Monthly Archives: February 2016

The Science of Concentration

The Science of Concentration

(An article published in the DNA newspaper some time back).

***You can drive yourself crazy trying to multitask or you can recognize your brain’s finite capacity for processing information, and achieve the satisfaction of a focused life***

Imagine that you have ditched your laptop and turned off  your smartphone. You are beyond the reach of Facebook, e-mail, text messages. You are in a Twitter-free zone, sitting in a taxicab with a copy of Rapt, a guide by Winfred Gallagher to the science of paying attention.

The book’s theme, which Gallagher chose after she learned that she had an especially nasty form of cancer, is borrowed from the psychologist William James: “My experience is what I agree to attend to.” You can lead a miserable life by obsessing on problems. You can drive yourself crazy trying to multitask and answer every e-mail message instantly.

Or, you can recognize your brain’s finite capacity for processing information, accentuate the positive and achieve the satisfactions of what Gallagher calls the focused life. It can sound wonderfully appealing, except that as you sit in the cab reading about the science of paying attention, you realize that…you are not paying attention to a word on the page.

The taxi’s television, which can’t be turned off, is showing a commercial of a guy in a taxi working on a laptop — and as long as he is jabbering about his new wireless card has made him productive during his cab ride, you can’t do anything productive during yours.

Why can’t you concentrate on anything except the desire to shut him up? And, even if you flee the cab, is there any realistic refuge anymore from the “age of distraction”?

I put these questions to Gallagher, and to one of the experts in her book, Robert Desimone, a neuroscientist at MIT who has been doing experiments somewhat similar to my taxicab TV experience. He has been tracking the brain waves of macaque monkeys and human beings as they stare at video screens looking for some flashing patterns.

When something bright or novel flashes, it tends to automatically win the competition for the brain’s attention, but that involuntary bottom-up impulse can be voluntarily overridden through a top-down process that Desimone calls “biased competition”. He and colleagues have found that neurons in the prefrontal cortex — the brain’s planning center — start oscillating in unison and send signals directing the visual cortex to heed something else.

These oscillations, called gamma waves, are created by neurons’ firing on and off at the same time. But, these signals can have trouble getting through in a noisy environment.

“It takes lot of your prefrontal brain power to force yourself not to process a strong input like a television commercial”, said Desimone, the director of the McGovern Institute for Brain Research at MIT.

Now that neuroscientists have identified the brain’s synchronizing mechanism, they have started work on therapies to strengthen attention. In the current issue of Nature, researchers from MIT, Penn and Stanford report that they directly induced gamma waves in mice by shining pulses of laser light through tiny optical fibres onto genetically engineered neurons.

Ultimately, Desimone said that it may be possible to improve your attention by using pulses of light to directly synchronize your neurons, a form of direct therapy that could help people with schizophrenia and attention-deficit problems.

In the nearer future, neuroscientists might also help you focus by observing your brain activity and providing biofeedback as you practice strengthening your concentration.

Gallagher advocates meditation to increase your focus, but she says there are also simpler ways to put the lessons of attention researchers to use. Once she learned how hard it was for the brain to avoid paying attention to sounds, particularly other people’s voices, she began carrying ear plugs with her.

She recommends starting your work day concentrating on your most important task for 90 minutes. At that point, your prefrontal cortex probably needs a rest, and you can answer e-mail, and sip caffeine before focusing again. But, until that first break, don’t get distracted by anything else, because it can take the brain 20 minutes to do the equivalent of re-booting after an interruption. “Multitasking is a myth,” Gallagher said.”You cannot do two things at once. The mechanism of attention is selection: it’s either this or that.”

“People don’t understand that attention is a finite resource, like money,” she said. “Do you want to invest your cognitive cash on endless net surfing or couch potatoing? You are constantly making choices, and your choices determine your experience.”

During her cancer treatment several years ago, Gallagher said, she managed to remain relatively cheerful by keeping in mind James’s mantra as well as a line from Milton: “The mind is its own place, and in itself, can make a heav’n of hell, a hell of heav’n.”

