Monthly Archives: March 2016

Prime Factors Facts

In number theory, we are concerned with natural numbers. Amongst those, the most important are the so-called prime numbers. A prime number is a number which cannot be divided by any other number except 1 and itself. A number which is not prime is composite. Composite numbers are “composed” of prime numbers; to be precise, the fundamental theorem of arithmetic says that any number can be written as product of powers of prime numbers uniquely up to rearrangement. So, prime numbers are the building blocks or atoms of all numbers !! By the way, 1 is considered neither prime nor composite.

So, what are the examples of prime numbers? Take the Sieve of Eratosthenes. (Google this !!!). The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, …

Surprisingly, as you pass 100, the “density of primes” starts lessening. If you go deeper into this, you are concerned with the “distribution of primes’…Many immortal mathematicians have devoted a major part of their lives to the study of primes…Carl Friedrich Gauss (one of his hobbies was to keep on finding more and more primes…); Bernhard Riemann who gave us a million dollar math problem on prime numbers (http://www.claymath.org ) etc.

If you are curious, or just want to take a one hour coffee break, refer to “A Brief History of Primes” by Prof. Manindra Agarwal of IIT, Kanpur http://claymath.msri.org/agrawal2002.mov

More later,

Nalin Pithwa

Number theory has numerous uses

One of the fun ways to get started in mathematics at an early age s via number theory. It does not require deep, esoteric knowledge of concepts of mathematics to get started, but as you explore and experiment, you will learn a lot and also you will have a ball of time writing programs in basic number theory. One of the best references I have come across is “A Friendly Introduction to Number Theory” by Dr. Joseph Silverman. It is available on Amazon India.

Well, number theory is not just pure math; as we all know, it is the very core of cryptography and security in a world transforming itself to a totally digital commerce amongst other rapid changes. Witness, for example, the current intense debate about opening up an iPhone (Apple vs. FBI) and some time back, there was the problem with AES Encrypted Blackberry messaging services in India.

Number theory is also used in Digital Signal Processing, the way to filter out unwanted “noise” from an information signal or “communications signal.” Digital Signal Processing is at the heart of modem technology without which we would not be able to have any real computer networks.

There was a time when, as G H Hardy had claimed that number theory is the purest of all sciences as it is untouched by human desire. Not any more !!!

Can you imagine a world without numbers ?? That reminds me of a famous quote: “God created the natural numbers, all the rest is man-made.” (Kronecker).

More later,

Nalin Pithwa

Equation of a circle

Consider a fixed complex number z_{0} and let z be any complex number which moves in such a way that its distance from z_{0} is always equal to r. This implies z would lie on a circle whose centre is z_{0} and radius r. And, its equation would be

|z-z_{0}|=r

or |z-z_{0}|^{2}=r^{2}

or (z-z_{0})(\overline{z}-\overline{z_{0}})=r^{2},

or z\overline{z}-z \overline{z_{0}}-\overline{z}z_{0}-r^{2}=0

Let -a=z_{0} and z_{0}\overline{z_{0}}-r^{2}=b. Then,

z\overline{z}+a\overline{z}+\overline{a}z+b=0

It represents the general equation of a circle in the complex plane.

Now, let us consider a circle described on a line segment AB (A(z_{1}), B(z_{2})) as its diameter. Let P(z) as its diameter. Let P(z) be any point on the circle. As the angle in the semicircle is \pi/2, so

\angle {APB}=\pi/2

\Longrightarrow (\frac{z_{1}-z}{z_{2}-z})=\pm \pi/2

\Longrightarrow \frac{z-z_{1}}{z-z_{2}} is purely imaginary.

\frac{z-z_{1}}{z-z_{2}}+\frac{\overline{z}-\overline{z_{1}}}{\overline{z}-\overline{z_{2}}}=0

\Longrightarrow (z-z_{1})(\overline{z}-\overline{z_{2}})+(z-z_{2})(\overline{z}-\overline{z_{1}})=0

Condition for four points to be concyclic:

Let ABCD be a cyclic quadrilateral such that A(z_{1}), B(z_{2}), C(z_{3}) and D(z_{4}) lie on a circle. (Remember the following basic property of concyclic quadrilaterals: opposite angles are supplementary).

