Showing posts with label Mathematics. Show all posts

September 28, 2008

Alert the Treasury, there's a number they can pick

On a day when my head boggled trying to figure out how much 700 million* is...

..via Neatorama, a blog I used to read often in 2006 or 2007 but have not visited in many months, I learn that a new Mersenne Prime Number has been discovered recently and it is 13 million digits long!

Previous post on Mersenne primes and other geeky math stuff here. :)

* Actually, one of the comments at Neatoram already said what I was going to wisecrack: "Don’t let the Treasury Department see the number, or they’ll want that instead of 700 billion!" :))

September 19, 2008

Of intuition, paradoxes, and incompatible beliefs

I had blogged earlier this week about David Foster Wallace's recent death. Through a post at India Uncut, I learned just now of a compilation of tributes to the author from writers and editors who knew him.

The one that caught my eye (and captured my mind?) is the one by author and Professor of Mathematics at Univ. of Wisconsin, Jordan Ellenberg, which I excerpt here:

Wallace's writing was driven by his struggle with contradictions. He was in love with the technical and analytic; but he saw that the simple dicta of religion and A.A. offered better weapons against drugs, despair, and killing solipsism. He knew it was supposed to be the writer's job to get inside other people's heads; but his chief subject was the predicament of being stuck fast inside one's own. Determined to record and neutralize the mediation of his own preoccupations and prejudices, he knew this determination was itself among those preoccupations, and subject to those prejudices. This is Phil 101 stuff, to be sure; but as any math student knows, the old problems you meet freshman year are some of the deepest you'll ever see. Wallace wrestled with the paradoxes just the way mathematicians do. You believe two things that seem in opposition. And so you go to work—step by step, clearing the brush, cataloging what you find there, separating what you know from what you believe, your intuition sounding at all times the nauseous alarm that somewhere you've made a mistake. And until you find the mistake, there's always a bit of hope—that your intuition is wrong, that your work isn't wasted, that what seems like a paradox really isn't one, that maybe the incompatible beliefs you hold can be satisfied all at once.
Usually it doesn't work out that way.

Indeed!

Apparently, Ellenberg is also an author, having written a novel called The Grasshopper King and a regular column in Slate called "Do The Math", in addition to several articles on mathematical topics for The New York Times, The Washington Post, The Boston Globe, Wired, Seed, and The Believer.

December 17, 2007

Fabulous Fibonacci

I started reading about perfect numbers (see previous post) because I ran into them in yet another book that I started over the weekend - The Fabulous Fibonacci Numbers by Alfred S. Posamentier and Ingmar Lehmann.

Of course most of us have heard about Fibonacci numbers that carry the name of the famous Italian mathematician, Leonardo of Pisa aka Leonardo Fibonacci (c. 1170 – c. 1250), who brought Arabic and Hindu numerals to the western world in his famous Book of Calculation, the Liber Abaci. My first introduction to them was perhaps through having to write my first algorithm and a computer code (in Fortran) for producing these numbers. But beyond that I rarely encountered them in my studies or reading and was astounded (yes..astounded) at some of the fascinating properties these numbers have. Transcribed below are some of the interesting ones I have read about so far.

1. Two consecutive Fibonacci numbers do not have common factors, i.e. they are ‘relatively prime’

2. F_nis composite (not a prime), if n is composite.

So, does that mean, Fn = prime, if n = prime? Not so..though amazing how many times it is true.

F₂ = 1 , not considered prime

F₃ = 2 = prime

F₅ = 5 = prime

F₇ = 13 = prime

F₁₁ = 89 = prime

F₁₃ = 233 = prime

F₁₉ = 4181 = 37 x 113, not prime

F₂₃ = 28657 = prime

…

10.
F₃, F₆, F₉, F₁₂, F₁₅, F₁₈…are all even and divisible by 2 or F₃

F₄, F₈, F₁₂, F₁₆, F₂₀, F₂₄… are all divisible by 3 or F₄

F₅, F₁₀, F₁₅, F₂₀,… are all divisible by 5 or F₅

And so on…

i.e. a Fibonacci number F_nm is divisible by _Fm

or in other words, if p is divisible by q, F_p is divisible by F_q.

