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.

Friday, August 27, 2010

Science in India and a few stories

The rich scientific and technical heritage, which our country possesses, is unknown to a major section of the people in our country. In this article, my endeavor is to put across few facts and achievements made by Indians in the field of science and technology. Let us start with the field of mathematics.

This dates way back to 1000 BC when Indian scientists first proposed what the number system is and how does it assist us in day-to-day life. In this respect, the Nalanda University (presently in Bihar and earlier known as state of Patliputra) did a commendable job in creating a high value learning atmosphere and was a unique institution of that era. Amongst the luminaries of that period was Apastamba (about 600 BC) who first proposed how to calculate the values of irrational numbers upto fifth decimal place so that the measurement of microscopic particles in those days could be made with appreciable accuracy. Then came the gift of “ZERO” (the number “0”) by Aryabhatta-I. He solved the mystery of “NOTHING” in mathematics. Actually, it is still a very interesting topic of research amongst mathematicians what this “0” is? His topics covered arithmetic, algebra, plane geometry and spherical trigonometry. He developed the whole concept of QUADRATIC equations, and continued fractions. Around 600 AD, Bhaskara (Saurashtra) played a crucial role in correct estimation of the number “” (called ‘pie’). His efforts lead to the expansion of trigonometric functions using polynomials and estimated the possible errors in these expansions. Around the same time, Brahmagupta stepped in with his brilliant postulates on the digit “0”. He gave reasons for all the possible operations, which were done using “0”, and still today they are not questioned. One of his quotes were,

“When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero”

By far, in the nineteenth century, India produced one greatest mind in the field of mathematical sciences, Srinivasa Aiyangar Ramanujan. In his short span of life (only 33 years, 1887-1920), he made some remarkable contributions in the field. His most significant contributions were:
1) Elliptic functions, continued functions and infinite series.
2) How to solve cubic equations, and his own method to solve quadratic equations.
3) Calculation of Euler’s constant upto 15th decimal places and his own investigations on Bernoulli’s number.
4) Proposition of Hypergeometric series, which is the basis of large problems encountered in the field of Science and engineering (especially Chemical and Electrical).
5) Foundation of “Indian Mathematical Society”.

There were a few more, genius of whose work, have not been quoted here like, Brahamdeva, Harish Chandra, Ramanujam, Yatirbhasava etc. etc.

The field of physical sciences is equally rich as their mathematics counterparts. It started again way back in 1000 BC and continued till the 15th century. However, all the evidences related to scientific developments taking place in India were destroyed during British colonization of India by East India Company. Moreover, India never saw an Industrial Revolution as it happened in the west. Therefore, in that respect we abysmally lagged far behind the west as far as our development on scientific and technical prowess was concerned during the 17th and 18th century. But, once the ruling power was taken over by the Queen (after 1857); there came some hope of revival on this front. Starting it all was, Meghnath Sircar (1833-1904), a lower middle class lad from the state of Bengal. Even though, his contributions did not lead to some cutting-edge research, he fought fiercely for establishment of Universities in India. Due to his sincere and pain-staking struggle, Calcutta University became the first University to be established on 24th January 1857 in modern India. It was essentially meant for exploiting the huge pool of talented young individuals of this country (as was thought by the Queen of the Britain). During its initial days, Calcutta University saw an unprecedented growth in terms of availability of foreign scientific literature, better experimental facilities and many more. In those days, this was only hub in the entire country to get access to the world class journals and articles (selected ones), which were easily available in the west. The presence of the facilities of this kind enthused the young generation to fulfill their desire to learn. Dr. Meghnath Sircar’s research mainly included application and understanding of the newly developed Maxwell’s equations at that time.

He was followed by great many researchers/scientists but most of them were from the state of Bengal and nearby. Most of them were classmates and showed tremendous camaraderie to take India to a commendably strong position in the field of Science and Technology. They were Meghnath Saha, Satyendra Nath Bose, Nikhilranjan Bose, J.C. Ghosh, Girijapathy Bhattacharya and many more like them. They came in contact with Netaji Subhash Chandra Bose during the freedom fighting movement who bolstered their confidence and encouraged them to create a better learning atmosphere. As all of you must have heard the name of Sarat Chandra Chatterjee (famous for his Devdas and other novels), also shared the same period at Calcutta University.