The above property means the following:

\arg (\frac{z_{4}-z_{1}}{z_{2}-z_{1}})+\arg (\frac{z_{2}-z_{3}}{z_{4}-z_{3}})=\pi

\Longrightarrow \arg (\frac{z_{4}-z_{1}}{z_{2}-z_{1}})(\frac{z_{2}-z_{3}}{z_{4}-z_{3}})=\pi

(\frac{z_{4}-z_{1}}{z_{2}-z_{1}})(\frac{z_{2}-z_{3}}{z_{4}-z_{3}}) is purely real.

Thus, points A(z_{1}), B(z_{2}), C(z_{3}), D(z_{4}) (taken in order) would be concyclic if the above condition is satisfied.

More later,

Nalin Pithwa

 

 

 

 

A tribute to Lloyd Shapley, Nobel Laureate mathematician

Lloyd Shapley, Nobel Laureate, mathematician, an inventor of theory of games passed away recently.

Source: Chapter 11, Lloyd, Princeton 1950 — Sylvia Nasat in “A Beautiful Mind”

When Lloyd Shapley first moved into the Graduate College, a few doors down from Nash in the fall of 1949, Lloyd Shapley had just turned twenty-six, five years and eleven days older than Nash. No one could have presented a stronger contrast with the childish, boorish, handsome, and uninhibited boy wonder from West Virginia.

Born and bred in Cambridge, Massachusetts, Shapley was one of five children of one of the most famous and revered scientists in America, the Harvard astronomer, Harlow Shapley. The senior Shapley was a public figure known to every educated household, and also one of the most politically active. In 1950, he was accorded the dubious honour of being the first prominent scientists to appear on the earliest of Senator John McCarthy’s famous list of crypto-communists.

Lloyd Shapley was a war hero. He was drafted in 1943. He refused an offer to become an officer. That same year, as a sergeant of the Army Air Corps, in Sheng-Du, China, Shapley got a Bronze Star for breaking the Japanese weather code. In 1945, he went back to Harvard, where he had begun to study mathematics before he was drafted, and finished his BA in mathematics in 1948.

When Shapley showed up at Princeton, von Neumann already considered him the brightest young star in game theory research. Shapley had spent the year after graduating from Harvard at the RAND Corporation, a think tank in Santa Monica that was attempting to use game theory applications to solve military problems, and came to Princeton while technically on leave from RAND. He was immediately recognized as brilliant and quite sophisticated in his thinking. One contemporary remembers that he “talked good math, knew a lot.” He did extraordinarily hard double crostics from The NewYork Times without using a pencil. He was a fiercely competitive and highly accomplished player of Kriegspiel and Go. Everybody knew “that ‘his game was strictly his own,” said another fellow student. “He went out of his way to find nonstandard moves. No one was going to anticipate them.” He was also well read. He played the piano beautifully. His manner suggested an acute awareness of pedigree and prospects. When Lefschetz wrote a letter telling him of a very generous grant if he came to Princeton, for example, Shapley replied loftily and hint of disdain, “Dear Leftschtez, The arrangements are satisfactory. Go ahead with the formalities, Shapley.”

Shapley was by no means as self-confident as his imperious note to Lefschetz implied. His appearance can only be described as rather strange. Tall, dark and so thin that his clothing hung from him like a scarecrow’s, Shapley reminded one young woman of a giant insect, another contemporary said he looked like a horse. His normally gentle demeanour and ironic banter hid a violent temper and a harshly self-critical streak. When challenged in some unexpected fashion, he could become hysterical, literally vibrating and shaking with fury. His perfectionism, which would later prevent him from publishing a large portion of his research, was extreme. He was, moreover, acutely self-conscious about being a few years older than some of the brilliant young men around the Princeton mathematics department.

