever falsified), and/or is superseded by a better theory.

Popper has always drawn a clear distinction between the logic of falsifiability and its applied methodology. The logic of his theory is utterly simple: if a single ferrous metal is unaffected by a magnetic field it cannot be the case that all ferrous metals are affected by magnetic fields. Logically speaking, a scientific law is conclusively falsifiable although it is not conclusively verifiable. Methodologically, however, the situation is much more complex: no observation is free from the possibility of error—consequently we may question whether our experimental result was what it appeared to be.

Thus, while advocating falsifiability as the criterion of demarcation for science, Popper explicitly allows for the fact that in practice a single conflicting or counter-instance is never sufficient methodologically to falsify a theory, and that scientific theories are often retained even though much of the available evidence conflicts with them, or is anomalous with respect to them. Scientific theories may, and do, arise genetically in many different ways, and the manner in which a particular scientist comes to formulate a particular theory may be of biographical interest, but it is of no consequence as far as the philosophy of science is concerned. Popper stresses in particular that there is no unique way, no single method such as induction, which functions as the route to scientific theory, a view which Einstein personally endorsed with his affirmation that ‘There is no logical path leading to [the highly universal laws of science]. They can only be reached by intuition, based upon something like an intellectual love of the objects of experience’. Science, in Popper's view, starts with problems rather than with observations—it is, indeed, precisely in the context of grappling with a problem that the scientist makes observations in the first instance: his observations are selectively designed to test the extent to which a given theory functions as a satisfactory solution to a given problem.
My personal favorite is this paragraph.
In the view of many social scientists, the more probable a theory is, the better it is, and if we have to choose between two theories which are equally strong in terms of their explanatory power, and differ only in that one is probable and the other is improbable, then we should choose the former. Popper rejects this. Science, or to be precise, the working scientist, is interested, in Popper's view, in theories with a high informative content, because such theories possess a high predictive power and are consequently highly testable. But if this is true, Popper argues, then, paradoxical as it may sound, the more improbable a theory is the better it is scientifically, because the probability and informative content of a theory vary inversely—the higher the informative content of a theory the lower will be its probability, for the more information a statement contains, the greater will be the number of ways in which it may turn out to be false. Thus the statements which are of special interest to the scientist are those with a high informative content and (consequentially) a low probability, which nevertheless come close to the truth. Informative content, which is in inverse proportion to probability, is in direct proportion to testability. Consequently the severity of the test to which a theory can be subjected, and by means of which it is falsified or corroborated, is all-important. 
Even after reading some of the articles on Popper, I am yet to completely understand and make my personal comment on his work. Nevertheless, I think he makes sense to me almost always and that encourages me to read more of him. Need to get hold of some of his written works. Suggestion in that regard would be most helpful.

To conclude here is what William W. Bartley has to say about Popper:
Sir Karl Popper is not really a participant in the contemporary professional philosophical dialogue; quite the contrary, he has ruined that dialogue. If he is on the right track, then the majority of professional philosophers the world over has wasted or is wasting their intellectual careers. The gulf between Popper's way of doing philosophy and that of the bulk of professional philosophers is as great as that between astronomy and astrology.

Sunday, June 13, 2010

One puzzle that starts a story

One fine day I happened to mistakenly enter my father's boss's office room. Once I introduced myself and apologized he welcomed me but after knowing I had just written my standard X board exams, he started to bombard me  with a series of puzzles and mathematical questions that afternoon as if it was punishment I had to endure for being in the wrong room at the wrong time. Also I sensed he was using me as a guinea pig  to test the  current schooling standards. I managed to tackle, all his question, but for one. I eventually cracked it that evening and the complete process made me move a notch up in my understanding of mathematics and made me realize how a puzzle can be one of the best teaching methodology. Since then I have asked many this puzzle and have in most cases got the answer in quick time. Much quicker than I managed.

What I will do now is state the puzzle here without further delay and wait for some weeks to get answers from whoever cares to read this post. Most likely some of you have come across it and and I am expecting some participation. In the process of solving the question, each of you, I am sure, will understand the wider implication of this puzzle and would agree to me when I say it could be a good starting point when one introduces a particular of branch of mathematics to students.

The question:
I have been asked to weight all weights up to 100. That is I might be asked to weight, lets say 49 kg or any integer weight under 100 (including). What would be the least number of weights I should have and which ones.
It's possible I was not able to frame the question in best possible way. Let me know if you have any queries.

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