Monthly Archives: October 2015

Fun with Chinese Remainder Theorem

A farmer is on the way to market to sell eggs when a meteorite hits his truck and destroys all of his produce. In order to file an insurance claim, he needs to know how many eggs were broken. He knows that when he counted the eggs by 2’s, there was 1 left over, when he counted the eggs by 3’s, there was 1 left over, when he counted the eggs by 4’s, there was 1 left over,

when he counted the eggs by 5’s, there was 1 left over, and when he counted them by 6’s, there was one left over, but when he counted them by 7’s, there was none left over. What is the smallest number of eggs that were in the truck?

More fun later…

Nalin Pithwa

Chinese Remainder Theorem — some history — RMO

The first recorded instance of the Chinese Remainder Theorem appears in a Chinese mathematical work from the late third or early fourth century. Somewhat surprisingly, it deals with the harder problem of three simultaneous congruences:

We have a number of things, but we do not know how  many. If we count them by threes, we have two left over. If we count them by fives, we have three left over. If we count them by sevens, we have two left over. How many things are there?

Sun Tzu Suan Ching (Master Sun’s Mathematical Manual)

Circa AD 300, volume 3, problem 26.

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Try to solve it and let me know your comments !

Nalin Pithwa

Aryabhata

Aryabhata was one of the greatest mathematicians and astronomers of all times, and certainly the first of the great mathematicians and astronomers responsible for the renaissance of mathematics and science in ancient India.

It is true that Indian mathematics and astronomy had great achievements to their credit even before Aryabhata’s time, but contributions in these fields were usually made by groups of saint-scholars, who combined in their writings mathematics, astronomy, astrology, religious rituals, science and spiritual knowledge, in a more or less integrated manner.

Aryabhata, for the first time, had the courage to break with this tradition. He studied thoroughly and completely all the mathematics and astronomy known in his time, systematized it, found the gaps in the knowledge of these disciplines and filled these gaps with his  own researches. He had the courage to reject the accepted values of astronomical parameters, if these were inconsistent with his and others’ observations. In this sense, he was a true scientist uninhibited by any consideration, other than that of search for truth.

He was dedicated to Lord Brahma, the creator of this universe, and he pursued mathematics and astronomy as he believed that researches in these disciplines would enable him to understand fully the universe created by his Lord. Instead of just worshipping Nature, he wanted to understand it and in this approach, he was very near in spirit to Kepler, Newton and later mathematicians and astronomers.

Aryabhata ushered in a revolution in Indian mathematics, astronomy and science, the effects of which were felt and continued to be for eight centuries and even after that.

Had political condition been more stable, Indian scientists would probably have continued to follow the path shown by Aryabhata, and India would have continued as a leader in the world of science and technology during the later centuries as well.

A study of his life and works inspires young Indian students, raises their scientific morale and gives them faith in themselves and in their ancient heritage. Aryabhata had a great scientific attitude of creativity, objectivity,, a relentless and single minded pursuit of truth, and he displayed courage in rejecting all false statements and from his desire to understand Nature. I hope that the great scientific values cherished by Aryabhata would continue to inspire all Indians.

More later,

Nalin Pithwa

 

 

Complex attitude!

Let us a discuss yet one more complex number based IITJEE mains problem.

Problem:

Let z_{1} and z_{2} be nth roots of unity, which subtend a right angle at the origin. Then, integer n must be of the form __________________. (fill in the blank).

Solution:

nth roots of unity are given by

\cos{\frac{2m\pi}{n}}+ i \sin{\frac{2m\pi}{n}}= e^{\frac{2m\pi}{n}}

where m=0,1,2, \ldots, n-1.

Let z_{1}=\cos{\frac{2m_{1}\pi}{n}}+ i\sin{\frac{2m_{1}\pi}{n}}=e^{2m_{1}\pi i/n}

Let z_{2}=e^{2m_{2}\pi i/n} where 0 \leq m_{1}, m_{2}< n, m_{1} \neq m_{2}

As the join of z_{1} and z_{2} subtends a right angle at the origin, we deduce that \frac{z_{1}}{z_{2}} is purely imaginary.