Nash was one of the first students Shapley met at the Graduate College. For a time, they shared a bathroom. Both of them attended Tucker’s game theory seminar every Thursday, now run by Kuhn and Gale while Tucker was at Stanford. The best way to describe the impression Nash made on Shapley when the two first talked about mathematics is to say that Nash took Shapley”s breath away. Shapley, could, of course, see what the others saw — the childishness, brattiness, obnoxiousness — but he saw a great deal more. He was dazzled by what he would later describe as Nash’s “keen, beautiful, logical mind.” Instead of being alienated by the younger’s man’s odd manner and weird behaviour, he interpreted these simply as signs of immaturity. “Nash was spiteful, a child with a social IQ of 12, but Lloyd did appreciate talent,” recalled Martin Shubik.

Shapley’s greatest eccentricity at the time was his claim that he was on a twenty-five hour sleep cycle. He worked and sleeped at extremely odd hours, often transposing day and night. “Every once in a while, he’d disappear from sight,” another student recalled. “That’s what he said. We accepted anything.” Waking Stanley up when he was lost to the world became an ongoing prank. “A group of us was attending a regular seminar at the institute given by de Rham and Kodaira. We were always very anxious to go but only three or four of us had cars. Lloyd Shapley was one, but there was one difficulty. Lloyd like to sleep late and was often asleep at two o’clock in the afternoon. So we had to devise all sorts of ways to wake him. We dropped hot candle wax on him. I devised another method. We played 45 rpm records of Lloyd’s favourite Chinese music without the little insert so that it oscillated all over the place (and made excruciating noise), Nash once tried to wake Shapley by climbing on his bed, straddling him and dropping wster in his ear with an eyedropper.

John McCarthy, one of the inventors of artificial intelligence, also befriended Shapley and apparently aroused Nash’s jealousy. One day McCarthy got an inquiry from a Philadelphia haberdashery about a massive shirt order he had placed. How good was his credit, the company wanted to know? McCarthy, who hadn’t placed any such order, immediately suspected Nash and asked Shapley if Nash was the culprit. Shapley confirmed that this was highly likely. McCarthy asked the company for the original order. Sure enough, a postcard came back with Nash’s unmistakable scrawl in green ink, the colour Nash always used. Shubik and McCarthy cornered Nash and confronted him. “There was no denying what he had done. We threatened him with postal inspectors. The post office refused to merely bawl him out. “If we do anything, we will prosecute him,” they said. Concluding that Nash had learned his lesson, Shubik and McCarthy dropped the matter. Another time, he rigged up McCarthy’s bed so that it would collapse when McCarthy tried to crawl under the covers.

It was Shapley, who reacted to Nash’s absurd behaviour with amused tolerance, who proposed that they might channel his mischievous impulses in a more intellectually constructive way. So, Nash, Shapley, Shubik, and McCarthy, along with another student named Mel Hausner, invented a game involving coalitions and double=-crosses. They called the game, So Long Sucker!. The game is played with a pile of different colored poker chips. Nash and others crafted a complicated set of rules, designed to force players to join forces with one another to advance, but ultimately to double-cross one another in order to win. The point of the game was to produce psychological mayhem, and, apparently, it often did. McCarthy, remembers losing his temper after Nash cold-bloodedly dumped him on the second-to-last round and Nash was absolutely astonished that McCarthy could get so emotional. “But, I didn’t need you anymore,” Nash kept saying, over and over.

By and large, Shapley tried to play the role of the mentor. He came to Nash’s aid, for example, when Tucker demanded that Nash include a concrete example of an equilibrium point in his thesis and Nash couldn’t think of a good one. Shapley spent weeks working out an elaborate but convincing example of Nash’s equilibrium concept involving three-handed poker, another Shapley speciality.