“When I woke up in the morning,” Gallagher said, “I’d ask myself: Do you want to lie here paying attention to the very good chance you’ll die and leave your children motherless, or do you want to get up and wash your face and pay attention to your work and your family and your friends? Hell or heaven — it’s your choice.”

A Little Note on Complex Numbers and Geometry for IITJEE Maths

Note:

  1. In acute triangle, orthocentre (H), centroid (G), and circumcentre (O) are collinear and HG:GO = 2:1
  2. Centroid of the triangle formed by points A(z_{1}), B(z_{2}), C(z_{3}) is (z_{1}+z_{2}+z_{3})/3.
  3. If the circumcentre of a triangle formed by z_{1}, z_{2} and z_{3} is origin, then its orthocentre is z_{1}+z_{2}+z_{3} (using 1).

Example 1:

Find the relation if z_{1}, z_{2}, z_{3}, z_{4} are the points of the vertices of a parallelogram taken in order.

Solution:

As the diagonals of a parallelogram bisect each other, the affix of the mid-point of AC is same as the affix of the mid-point of BD. That is,

\frac{z_{1}+z_{3}}{2}=\frac{z_{2}+z_{3}}{2}

or z_{1}+z_{3}=z_{2}+z_{4}

Example 2:

if z_{1}, z_{2}, z_{3} are three non-zero complex numbers such that z_{3}=(1-\lambda)z_{1}+\lambda z_{2} where \lambda \in \Re - \{ 0 \} then prove that the points corresponding to z_{1}, z_{2}, and z_{3} are collinear.

Solution:

z_{3}=(1-\lambda)z_{1}+z_{2} = latex \frac{(1-\lambda)z_{1}+\lambda z_{2}}{1-\lambda +\lambda}$.

Hence, z_{3} divides the line joining A(z_{1}) and B(z_{2}) in the ratio \lambda : (1-\lambda). Thus, the given points are collinear.

Homework:

  1. Let z_{1}, z_{2}, z_{3} be three complex numbers and a, b, c be real numbers not all zero, such that a+b+c=0 and az_{1}+bz_{2}+cz_{3}=0. Show that z_{1}, z_{2}, z_{3} are collinear.
  2. In triangle PQR, P(z_{1}), Q(z_{2}), and R(z_{3}) are inscribed in the circle |z|=5. If H(z^{*}) be the orthocentre of triangle PQR, then find z^{*}.

More later,

Nalin Pithwa

 

TV and video games or mobile games can spoil a kid’s concentration power

Vineeta Pandey

DNA India, Aug 5, 2010.

Study says attention problems may linger till child attains adolescence.

Children may find television viewing and playing video games more fun than playing with other children.

But this temporary, quick-fix solution comes with a whole set of long-lasting problems.

A study published in American Journal of Pediatrics said that viewing television and playing video games can cause serious attention problems among children. What’s worse is that the problems may linger till they attain adolescence and, in some cases, continue even in their youth.

The study says that this can be because most television programmes involve rapid changes in focus and frequent exposure to television has the potential to harm children’s abilities to sustain focus on tasks that are not inherently attention-grabbing.

Also, since most TV shows are exciting, children who frequently watch them have more difficulty paying attention to less exciting tasks like school work. Similar is the case with children playing video games.

Delhi-based psychologist Dr Aruna Broota said, “Frequent television viewing leads to attention problems among children. They lose interest in studies, books and reading newspapers. Even if they read, they tend to lose interest fast and often do not complete the full story or book. This is because on TV events jump from one theme to the next. Children’s emotions get blunted as a result of watching cartoons, etc, which are thrilling and exciting.”

“Similarly, video games that are often seen as gadgets to help gain concentration among children can, in fact, lead to concentration problems if played for more than half-an-hour,” Broota said.

Attention problems, often manifested in the form of attention-deficit/ hyperactivity disorder, are associated with negative outcomes among children and adolescents, which include poor performance in school and increased aggression.

