Guess the value. It's near - linear!!!

Mathematics

Solve for \(x\)

\( ... \)

Womb, Matrix, and List

J. J. Sylvester coined the term "matrix" in 1848. But what is the etymon of matrix?

From Online etymological dictionary:

matrix 1373, from O.Fr. matrice, from L. matrix (gen. matricis) "pregnant animal," in L.L. "womb," also "source, origin," from mater (gen. matris) "mother." Sense of "place or medium where something is developed" is first recorded 1555; sense of "embedding or enclosing mass" first recorded 1641. Logical sense of "array of possible combinations of truth-values" is attested from 1914.

How is womb connected to matrix? In the sense of origin? Or in the sense of source?

But wait, there is another word - matriculate. The etymon is matriculare - to register; to create a list. Could it be possible that Sylvester had coined matrix from matriculare? I do not know. Incidentally, the online etymological dictionary attributes the confusion to translation from Greek to Latin.

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Expression of Beauty

It was a classroom. The sixth standard classroom. The mathematics teacher was strolling. From left to right. After few cycles, from front to back. He had given an exercise to the students. Two, in fact. One was an algebra problem and the other was a geometrical construction. The eyes of the students were showing the signs of fear, excitement, joy, tension, wonder, etc. depending upon the mind to which they belonged. But the teacher was ruthless.

If he saw something silly, he would stop and give a special treatment of impulse-strike, aided by his strong knuckles, on the head. He was sincere in trying to explain things. You had to tell him that you did not understand. He would explain. Again tell him that you did not understand. He would explain. Again tell him that you did not understand. He was quite patient. He would explain again. Explain again and again. Tell him you had understood and do a simple and silly mistake. You would then know what others were always saying - that he was a good volley-ball player. The only thing was the hit would leave a mark on your head.

He was interested in answers as well as how it was written (algebra) and drawn (geometry construction). The line that comes between numerator and denominator would often become a line that brings you either good pat or hit. The circles and lines should be circles and lines of same thickness. If you were going to draw a line second time to get good impression, you better do it for that line completely. If some portion of the line had more strokes than the other, the stroke you left would emerge on your head or palm.

Once a parent came and asked him in front of us, why did he reduce some marks though their son had correctly constructed the geometrical object. He said,

"Mathematics is a beautiful subject. When you write about such a beautiful subject, there is no place for any ugliness there. Whether you write or talk or communicate or learn, you have to do justice to the beauty of the content. You have to discipline yourself to appreciate the subject. Just knowing how to construct will not do. It is not enough for me."

That was Chandran Sir.

I did not understand completely then. That was the age where you would associate beauty to visuals and sounds; not to few symbols and their meanings. Those words and their contents have grown along various directions over a period of time destroying the artificial markings of human exploration. They go beyond subjects and ideas. Stressing the importance of how you could express something beautiful in a beautiful manner is such a wonderful feeling to pass it on to the next generation.

I liked him then. I like him now.

Happy Teachers' day.

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Gian-Carlo Rota's 10 lessons

Kovailputhalvan has sent me this link to Gian-Carlo Rota's 10 lessons of an MIT Education. Thanks Kovaiputhalvan.

Few snippets from those 10 lessons:

--- Students join forces on the problem sets, and some students benefit more than others from these weekly collective efforts. The most brilliant students will invariably work out all the problems and let other students copy, and I pretend to be annoyed when I learn that this has happened. But I know that by making the effort to understand the solution of a truly difficult problem discovered by one of their peers, students learn more than they would by working out some less demanding exercise.

--- Half a century ago, the philosopher Gilbert Ryle discussed the difference between "knowing how" courses are those in mathematics, the exact sciences, engineering, playing a musical instrument, even sports. "Knowing what" courses are those in the social sciences, the creative arts, the humanities, and those aspects of a discipline that are described as having social value.

--- It is my theory that "knowing how" is revered because it can be tested.

(This knowing-how-and-knowing-what reminds me of Snow's Two Cultures!! Imprinting such "divisive" ideas on human psyche may not be good for education.)