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-Nalin Pithwa

 

 

 

 

 

 

A mathematician is at once an adult as well as a child

All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But, they also live in the common world where things are ambiguous, transient, subject to vicissitudes. Mathematicians go backwards and forwards from one world to another. They are adults in the crystalline world, infants in the real one. — S. Cappell, Courant Institute of Mathematical Sciences, 1996.

A movie on mathematician and astronomer Aryabhatta

(From a recent print edition of the daily newspaper, The Hindu)

Veteran Bollywood actor and film maker Manoj Kumar is working on his next directorial venture which would be based on mathematician and astronomer Aryabhatta.

“I am going to return as a director soon. I am working on a film on Aryabhatta. It is an interesting story to tell viewers about his contributions,” said Kumar, who was announced the winner of the Dadasaheb Phalke Award.

“Three years we have been working on script, as it is a difficult subject to deal with,” the 78-year-old added. Aryabhatta was a pioneer during the classical age of Indian mathematics and Indian astronomy.

Kumar is scouting for new comers for the film.

“We are working with Kishore Namit Kapoor who has trained some of the most prominent film actors. He will help  us with it (casting),” Kumar said. — PTI.

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Remark: A film on Srinivasa Ramanujan is also being made by another director !

Cheers,

Nalin Pithwa

Equation of a line : Geometry and Complex Numbers

complexstlineEquation of the line passing through the point z_{1} and z_{2}:

Ref: Mathematics for Joint Entrance Examination JEE (Advanced), Second Edition, Algebra, G Tewani.

There are two forms of this equation, as given below:

\left | \begin{array}{ccc} z & \overline{z_{1}} & 1 \\ z_{1} & \overline{z_{2}} & 1 \\ z_{2} & \overline{z_{2}} & 1 \end{array} \right |=0

and \frac{z-z_{1}}{\overline{z}-\overline{z_{1}}}=\frac{z_{1}-z_{2}}{\overline{z_{}}-\overline{z_{2}}}

Proof:

Let z_{1}=x_{1}+iy_{1} and z_{2}=x_{2}+iy_{2}. Let A and B be the points representing z_{1} and z_{2} respectively.

Let P(z) be any point on the line joining A and B. Let z=x+iy. Then P \equiv (x,y), A \equiv (x_{1}, y_{1}) and B \equiv (x_{2},y_{2}). Points P, A, and B are collinear.

See attached JPEG figure 1.

The figure shows that the three points A, P  and B are collinear.

Shifting the line AB at the origin as shown in the figure; points O, P, Q are collinear. Hence,

\arg(z-z_{2})=\arg(z_{1}-z_{2}) or

\arg {\frac{z-z_{2}}{z_{1}-z_{2}}}=0

\Longrightarrow \frac{z-z_{2}}{z_{1}-z_{2}} is purely real.

\frac{z-z_{2}}{z_{1}-z_{2}}=\frac{\overline{z-z_{2}}}{z_{1}-z_{2}}

or, \frac{z-z_{2}}{z_{1}-z_{2}}=\frac{\overline{z}-\overline{z_{2}}}{\overline{z_{1}}-\overline{z_{2}}} call this as Equation 1.

\left | \begin{array}{ccc} z & \overline{z} & 1 \\ z_{1} & \overline{z_{1}} & 1\\ z_{2} & \overline{z_{2}} & 1 \end{array} \right |=0. Call this as Equation 2.

Hence, from (2), if points z_{1}, z_{2}, z_{3} are collinear, then

\left | \begin{array}{ccc} z_{1} & \overline{z_{1}} & 1 \\ z_{2} & \overline{z_{2}} & 1 \\ z_{3} & \overline{z_{3}} & 1 \end{array} \right |=0.