The study says that exposure to television and video games was associated with greater attention problems among late adolescents and young adults. This indicated that a child’s attention span continued to remain affected irrespective of whatever age he or she was addicted to watching TV or playing video games.

Similar studies in TV viewing and playing video games have been linked with problems such as high blood pressure and disturbed sleep among children.

More tips on concentration and studies, later,

Nalin Pithwa

Tagore on Time Management

The butterfly counts not months, but moments, and has time enough. — Rabindranath Tagore, Poet Laureate.

Pursuit of Mathematics and Creativity in Mathematics

Reference: Adventures of a mathematician — Stanislaw Ulam.

While still a schoolboy in Lwów, then a city in Poland, Stanislaw Ulam signed his notebook “S. Ulam, astronomer, physicist and mathematician.”

Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam’s early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed
talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day —- The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam’s highly verbal and collaborative style early on. Ulam’s early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam’s lifelong, highly creative quest for new mathematical and scientific problems.

….

Becoming a mathematician in Poland

When I try to remember how I started to develop my interest in science I have to go back to certain pictures in a popular book on astronomy I had. It was a textbook called Astronomy of Fixed Stars, by Martin Ernst, a professor of astronomy at the University of Lwów. In it was a reproduction of a portrait of Sir Isaac Newton. I was nine or ten at the time, and at that age a child does not react consciously to the beauty of a face. Yet I remember distinctly that I considered this portrait— especially the eyes—as something marvelous. A mixture of physical attraction and a feeling of the mysterious emanated from his face. Later I learned it was the Geoffrey Kneller portrait of Newton as a young man, with hair to his shoulders and an open shirt. Other illustrations I distinctly remember were of the rings of Saturn and of the belts of Jupiter. These gave me a certain feeling of wonder, the flavor of which is hard to describe since it is sometimes associated with nonvisual impressions such as the feeling one gets from an exquisite example of scientific reasoning. But it reappears, from time to time, even in older age, just as a familiar scent will reappear. Occasionally an odor will come back, bringing coincident memories of childhood or youth.

Reading descriptions of astronomical phenomena today brings back to me these visual memories, and they reappear with a nostalgic (not melancholy but rather pleasant) feeling, when new thoughts come about or a new desire for mental work suddenly emerges.

The high point of my interest in astronomy and an unforgettable emotional experience came when my uncle Szymon Ulam gave me a little telescope. It was one of the copper or bronze tube variety and, I believe, a refractor with a twoinch objective. To this day, whenever I see an instrument of this kind in antique shops, nostalgia overcomes me, and after all these decades my thoughts still turn to visions of the celestial wonders and new astronomical problems.
At that time, I was intrigued by things which were not well understood—for example, the question of the shortening of the period of Encke’s comet. It was known that this comet irregularly and mysteriously shortens its threeyear
period of motion around the sun. Nineteenth century astronomers made several attempts to account for
this as being caused by friction or by the presence of some new invisible body in space. It excited me that nobody really knew the answer. I speculated whether the $latex 1/r^{2}$ law of attraction of Newton was not quite exact. I tried to imagine how it could affect the period of the comet if the exponent was slightly different from 2,
imagining what the result would be at various distances. It was an attempt to calculate, not by numbers and symbols, but by almost tactile feelings combined with reasoning, a very curious mental effort.

No star could be large enough for me. Betelgeuse and Antares were believed to be much larger than the sun (even though at the time no precise data were available) and their distances were given, as were parallaxes of many stars. I had memorized the names of constellations and the individual Arabic names of stars and their distances and luminosities. I also knew the double stars.

In addition to the exciting Ernst book another, entitled Planets and the Conditions of Life on Them, was strange. Soon I had some eight or ten astronomy books inmy library, including the marvelous NewcombEngelmann
Astronomie in German. The BodeTitius formula or “law” of planetary distances also fascinated me,
inspiring me to become an astronomer or physicist. This was about the time when, at the age of eleven or so, I inscribed my name in a notebook, “S. Ulam, astronomer, physicist, mathematician. My love for astronomy has never ceased; I believe it is one of the avenues that brought me to mathematics.” 

