The most beautiful thing that we can experience is the mysterious. It is the source of all true art and sciences.

— Albert Einstein, in What I believe, 1930.

Mathematics demystified

August 16, 2019 – 6:15 am

The most beautiful thing that we can experience is the mysterious. It is the source of all true art and sciences.

— Albert Einstein, in What I believe, 1930.

August 16, 2019 – 5:25 am

(From A Beautiful Mind by Sylvia Nasar)

The first bite of mathematical apple probably occurred when Nash at around age thirteen or fourteen read E. T. Bell’s extra ordinary book Men of Mathematics — an experience he alludes to in his autobiographical essay (of Nobel Prize, Economics) Bell’s book, which was published in 1937, would have given Nash the first glimpse of real mathematics, a heady realm of symbols and mysteries entirely unconnected to the seemingly arbitrary and dull rules of arithmetic and geometry taught in school or even in the entertaining but ultimately trivial calculations that Nash carried out in the course of chemistry and electrical experiments.

Men of Mathematics consists of lively — and, as it turns out, not entirely accurate — biographical sketches. Its flamboyant author, a professor of mathematics at California Institute of Technology, declared himself disgusted with “the ludicrous untruth of the traditional portrait of the mathematician” as a “slovenly dreamer totally devoid of common sense.” He assured his readers that the great mathematicians of history were an exceptionally virile and even adventuresome breed. He sought to prove his point with vivid accounts of infant precocity, monstrously insensitive educational authorities, crushing poverty, jealous rivals, love affairs, royal patronage, and many varieties of early death, including some resulting from duels. He even went so far in defending mathematicians as to answer the question : “How many of the great mathematicians have been perverts?” None, was his answer. ‘Some lived celibate lives, usually on account of economic disabilities, but the majority were happily married…The only mathematician discussed here whose life might offer something of interest to a Freudian is Pascal.’ The book became a bestseller as soon as it appeared.

What makes Bell’s account not merely charming, but intellectually seductive, are his lively descriptions of mathematical problems that inspired his subjects when they were young, and his breezy assurance that there were still deep and beautiful problems that could be solved by amateurs, boys of fourteen, to be specific. It was Bell’s essay on Fermat, one of the greatest mathematicians of all time, but a perfectly conventional seventeenth century French magistrate, whose life was “quiet, laborious and uneventful,” that caught Nash’s eye. The main interest of Fermat, who shares the credit for inventing calculus with Newton and analytic geometry with Descartes, was number theory — “the higher arithmetic.” Number theory, investigates the natural relationships of those common whole numbers 1, 2, 3, 4, 5…which we utter almost as soon as we learn to talk.

For Nash, proving a theorem known as Fermat’s (Little) Theorem about prime numbers, those mysterious integers that have no divisors besides themselves and one produced an epiphany of sorts. Often mathematical geniuses, Albert Einstein and Bertrand Russell among them recount similar revelatory experiences in early adolescence. Einstein recalled the “wonder” of his first encounter with Euclid at age twelve:

“Here were assertions, as for example the intersection of three altitudes of a triangle at one point which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me.”

Nash does not describe his feelings when he succeeded in devising a proof for Fermat’s assertion that if n is any whole number and p any prime number, then n multiplied by itself p times minus p is divisible by p. But, he notes the fact in his autobiographical essay, and his emphasis on this concrete result of his initial encounter with Fermat suggests that the thrill of discovering and exercising his own intellectual powers — as much as any sense of wonder inspired by hitherto unsuspected patterns and meanings — was what made this moment such a memorable one. That thrill has been decisive for many a future mathematician. Bell describes how success in solving a problem posed by Fermat led Carl Friedrich Gauss, the renowned German mathematician, to choose between two careers for which he was similarly talented. ‘It was this discovery …which induced the young man to choose mathematics instead of philology as his life work.”…

For those readers who are interested:

- Who wants to be a mathematician:

http://www.ams.org/publicoutreach/students/wwtbam/wwtbam

2. Resonance Journal (India):

https://www.ias.ac.in/Journals/Resonance_%E2%80%93_Journal_of_Science_Education/

3. Ramanujan School of Mathematics; Super30 of Prof Anand Kumar:

http://www.super30.org/rsm.html

Cheers,

Nalin Pithwa

August 30, 2015 – 12:07 am

**Problem.**

The biographer and friend of the immortal physicist Albert Einstein, A. Moszkowski, wished to distract his friend during an illness and suggested the following problem:

The problem he posed was this: “Take the position of the hands of a clock at 12 noon. If the hour hand and the minute hand were interchanged in this position the time would would still be correct. But, at other times (say at 6 o”clock) the interchange would be absurd, giving a position that never occurs in ordinary clocks: the minute cannot be on 6 when the hour hand points to 12. The question that arises is when and how often do the hands of a clock occupy positions in which interchanging the hands yields a new position that is correct for an ordinary clock?

“Yes,” replied Einstein, “this is just the type of problem for a person kept to his bed by illness; it is interesting enough and not so very easy. I am afraid thought that the amusement won’t last long because I already have my fingers on a solution.”

“Getting up in bed, he took a piece of paper and sketched the hypothesis of the problem. And, he solved it in no more time than it took me to state it.”

How is the problem tackled?

**Solution:**

We measure the distance of the hands around the dial from the point 12 in sixtieths of a circle.

Suppose one of the required positions of the hands was observed when the hour hand moved x divisions, from 12, and the minute hand moved q divisions. Since the hour hand passes over 60 divisions in 12 hours, or 5 divisions every hour, it covered the x divisions in x/5 hours. In other words, x/5 hours passed after the clock indicated 12 o’clock. The minute hand, covered y divisions in y minutes, that is, in y/60 hours. In other words, the minute hand passed the figure 12 a total of y/60 hours ago, or

hours after both hands stood at twelve. This number is whole(from 0 to 11) since it shows how many whole hours have passed since twelve.

When the hands are interchanged, we similarly find that

whole hours have passed from 12 o’clock to the time indicated by the hands. This is a whole number from 0 to 11.

And, so we have the following system of equations:

where m and n are integers (whole numbers) that can vary between 0 and 11. From this system, we find

By assigning m and n the values from 0 to 11, we can determine all the required positions of the hands. Since each of the 12 values of m can be correlated with each of the 12 values of n, it would appear that the total number of solutions is equal to 12 times 12, that is, 144. Actually, however, it is 143 because when and also when we obtain the same position of the hands.

Put , we have

and the clock shows 12, as in the case of .

We will not discuss all possible positions, but only two.

First example:

and

and the clock reads 1 hour minutes; the hands, have merged by this time and they can of course be interchanged (as in all other cases of coincidence of the hands).

Second Example.

.

and .

The respective times are 8 hours 28.53 minutes and 5 hours 42.38 minutes.

More clock problems later,

Nalin Pithwa