Equation (2) can also be written as

(\overline{z_{1}}- \overline{z_{2}}) - (z_{1}-z_{2})\overline{z}+z_{1}\overline{z_{2}}-z_{2}\overline{z_{1}}=0

\Longrightarrow i(\overline{z_{1}}-\overline{z_{2}})z-(z_{1}-z_{2})\overline{z} + z_{1}\overline{z_{2}}-z_{2}\overline{z_{1}}=0

\Longrightarrow \overline{a}z + a\overline{z}+b=0 let us call this Equation 3.

where a=-i(z_{1}-z_{2}) and b=i(z_{1}\overline{z_{2}}-z_{2}\overline{z_{1}})=i 2i \times \Im (z_{1} \overline{z_{2}}), which in turn equals

-2 \times \Im(z_{1}\overline{z_{2}}), which is a real number.

Slope  of the given line

In Equation (3), replacing z by x+iy, we get (x+iy)\overline{a} + (x-iy)a+b=0,

\Longrightarrow (a+\overline{a})x + iy(\overline{a}-a)+b=0

Hence, the slope = \frac{a+\overline{a}}{i(a-\overline{a})}=\frac{2 \Re(a)}{2i \times \Im(a)}=-\frac{\Re(a)}{\Im(a)}

Equation of a line parallel to the line z \overline{a}+\overline{z}a+b=0 is z \overline{a} + \overline{z} a + \lambda=0 (where \lambda is a real number).

Equation of a line perpendicular to the line z\overline{a}+\overline{z}a+b=0 is z\overline{a}+\overline{z} a + i \lambda=0 (where \lambda is a real number).

Equation of a perpendicular bisector

Consider a line segment joining A(z_{1}) and B(z_{2}). Let the line L be its perpendicular bisector. If P(z) be any point on L, then we have (see attached fig 2)

PA=PB \Longrightarrow |z-z_{1}|=|z-z_{2}|

or |z-z_{1}|^{2}=|z-z_{2}|^{2}

or (z-z_{1})(\overline{z}-\overline{z_{1}})=(z-z_{2})(\overline{z}-\overline{z_{2}})

or

Here, a= z_{2}-z_{1} and b=z_{1}\overline{z_{1}}-z_{2} \overline{z_{2}}

Distance of a given point from a given line:

(See attached Fig 3).

Let the given line be z \overline{a} + \overline{z} a + b=0 and the given point be z_{c}. Then,

z_{c}=x_{c}+iy_{c}

Replacing z by x+iy in the given equation, we get

x(a+\overline{a})+iy(\overline{a}-a)+b=0

Distance of (x_{c},y_{c}) from this line is

\frac{|x_{c}(a+\overline{a})+iy_{c}(\overline{a}-a)+b|}{\sqrt{(a+\overline{a})^{2}-(a-\overline{a})^{2}}}

which in turn equals

\frac{z_{c}\overline{a}+\overline{z_{c}}a+b}{\sqrt{4(\Re(a))^{2}+4(\Im(a))^{2}}} which is equal to finally

\frac{|z_{c}\overline{a}+\overline{z_{c}}a+b|}{2|a|}.

More later,

Nalin Pithwa

 

 

 

 

 

 

 

 

 

 

Learn music to be better at mathematics !

Music soothes the soul. Music calms the soul. Albert Einstein used to play the violin. John Nash, Jr. used to listen to classical western music, and would whistle Bach even while churning Math in his head. More importantly, music lets the subconscious mind free to work on the intense math or even advanced programming or engineering problems that one is working on.

I just read in yesterday’s newspaper, The Hindu (print version) (an article by Allan Moses Rodricks) that the following are the benefits of learning music:

  • sharpens the brain
  • boosts memory
  • enhances skills
  • builds discipline
  • social interaction
  • boosts self-esteem
  • cultural exchange
  • betters language
  • enhances creativity
  • betters expression

To the above, I may add, it is catharsis or stress-buster. Once Einstein had made some mistake — the story goes that he played his violin for several hours and relieved and rejuvenated himself. I do not play any musical instrument, but like to be able to appreciate western classical music. Or, just hear some Indian musical instrument. Like mathematical training, musical training should be given from a young age…