From today’s perspective Lwów may seem to have been a provincial city, but this is not so. Frequent lectures by scientists were held for the general public, in which such topics as new discoveries in astronomy, the new physics and the theory of relativity were covered. These appealed to lawyers, doctors, businessmen, and other laymen.

I had mathematical curiosity very early. My father had in his library a wonderful series of German paperback books—Reklam, they were called. One was Euler’Algebra. I looked at it when I was perhaps ten or eleven, and it gave me a mysterious feeling. The symbols looked like magic signs; I wondered whether one day I
could understand them. This probably contributed to the development of my mathematical curiosity. I discovered by myself how to solve quadratic equations. I remember that I did this by an incredible concentration and almost painful and not quiteconscious effort. What I did amounted to completing the square in my head without paper or pencil.

In high school, I was stimulated by the notion of the problem of the existence of odd perfect numbers. An integer is perfect if it is equal to the sum of all its divisors including one but not itself. For instance: 6 = 1 + 2 + 3 is perfect. So is 28 = 1 + 2 + 4 + 7 + 14. You may ask: does there exist a perfect number that is odd? The answer is unknown to this day.

In general, the mathematics classes did not satisfy me. They were dry, and I did not like to have to memorize certain formal procedures. I preferred reading on my
own.

At about fifteen I came upon a treatise on the infinitesimal calculus in a book by Gerhardt Kowalevski. I did not have enough preparation in analytic geometry or even in trigonometry, but the idea of limits, the definitions of real numbers, the notion of derivatives and integration puzzled and excited me greatly. I decided to read a page
or two a day and attempt to learn the necessary facts about trigonometry and analytic geometry from other books.
I found two other books in a secondhand bookstore. These intrigued and fascinated me more than anything else for many years to come: Sierpinski’s Theory of Sets and a monograph on number theory. At the age of seventeen I knew as much or more elementary number theory than I do now.

I also read a book by the mathematician Hugo Steinhaus entitled What Is and What Is Not Mathematics and in Polish translation Poincaré’s wonderful La Science et l’Hypothèse, La Science et la Méthode, La Valeur de la Science, and his Dernières Pensées. Their literary quality, not to mention the science, was admirable. Poincaré molded portions of my scientific thinking. Reading one of’ his books today demonstrates how many wonderful truths have remained, although everything in mathematics has changed almost beyond recognition and in physics perhaps even more so. I admired Steinhaus’s book almost as much, for it gave many examples of actual mathematical problems.

The mathematics taught in school was limited to algebra, trigonometry, and the very beginning of analytic geometry. In the seventh and eighth classes, where the students were sixteen and seventeen, there was a course on elementary logic and a survey of history of philosophy. The teacher, Professor Zawirski, was a real scholar, a lecturer at the University and a very stimulating man. He gave us glimpses of recent developments in advanced modern logic. Having studied Sierpinski’s books on the side, I was able to engage him in discussions of set theory during recess and in his office. I was working on some problems on transfinite numbers and on the problem of the continuum hypothesis.

I also engaged in wild mathematical discussions, formulating vast and new projects, new problems, theories and methods bordering on the fantastic, with a boy named Metzger, some three or four years my senior. He had been directed toward me by friends of’ my father who knew that he too had a great interest in mathematics.

In the fall of 1927 I began attending lectures at the Polytechnic Institute in the Department of General Studies, because the quota of Electrical Engineering already was full. The level of the instruction was obviously higher than that at high school, but having read Poincaré and some special mathematical treatises, I naively expected
every lecture to be a masterpiece of style and exposition. Of course, I was disappointed.