I will start with Meghnath Saha whose work was mainly focused on ionization of atoms at high temperature environment as it is found in stellar matter (sun, stars etc.). His work led to determination of temperature of different stellar objects in terms of all existing physical parameters. In 1919, the Premchand Raychand Scholarship took Saha to Europe for about two years. He spent five months in Imperial College, London, where he developed his theory further with the help of Professor A.Fowler. In November 1921 Saha returned to Calcutta as Khaira Professor of Physics, a new chair created from the endowment of Kumar Guruprasad Singh of Khaira. He left for Allahabad in 1923, to return in 1938 as Palit Professor of Physics at University of Calcutta. He was elected Fellow of the Royal Society in 1927 at the age of thirty-four. Saha took keen interest in the establishment of Indian Association for the Cultivation of Science (IACS which is presently located at Jadavpur, Calcutta), and became its first full-time director in 1952. He also established what is now known as the Saha Institute of Nuclear Physics (SINP) in 1948 and installed a 38-inch cyclotron or atom smasher, the first of its kind outside America and Europe. “Saha’s ionization formula” is something which one cannot escape while studying basic Nuclear physics.

Thereafter, it was the genius Satyendra Nath Bose. Satyendra Nath Bose was born on the first of January 1894 in Calcutta. His father Surendra Nath was employed in the Engineering Department of the East India Railway. Satyendra Nath was the eldest of his seven children; the rest were all daughters. Though Surendra Nath Bose lost his wife at an early age, without losing heart, he brought up his children well. It is said that, when Satyendra Nath was hardly three years old, a Bengali astrologer made this prediction: "This child will face many obstacles all through his life; nevertheless he will overcome them with his exceptional intelligence and attain great fame." The father, naturally, took a special interest in his son's progress. Though he had seven children he took care to see that nothing came in the way of the boy's education. At this young age, he never seemed to be eyeing for anything big but once he was trapped in the competitive environment of Calcutta University, he molded himself from an ordinary student to an extraordinary one. During his stay at the Presidency College, Calcutta he caught the eyes of all the Professors and teachers there. Once in paper of mathematics, he scored more than 100 (exactly 110) and the teacher was bound to award him the extra marks as he solved couple of problems (correctly!!!) in more than three ways. He joined Calcutta University in 1916 as lecturer and remained there till 1921 (in his first stint). After that, he joined Dacca University where he did something which created history. In the spring of 1924, he sent a six-paged letter to the great Sir Albert Einstein (then in Germany), in which discussed few issues related to “Planck’s law of radiation” and “Light Quantum Theory” which were themselves in an infant stage. Sir Einstein was so impressed with the content of the letter that he, himself translated it in German and got it published in the famous German Journal “Zeitschrift fur Physik”. Even though Satyendra Nath had only Master’s degree, the German government got ready to bear all the expenses for his trip. In Germany, he worked with Sir Albert Einstein and gave the scientific community “Bose-Einstein Statistics” which was absolutely a new way to perceive the concept of “Light”. They also gave a theory for a unique kind of condensation known as “Bose-Einstein” condensation. In fact, those particles, which follow Bose-Einstein Statistics, are known as Bosons.

(Bose first visited Paris in 1924. He stayed there for a year. He conducted research in the Madame Curie Laboratory, which had special facilities. Here he became acquainted with several physicists. The next year, he left Paris for Berlin to join Einstein and worked with him. There he came into close contact with noted scientists like Schroedinger and Heisenberg. He participated in all the meetings and discussions held there. While Bose was in Berlin, the post of a professor fell vacant in Dacca University. J. C. Ghosh and other friends persuaded him to apply for the post. Bose had not yet got his doctorate. It was, therefore, difficult for him to secure the professorship. A recommendation from Albert Einstein to select him would have made things easy for him. So, with great hesitation, Bose approached Einstein. Einstein was surprised. He said, “You are so proficient in your subject; is there a need for any other certificate or recommendation?” He wrote a letter to the authorities of Dacca University in which he said, “Can you find another scientist as proficient as Satyendra Nath? He is quite fit for the post” It had the desired effect. In 1926, Satyendranath Bose was appointed Professor and Head of the Department of Physics).
… more to follow

Silicon: life beyond carbon

I visited Drumheller (Alberta, Canada) last month to see the fossil of dinosaurs at Royal Tyrell Museum. I am attaching some of the pictures. Most of the fossils were bones of different parts (of the dinosaurs), different species and different times. The bones that I saw were hard to believe that they are actually "bones". Mainly because we are used to see bones, which are white or yellowish or brownish at some extent. During that visit, I could not distinguish between an EGG and a ROCK kept side by side and I identified incorrectly (unbelievable ehh...?).
Believe me they (the bones) look like rocks.....hard solid rocks. My post starts now.