Cheers,

Nalin Pithwa

Ivy University Myths

Excellent lecture by a Princeton Professor: “Where you go is not where you’ll be ” This is the universal anxiety for all parents and students in Asian countries where there are li…

Source: Ivy University Myths

Mathematics and German Business: Many professional uses of pure and applied mathematics

Mathematics and German business

 

Oberwolfach/Heidelberg, 1 April 2008

Mathematics as a key technology for German business

Bernd Pischetsrieder and Gert-Martin Greuel present Springer book to German Education Minister Annette Schavan and Baden-Württemberg’s Minister of Culture Helmut Rau

 

The Oberwolfach Stiftung and the Mathematisches Forschungsinstitut Oberwolfach (MFO) are launching the Springer title Mathematik – Motor der Wirtschaft today.  The book features articles by renowned business figures, and will be presented by the German Federal Minister of Education and Research, Annette Schavan, at a gala event.

In their articles, various heads of major German companies – Allianz, Daimler, Lufthansa, Linde, and TUI, to name but a few – sum it up in a nutshell:  Mathematics is everywhere, and our economy would not work without it.  SAP’s CEO, Henning Kagermann, puts it like this:  “Corporate management without mathematics is like space travel without physics.  Numbers aren’t the be all and end all in business life.  But without mathematics, we would be nothing.”  The list of authors in the Springer book reads like a Who’s Who of German Dax companies.

The Year of Mathematics is the German federal government’s ninth science year, and was officially launched by the Education Minister Annette Schavan in January.  It presents a welcome opportunity to underline the importance of mathematics in leading companies.  “We need more mathematicians in Germany, more engineers and highly qualified experts who can bring mathematical solutions into specific products, processes and services.  Mathematics offers exciting career opportunities for young people.  That’s why we’re showing specifically how fascinating and diverse this science is during the Year of Mathematics,” Schavan said.

The book Mathematik – Motor der Wirtschaft came about in close cooperation between the Oberwolfach Stiftung and the Mathematisches Forschungsinstitut Oberwolfach.  The practical articles by leading German businesspeople show that mathematics is not just about abstract research topics, but is also used as a daily tool in all branches of the economy:  from insurance mathematics through high-tech applications in the pharmaceuticals industry, designing new cars, flight safety to risk management.  And good mathematical skills are even a prerequisite for furthering one’s career, as the authors demonstrate.

The internationally renowned Mathematische Forschungsinstitut Oberwolfach hosts some 2,500 mathematicians from around the world every year, who use it as a conference and research center.  The MFO’s scientific programs cover the entire range of mathematics, including its applications in natural sciences and technology.
The Oberwolfach Stiftung is a foundation that finances the MFO, alongside regional and national government funding.  The foundation’s board consists of leading representatives from the worlds of business and mathematics.

Springer Science+Business Media is one of the world’s leading suppliers of scientific and specialist literature.  It is the second-largest publisher of journals in the science, technology, and medicine (STM) sector, the largest publisher of STM books and the largest business-to-business publisher in the German-language area.  With just under 100 journals and an annual 450 book titles on mathematics, Springer is the world’s number one in the field.  The group publishes over 1,700 journals and more than 5,500 new books a year, as well as the largest STM eBook Collection worldwide.  Springer has operations in about 20 countries in Europe, the USA, and Asia, and some 5,000 employees.

Mathematik – Motor der Wirtschaft
Greuel, Gert-Martin; Remmert, Reinhold; Rupprecht, Gerhard (Eds.)
2008, XX, 125 pages, hardcover
ISBN 978-3-540-78667-2

Contact and review copies:
Renate Bayaz
Tel. +49 171 866 8118

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I  am just sharing the above information with those who question the utility of mathematics especially parents and students who feel mathematics just leading to a teaching position. The awareness of mathematics as a profession in the industry is not at all there in India compared to the West. I hope to put up/share this information.

Regards,

Nalin Pithwa