As I knew many of the subjects in mathematics from my studies, I began to attend a second year course as an auditor. It was in set theory and given by a young professor fresh from Warsaw, Kazimir Kuratowski, a student of Sierpinski, Mazurkiewicz, and Janiszewski. He was a freshman professor, so to speak, and I a freshman student. From the very first lecture I was enchanted by the clarity, logic, and polish of his exposition and the material he presented. From the beginning I participated more actively than most of the older students in discussions with Kuratowski, since I knew something of the subject from having read Sierpinski’s book. I think he quickly noticed that I was one of the better students; after class he would give me individual attention. This is how I started on my career as a mathematician, stimulated by Kuratowski.

Soon I could answer some of the more difficult questions in the set theory course, and I began to pose other problems. Right from the start I appreciated Kuratowski’s patience and generosity in spending so much time with a novice. Several times a week I would accompany him to his apartment at lunch time, a walk of about twenty
minutes, during which I asked innumerable mathematical questions. Years later, Kuratowski told me that the questions were sometimes significant, often original, and
interesting to him.
My courses included mathematical analysis, calculus, classical mechanics, descriptive geometry, and physics. Between classes, I would sit in the offices of some of the mathematics instructors. At that time I was perhaps more eager than at any other time in my life to do mathematics to the exclusion of almost any other activity.

It was there that I first met Stanislaw Mazur, who was a young assistant at the University. He came to the Polytechnic Institute to work with Orlics, Nikliborc and Kaczmarz, who were a few years his senior.
In conversations with Mazur I began to learn about problems in analysis. I remember long hours of sitting at a desk and thinking about the questions which he broached to me and discussed with the other mathematicians. Mazur introduced me to advanced ideas of real variable function theory and the new functional analysis.

We discussed some of the more recent problems of Banach, who had developed a new approach to this theory.
Banach himself would appear occasionally, even though his main work was at the University. I met him during this first year, but our acquaintance began in a more meaningful, intimate, and intellectual sense a year or two later.

At the beginning of the second semester of my freshman year, Kuratowski told me about a problem in set theory that involved transformations of sets. It was connected with a well known theorem of Bernstein: if 2A = 2B, then A = B, in the arithmetic sense of infinite cardinals. This was the first problem on which I really spent arduous hours of thinking. I thought about it in a way which now seems mysterious to me, not consciously or explicitly knowing what I was aiming at. So immersed in some aspects was I, that I did not have a conscious overall view. Nevertheless, I managed to show by means of a construction how to solve the problem, devising a method of representing by graphs the decomposition of sets and the corresponding transformations. Unbelievably, at the time I thought I had invented the very idea of graphs.

I wrote my first paper on this in English, which I knew better than German or French. Kuratowski checked it and the short paper appeared in 1928 in Fundamenta Mathematicae, the leading Polish mathematical journal which he edited. This gave me self confidence.

I still was not certain what career or course of work I should pursue. The practical chances of becoming a professor of mathematics in Poland were almost nil—there were few vacancies at the University. My family wanted me to learn a profession, and so I intended to transfer to the Department of Electrical Engineering for my
second year. In this field the chance of making a living seemed much better.

Before the end of the year Kuratowski mentioned in a lecture another problem in set theory. It was on the existence of set functions which are “subtractive” but not completely countably additive. I remember pondering the question for weeks. I can still feel the strain of thinking and the number of attempts I had to make. I gave myself an ultimatum. If I could solve this problem, I would continue as a mathematician. If not, I would change to electrical engineering.
After a few weeks I found a way to achieve a solution. I ran excitedly to Kuratowski and told him about my solution, which involved transfinite induction. Transfinite induction had been used by mathematical workers many times in other connections; however, I believe that the way in which I used it was novel.

I think Kuratowski took pleasure in my success, encouraging me to continue in mathematics. Before the end of my first college year I had written my second paper, which Kuratowski presented to Fundamenta. Now, the die was cast. I began to concentrate on the “impractical” possibilities of an academic career. Most of what people
call decision making occurs for definite reasons. However, I feel that for most of us what is ultimately called a “decision” is a sort of vote taken in the subconscious, in which the majority of the reasons favoring the decision win out.