All of us know, life was not like this 200 millions years ago. What about the earth and life on earth before that? How was it? I do not know and no one is SURE about the answer. Basic elements that constitutes life are mainly Carbon, Oxygen, Hydrogen. (Of course many elements like nitrogen, calcium, phosphorous etc etc.. also are vital part of life on earth.. I think all of you will agree that the way these elements make bonds now, were different long back ago. Let us think of an alternative to this system of carbon-hydrogen based life.

I remember a line in Sriramcharitmanas by Tulsidaas, a well known and one of the most popular Hindu script, which reads... छिति जल पावक गगन समीरा, पंच रचित अति अधम शरीरा। it means our body is made up of SOIL, WATER, FIRE, SKY AND AIR. In simple words, it is said in Hindu mythology that our body is made up of soil. This is also interpreted as follows. after death our body is disposed and it goes into these above-said five components. Sometimes I wander whether it gives us some more information other than this. Is it possible to have a life with soil? There is no answer to this but there are hints from the mother nature to decode. Silicon and silicates are main constituent of soil and is in abundance in the nature on earth. Can we imagine a life system with silicon and its derivatives as the basic unit instead of carbon and its derivatives as we see it in present.

Keeping all these questions in mind I started searching what our scientific knowledge says about this. While searching, I saw a literature by Prof. E. REYNOLDS published in NATURE in year 1893. That paper is about looking at silicon analogues of carbon compound. Periodic table is of help for this kind of comparison. I am writing some of the matter from that paper because I think it is important. "..Carbon, whether combined with hydrogen, oxygen or nitrogen or with all three, is the great element of organic nature, while silicon, in union with oxygen and various metals, not only forms about one-third of the solid crust of the earth, but is unquestionably the most important element of inorganic nature....." he further writes.. "....The chief functions of carbon are those which are performed at comparatively low temperatures;hence carbon is essentially the element of the present epoch. on the other hand, the activities of silicon are most marked at very high temperatures;hence it is the element whose chief work in nature was performed in the distant past, when the temperature of this earth was far beyond that at which the carbon compounds of organic life could exist......."

Silicon's ability to make stable bonds with elements like nitrogen, hydrogen and oxygen indicates the possibility of a silicon based life possibly in other atmosphere other than oxygen (as it is now). Other possible atmosphere could be SULFUR. I suggest sulfur because it shows properties similar to oxygen due to their presence in the same group (group 16). Also sulfur is related to silicon in period third in the same way as oxygen is related to carbon in the second period. Apart from this silicon is capable of making long chains, multiple bonds and good bond strength with oxygen. Moreover, silicon also has an edge over carbon because of its solubility in aluminum. as it is said "silicon dissolves freely in aluminum". In a way Aluminum resembles nitrogen in directly forming bond with silicon at higher temperatures (here the comparison is between Si-Al and C-N). Si-Al dissolution and ability to stay together with ease and stability can be understood in terms of there coexistence in the crust of earth which has been named with them as SIAL (refer to the other part of crust called "Sima", consisting of silicates and magnesium).

Before saying final words for this post, I would like to make two points about the importance of silicon (and its derivatives) in our life. First, a complex derivative of silicon made up of Silicon, oxygen, hydrogen and carbon is used in many medical application including cosmetic implants. This material is called "polymerized siloxene" or more commonly known as "silicone". This material shows know side effects in our body if implanted in our body. Secondly, recent research shows the importance of SILICON in our nutrition for strong and healthy bones (Journal of Nutrition and health aging, Vol 11, page 99 (2007)).

Finally, Its time to think on the possibility of a life with silicon, aluminum and oxygen as basic building blocks instead of carbon, nitrogen and hydrogen. Life in terms of this silicon and group can be thought to be present in past on earth or may be the world outside our reach.  Science gave the answer about our evolution, courtesy Charles Darwin. But it reminds me of a statement by George Bernard Shaw.
Science never solves a problem without creating ten more.