\Longrightarrow \frac{e^{2m_{1}\pi i/n}}{e^{2m_{2}\pi i/n}}=ik, for some real k

\Longrightarrow e^{2(m_{1}-m_{2})\pi i/n}=ik

\Longrightarrow n=4(m_{1}-m_{2}). Thus, n must be of the form 4k.

More later,

Nalin Pithwa

 

Compensating Errors

The class had been given a sum to do, involving three positive whole numbers (‘positive’ here means ‘greater than zero). During the break, two classmates compared notes.

“Oops, I added the three numbers instead of multiplying them,” said George.

“You are lucky, then,” said Henrietta. “It’s the same answer either way.”

What were the three numbers? What would they have been if there had been only two of them, or four of them, again with their sum equal to their product?”

More later,

Nalin Pithwa

The importance of Recreational Mathematics

The following article is from today’s newspaper, The New York Times (by Manil Suri, Prof of Mathematics, University of Maryland, Baltimore County):

Baltimore — IN 1975, a San Diego woman named Marjorie Rice read in her son’s Scientific American magazine that there were only eight known pentagonal shapes that could entirely tile, or tessellate, a plane. Despite having had no math beyond high school, she resolved to find another. By 1977, she’d discovered not just one but four new tessellations — a result noteworthy enough to be published the following year in a mathematics journal.

The article that turned Ms. Rice into an amateur researcher was by the legendary polymath Martin Gardner. His “Mathematical Games” series, which ran in Scientific American for more than 25 years, introduced millions worldwide to the joys of recreational mathematics. I read him in Mumbai as an undergraduate, and even dug up his original 1956 column on “hexaflexagons” (folded paper hexagons that can be flexed to reveal different flowerlike faces) to construct some myself.

Recreational math” might sound like an oxymoron to some, but the term can broadly include such immensely popular puzzles as Sudoku and KenKen, in addition to various games and brain teasers. The qualifying characteristics are that no advanced mathematical knowledge like calculus be required, and the activity engage enough of the same logical and deductive skills used in mathematics.

Unlike Sudoku, which always has the same format and gets easier with practice, the disparate puzzles that Mr. Gardner favored required different, inventive techniques to crack. The solution in such puzzles usually pops up in its entirety, through a flash of insight, rather than emerging steadily via step-by-step deduction as in Sudoku. An example: How can you identify a single counterfeit penny, slightly lighter than the rest, from a group of nine, in only two weighings?

Mr. Gardner’s great genius lay in using such basic puzzles to lure readers into extensions requiring pattern recognition and generalization, where they were doing real math. For instance, once you solve the nine coin puzzle above, you should be able to figure it out for 27 coins, or 81, or any power of three, in fact. This is how math works, how recreational questions can quickly lead to research problems and striking, unexpected discoveries.

A famous illustration of this was a riddle posed by the citizens of Konigsberg, Germany, on whether there was a loop through their town traversing each of its seven bridges only once. In solving the problem, the mathematician Leonhard Euler abstracted the city map by representing each land mass by a node and each bridge by a line segment. Not only did his method generalize to any number of bridges, but it also laid the foundation for graph theory, a subject essential to web searches and other applications.

With the diversity of entertainment choices available nowadays, Mr. Gardner’s name may no longer ring a bell. The few students in my current batch who say they still do mathematical puzzles seem partial to a website called Project Euler, whose computational problems require not just mathematical insight but also programming skill.

This reflects a sea change in mathematics itself, where computationally intense fields have been gaining increasing prominence in the past few decades. Also, Sudoku-type puzzles, so addictive and easily generated by computers, have squeezed out one-of-a-kind “insight” puzzles, which are much harder to design — and solve. Yet Mr. Gardner’s work lives on, through websites that render it in the visual and animated forms favored by today’s audiences, through a constellation of his books that continue to sell, and through biannual “Gathering 4 Gardner” recreational math conferences.