The mathematics offices of the Polytechnic Institute continued to be my hangout. I spent mornings there, every day of the week, including Saturdays. (Saturdays were not considered to be part of the weekend then; classes were held on Saturday mornings.)

Mazur appeared often, and we started our active collaboration on problems of function spaces. We found a solution to a problem involving infinitely dimensional vector spaces. The theorem we proved—that a transformation preserving distances is linear—is now part of the standard treatment of the geometry of function
spaces. We wrote a paper which was published in the Compte Rendus of the French Academy.

It was Mazur (along with Kuratowski and Banach) who introduced me to certain large phases of mathematical thinking and approaches. From him I learned much about the attitudes and psychology of research. Sometimes we would sit for hours in a coffee house. He would write just one symbol or a line like y = f(x) on a piece of paper, or on the marble table top. We would both stare at it as various thoughts were suggested and discussed. These symbols in front of us were like a crystal ball to help us focus our concentration. Years later in America, my friend Everett and I often had similar sessions, but instead of a coffee house they were held in an office with a blackboard. Mazur’s forte was making what he called “observations and remarks.” These stated—usually in a concise and precise form—some properties of notions. Once made, they were perhaps not so difficult to verify, for sometimes they were peripheral to the usual formulations and had gone unnoticed. They were often decisive in solving problems.

,,,

Banach enjoyed long mathematical discussions with friends and students. I recall a session with Mazur and Banach at the Scottish Café which lasted seventeen hours without interruption except for meals. What impressed me most was the way he could discuss mathematics, reason about mathematics, and find proofs in these conversations. Since many of these discussions took place in neighborhood coffee houses or little inns, some mathematicians also dined there frequently. It seems to me now the food must have been mediocre, but the drinks were plentiful. The tables had white marble tops on which one could write with a pencil, and, more important, from which notes could be easily erased. There would be brief spurts of conversation, a few lines would be written on the table, occasional laughter would come from some of the participants, followed by long periods of silence during which we just drank coffee and stared vacantly at each other. The café clients at neighboring tables must have been puzzled by these strange doings. It is such persistence and habit of concentration which somehow becomes the most important prerequisite for doing genuinely creative mathematical work.

*********************************************************************************************

Regards,

Nalin Pithwa

Mathematics in taste buds !!!

Sweet and sour:

Chemists frequently need to describe how acidic or alkaline a particular solution is: that is, to give a quantitative measure of its concentration of hydrogen ions, H^{+}. The scale generally used is the pH scale; pH means potential for hydrogen.

The pH ranges from 0 to 14; a pH below 7 indicates that the solution is acidic, and a pH greater than 7 indicates that the solution is alkaline. A neutral solution such as pure water has a pH of 7. (Strictly speaking, the temperature must be mentioned when we quote the pH value. For pure water, pH equals 7 only at 25 deg celsius.) Full strength sulphuric acid H_{2}SO_{4} has a pH of 0, as does concentrated nitric acid (HNO_{3}), or concentrated hydrochloric acid (HCl).

The pH scale was invented in 1909, by a Dutch biochemist, Soren Sorenson. The scale is logarithmic in nature; pH is defined by the equation

pH = - \log_{10}{H^{+}}

where H^{+} refers to the concentration of H^{+} ions, expressed in moles per liter. A solution with a pH of 4 contains 10^{-4} moles of H^{+} ions per liter, and so on.

The pH may be  measured by using an electronic pH meter, or by means of special dyes called acid-base indicators. The dyes respond to the acidity level by changing colour in a way that depends on the pH. Another measuring device is pH paper. This has indicators within it, which change colour at different pHs.  When dipped into a solution, the paper colour indicates the approximate pH of the solution.

Many chemical reactions depend on pH, indeed, they may be highly sensitive to pH. Agricultural practices have to take soil pH into consideration, because certain crops grow best in an acidic soil (pH < 7) whereas others grew best in an alkaline soil pH > 7. When the soil is found too acidic, lime may be added to make it more alkaline (“sweeten it” to use the gardener’s terminology). The fallen leaves of certain trees contribute to the soil’s acidity and the consequent lack of growth of plants in the shade of the tree. This happens, for example, with eucalyptus trees.