--- The idea of genius elaborated during the Romantic Age (late 18th and 19th centuries) has done harm to education. It is demoralizing to give a young person role models of Beethoven, Einstein, and Feynman, presented as saintly figures who moved from insight to insight without a misstep. Scientific biographies often fail to give a realistic description of personality, and thereby create a false idea of scientific work.

--- Young people will correct any fantasies they have about genius. As they start doing research with their professors they learn another healthy lesson, namely, a professor may well behave like a fumbling idiot.

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Realizing the Belief ...

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

~John Louis von Neumann

Paul R. Halmos (1916-2006)

Prof. Paul R. Halmos died on October 2, 2006. George Pólya Award, for articles of expository excellence published in the College Mathematics Journal, was given to him in 1983 for his article The Thrills of Abstraction.

From The Thrills of Abstraction:

My wife and I were invited to a party recently, a party attended by four other couples, making a total of ten people. Some of those ten knew some of the others, and some did not, and some were polite, and some were not. As a result a certain amount of handshaking took place in an unpredictable way, subject only to two obvious conditions: no one shook his or her own hand and no husband shook his wife's hand. When it was all over, I became curious and I went around the party asking each person: " How many hands did you shake? ... And you? ... And you?" What answers could I have received? Conceivably some people could have said "None", and others could have given me any number between 1 and 8 inclusive. That's right, isn't it? Since self-handshakes and spouse-handshakes were ruled out, 8 is the maximum number of hands that any one of the party of 10 could have shaken.

I asked nine people (everybody, including my own wife), and each answer could have been any one of the nine numbers 0 to 8 inclusive. I was interested to note, and I hereby report, that the nine different people gave me nine different answers; someone said 0, someone said 1, and so on, and, finally, someone said 8. When it was all over, my curiosity was satisfied: I knew all the answers. Next morning, I told the story to my colleagues at the office, exactly as I told it now, and I challenged them, on the basis of the information just given, to tell me how many hands my wife shook.

If you try to find the answer for that question, you will find a kind of similarity, in approach, with the following problem.

C thinks of two consecutive numbers between 1 and 10. C tells one number to A, and the another to B. Now for the conversation between A and B:

A: I do not know your number
B: I do not know your number
A: I do not know your number
B: I know your number
A: I too know your number

What are those numbers?

References

1. An obituary at the website of the Mathematical Association of America
2. A brief biography: http://www-history.mcs.st-and.ac.uk/history/Biographies/Halmos.html
3. Wikipedia http://en.wikipedia.org/wiki/Paul_Halmos
4. Paul R. Halmos, The Thrills of Abstraction, Two-Year Coll. Math. J. 13 (1982), 243-251.
5. Number puzzle

6174

1. Start with a four-digit number whose digits are not all equal.

2. Arrange the digits in ascending, say B, and descending order, say A.

3. Find A-B.

4. Go to step-2.

The above process terminates on the number 6174 after seven or fewer steps.

Example:

5063 --> 6530-0356=6174 --> 7641-1467=6174

The Indian mathematician Dattathreya Ramachandra Kaprekar (1905-1988) published the above property of the number 6174 in 1955 [#]. There are other interesting Kaprekar numbers too [#].

[#] Kaprekar, D.R., An Interesting Property of the Number 6174, Scripta Mathematica 15, 1955, pp. 244-245.

[#] Links:
Mathews: Kaprekar Process
Kaprekar Number
http://mathworld.wolfram.com/KaprekarNumber.html

Lilavati

Bhaskara II was born in 1114 or 1115 and died around 1185. His famous work Lilavati is said to be named after his daughter, to console her for an astrological forecast that went wrong.

The story goes that Bhaskara used his knowledge to choose the most propitious date and time for his daughter's wedding. As the time approached, one of her pearls fell into the water clock as she leaned over it, stopping the outflow of water. Before anyone noticed, however, the critical time passed and the wedding had to be called off. The hapless Lilavati never married, and now she is remembered only through the book that bears the name.
(From Mathematics and Its History by John Stillwell)
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I have also read somewhere about a legend which says that the wedding indeed took place but her husband died soon after.