In his final article for Scientific American, in 1998, Mr. Gardner lamented the “glacial” progress resulting from his efforts to have recreational math introduced into school curriculums “as a way to interest young students in the wonders of mathematics.” Indeed, a paper this year in the Journal of Humanistic Mathematics points out that recreational math can be used to awaken mathematics-related “joy,” “satisfaction,” “excitement” and “curiosity” in students, which the educational policies of several countries (including China, India, Finland, Sweden, England, Singapore and Japan) call for in writing. In contrast, the Common Core in the United States does not explicitly mention this emotional side of the subject, regarding mathematics only as a tool.

Of course, the Common Core lists only academic standards, and leaves the curriculum to individual districts — some of which are indeed incorporating recreational mathematics. For instance, math lesson plans in Baltimore County public schools now usually begin with computer-accessible game and puzzle suggestions that teachers can choose to adopt, to motivate their classes.

The body of recreational mathematics that Mr. Gardner tended to and augmented is a valuable resource for mankind. He would have wanted no greater tribute, surely, than to have it keep nourishing future generations.

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More later,

Nalin Pithwa

Graphs of trig raised to trig

Question: Consider the function

y=f(x)=x^{x}. Can you graph it? It is variable raised to variable. Send me your observations.

Now, consider the functions:

(\tan \theta)^{\tan \theta}, (\tan \theta)^{\cot \theta},

(\cot \theta)^{\tan {\theta}}, (\cot \theta)^{\cot \theta}.

Can you graph these? What is the difference between these and the earlier generalized case?

Now, consider the function:

Let 0 \deg < \theta < 45 \deg.

Arrange t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}

t_{3}=(\cot \theta)^{\tan \theta} and t_{4}=(\cot \theta)^{\cot \theta}

in decreasing order.

Kindly send your comments/observations.

More later,

Nalin Pithwa

Complex ain’t so complex ! Learning to think!

Problem:

If 1, \omega, \omega^{2}, \omega^{3}, \ldots, \omega^{n} are the nth roots of unity, then find the value of (2-\omega)(2-\omega^{2})(2-\omega^{3})\ldots (2-\omega^{n-1}).

Solution:

Learning to think:

Compare it with what we know from our higher algebra — suppose we have to multiply out:

(x+a)(x+b)(x+c)(x+d). We know it is equal to the following:

x^{4}+(a+b+c+d)x^{3}+(ab+ac+ad+bc+bd+cd)x^{2}+(abc+acd+bcd+abd)x+abcd

If we examine the way in which the partial products are formed, we see that

(1) the term x^{4} is formed by taking the letter x out of each of the factors.

(2) the terms involving x^{3} are formed by taking the letter x out of any three factors, in every possible way, and one of the letters a, b, c, d out of the remaining factor

(3) the terms involving x^{2} are formed by taking the letter x out of any two factors, in every possible way, and two of the letters a, b, c, d out of the remaining factors

(4) the terms involving x are formed by taking the letter x out of any one factor, and three out of the letters a, b, c, d out of the remaining factors.

(5) the term independent of x is the product of all the letters a, b, c, d.

Further hint:

relate the above to sum of binomial coefficients.

and, you are almost done.

More later,

Nalin Pithwa

Percentage play

Alphonse bought two bicycles. He sold one to Bettany for 300 pounds making a loss of 25%, and one to Gemma also for 300 pounds making a profit of 25%. Overall, did he break even? If not, did he make a profit or loss, and by how much?

More later,

Nalin Pithwa

If you thought mathematicians were good arithmetic…

Ernst Kummer was a German algebraist, who did some of the best work on Fermat’s Last Theorem before the modern era. However, he was poor at arithmetic, so he always asked his students to do the calculations for him. On one occasion, he needed to work out 9 \times 7. “Umm…nine times seven is…nine times seven is…”

“Sixty one” suggested one student. Kummer wrote this on the blackboard.

“No, Professor, it should be sixty-three” suggested another student.

“Come, come, gentleman,” said Kummer, “it can’t be both. It must be one or the other”.

More later,

Nalin Pithwa