More later, can u find more such examples and share with us?

Nalin Pithwa

 

Geometry with complex numbers — section formula

It ain’t complex, it’s simple !!

Section Formula:

If P(z) divides the line segment joining A(z_{1}) and B(z_{2}) internally in the ratio m:n, then

z = \frac{mz_{2}+nz_{1}}{m+n}

If the division is external, then z=\frac{mz_{2}-nz_{1}}{m-n}

Proof:

Let z_{1}=x_{1}+iy_{1}, z_{2}=x_{2}+iy_{2}. Then, A \equiv (x_{1},y_{1}) and B \equiv (x_{2},y_{2}).

Let z = x+iy. Then, P \equiv (x,y). We know from co-ordinate geometry,

x = \frac{mx_{2}+nx_{1}}{m+n} and y=\frac{my_{2}+my_{1}}{m+n}

Hence, complex number of P is

z = \frac{mx_{2}+nx_{1}}{m+n}+i\frac{my_{2}+my_{1}}{m+n}

\frac{m(x_{2}+iy_{2})+n(x_{1}+iy_{1})}{m+n}

mz_{2}+nz_{1}

more later,

Nalin Pithwa

 

Square root of a complex number

Let a+ib be a complex number such that \sqrt{a+ib} = x+iy where x and y are real numbers. Then,

\sqrt{a+ib}=x+iy

or (a+ib)=(x+iy)^{2}

or a+ib=(x^{2}-y^{2})+2ixy

On equating real and imaginary parts, we get

x^{2}-y^{2}=a

2xy=b

Now, (x^{2}+y^{2})^{2}=(x^{2}-y^{2})^{2}+4x^{2}y^{2}

or (x^{2}+y^{2})^{2}=a^{2}+b^{2}

or (x^{2}+y^{2})=\sqrt{a^{2}+b^{2}} since x^{2}+y^{2} \geq 0

From the above, we get

x^{2}=(1/2)(\sqrt{a^{2}+b^{2}}+a) and y^{2}=(1/2)(\sqrt{a^{2}+b^{2}}-a)

which in turn implies x= \pm \sqrt {(1/2)(\sqrt{a^{2}+b^{2}}+a)}

and y=\pm \sqrt{(1/2)(\sqrt{a^{2}+b^{2}}-a)}

If b is positive, then by the relation 2xy=b, x and y are of the same sign. Hence,

\sqrt{a+ib}=\pm (\sqrt{(1/2)(\sqrt{a^{2}+b^{2}}+a)}+i\sqrt{(1/2)(\sqrt{a^{2}+b^{2}}-a)})

If b is negative, then by the relation 2xy=b, x and y are of different signs. Hence,

\sqrt{a+ib}=\pm (\sqrt{(1/2)(\sqrt{a^{2}+b^{2}}+a)}-i\sqrt{(1/2)(\sqrt{a^{2}+b^{2}}-a)}).

Note: When you have to actually, find the square root of a particular complex number or even a complex expression, carry out the above steps and don’t just mug up  the formula and try to substitute! There are a thousands of such derivations in math, with fancy formulae, so it is better to gain a deep understanding of the proofs rather than mug up  techniques or tips or tricks !!

Try this homework now:

  1. Find the square root of 1+2i
  2. Find all possible values of \sqrt{i}+\sqrt{-i}
  3. Solve for z: z^{2}-(3-2i)z=(5i-5)

Have fun the complex way 🙂

Nalin Pithwa

 

 

 

 

 

 

 

Inclusion Exclusion Principle theorem and examples

Reference: Combinatorial Techniques by Sharad Sane, Hindustan Book Agency.

Theorem: 

The inclusion-exclusion principle: Let X be a finite set and let and let P_{i}: i = 1, 2, \ldots n  be a set of n properties satisfied by (s0me of) the elements of X. Let A_{i} denote the set of those elements of X that satisfy the property P_{i} . Then, the size of the set \overline{A_{1}} \bigcup \overline{A_{2}} \bigcup \ldots \bigcup \overline{A_{n}} of all those elements that do not satisfy any one of these properties is given by

\overline{A_{1}} \bigcup \overline{A_{2}} \bigcup \ldots \bigcup \overline{A_{n}} = |X| - \sum_{i=1}^{n}|A_{n}|+ \sum_{1 \leq i <j \leq n}|A_{i} \bigcup A_{j}|- \ldots + \{ (-1)^{k} \sum_{1 \leq i_{1} < i_{2}< \ldots < i_{k} \leq n}|A_{i_{1} \bigcup A_{i_{2}}} \ldots \bigcup A_{i_{k}}|\}+ \ldots+ (-1)^{n}|A_{1} \bigcup A_{2} \ldots \bigcup A_{n}|.

Proof:

The proof will show  that every object in the set X is counted the same number of times on both the sides. Suppose x \in X and assume that x is an element of the set on the left hand side of above equation. Then, x has none of the properties P_{i}. We need to show that in this case, x is counted only once on the right hand side. This is obvious since x is not in any of the A_{i} and x \in X. Thus, X is counted only once in the first summand and is not counted in any other summand since x \notin A_{i} for all i. Now let x have k properties say P_{i_{1}}, P_{i_{2}}, \ldots, P_{i_{k}} (and no  others). Then x is counted once in X. In the next sum, x occurs {k \choose 2} times and so on. Thus, on the right hand side, x is counted precisely,

{k \choose 0}-{k \choose 1}+{k \choose 2}+ \ldots + (-1)^{k}{k \choose k}

times. Using the binomial theorem, this sum is (1-1)^{k} which is 0 and hence, x is not counted on the right hand side. This completes the proof. QED.

More later,

Nalin Pithwa

 

 

 

 

 

 

 

 

 

The inclusion-exclusion principle for RMO and IITJEE Maths

Reference: Combinatorial Techniques by Sharad Sane:

The principle and its applications:

The inclusion-exclusion principle, is among the most basic techniques of combinatorics. Suppose we have a set X with subsets A and B. Then, the number of elements that are in A or B (or both), that is, the cardinality of A \cup B is given by |A|+|B|- |A \cap B|. The elements that are in both A and B were counted twice. To get rid of the over-counting, we must subtract. If \overline{A} and \overline{B} denote the complements of A and B respectively, then how many elements does the set \overline{A} \cup \overline{B} have? This number is |X|-|A|-|B|+|A \cap B|. The explanation is as before. From the set of all the elements we get rid of those that are in A or B. In doing so, we subtracted the elements of A \cap B twice. This has to be corrected by adding such elements once. Essentially, this way of over-counting (inclusion) correcting it using under-counting (exclusion) and again correcting (over-correcting) and so on is referred to as the inclusion-exclusion technique. As another example, consider the question of finding how many positive integers up to 100 are not divisible by 2, 3 or 5. We see that there are 50 integers that are multiple of 2, 33 that are multiples of 3 and 20 that are multiples of 5. This certainly amounts to over-counting as there are integers that are divisible by two of the given three numbers 2, 3 and 5. In fact, the number of integers divisible by both 2 and 3 is 16, the number of integers divisible by both 2 and 5, that is divisible by 10 is 10, the number of integers divisible by both 3 and 5 is the number of integers below 100 and divisible by 15 and that number is 6. Finally, the number of integers divisible by all three 2, 3, and 5 is just 3. Hence, the number of integers NOT divisible by any one of 2, 3 or 5 is

100-(50+33+20)+(16+10+6)-3=26

The technique used above is called the sieve method. This technique was known to the Greeks and is in fact, known as the Sieve of Eratosthenes. (Remember: in high school, you have used this method to find primes from 1 to 100).

Also, note that I am reminded of the technique of telescoping series when I think about inclusion-exclusion. There seems to be some similarity in spirit.

More later,

Nalin Pithwa