15. F_n / F_{n+1 ~} golden ratio, phi = 0.618….a

F_n / F_n+2~ 0.38197….b

F_n / F_n+3à 0.23606 …c

Note: a + b = 1.0 and b + c = a = phi!

More about the golden ratio, phi, later.

16. Natural number n can be written as an ordered sum of ones and twos in F_n+1 ways.

Eg: 5 can be written as 1+1+1+1+1

1+1+1+2

1+1+2+1

1+2+1+1

2+1+1+1

1+2+2

2+1+2

2+2+1 …. In 8 = F₆ ways.

Two items that show an interesting relation between Fibonacci numbers and Pythagoras triples:

17. Take any four consecutive Fibonacci numbers

- Product of middle two #s and double it = x

- Multiply outer two #s = y

- Add squares of inner two #s = z

Then, x, y and z form a Pythagoras triple i.e. x² + y² = z²

18. Babylonians had the following formula for obtaining Pythagoras triples (a, b, c)

a = m² - n²

b = 2mn

c = m² + n²

where m and n are natural numbers.

(a, b, c) are called primitive, if abc are relatively prime i.e. there are no common factors between a, b, c. This occurs when m and n are two relatively prime, with m > n and of different parity i.e. one is odd and the other even.

If m and n take on values of Fibonacci numbers, then C is a Fibonacci number too.

December 16, 2007

Attaining perfection

We knew numbers could be odd.. but who knew numbers can be deficient, weird, amicable, friendly, sociable, solitary, sublime, frugal, and extravagant.

But we all knew it is easier to be semi- or pseudo-perfect than to be perfect though you could aim to be hyper-perfect! However, it is very difficult to be sublime! (There are only two known sublime numbers, 12 and 6086555670238378989670371734243169622657830773351885970528324860512791691264.)

Being friendly or better still an amicable pair seems like fun though. And if you are weird, you are abundant but not semiperfect.

So kids, learn your math! Numbers and mathematics can teach us so much about people and life. :)

--
* You can stop reading now if you are not interested in geeky mathematic details. From wikipedia, I am transcribing here some of interesting information about perfect numbers and their relation to prime numbers - specifically Mersenne primes.

A semiperfect or pseudoperfect number is a number that is equal to the sum of all or some of its proper divisors. The first few semiperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, .. (not all even ... but the smallest odd semiperfect number is 945.)

A perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128.

Greek mathematicians knew only these four perfect numbers. Euclid discovered that the first four perfect numbers are generated by the formula 2ⁿ⁻¹(2ⁿ − 1), where n = 2, 3, 5, 7. Noticing that 2ⁿ − 1 is a prime number in each instance, Euclid proved that the formula 2ⁿ⁻¹(2ⁿ − 1) gives an even perfect number whenever 2ⁿ − 1 is prime. In order for 2ⁿ − 1 to be prime, it is necessary but not sufficient that n should be prime.

The Arabic mathematician Ibn al-Haytham had conjectured (around 1000 AD) that every even perfect number is of the form 2ⁿ⁻¹(2ⁿ − 1) where 2ⁿ − 1 is prime but it was not until the 18th century that Leonhard Euler proved that the formula 2ⁿ⁻¹(2ⁿ − 1) will yield all the even perfect numbers.

Prime numbers of the form 2ⁿ − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. As of September 2007, only 44 Mersenne primes are known, which means there are 44 perfect numbers known, the largest being 2^32,582,656 × (2^32,582,657 − 1) with 19,616,714 digits.

And speaking of prime numbers, of which the Mersennes are just a small subset:

The prime number theorem describes the asymptotic distribution of the prime numbers. It states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N.

The prime counting function is the function pi(x) giving the number of primes less than or equal to a given number x.

The growth rate of the prime-counting function was conjectured in the late 18th century by Gauss and by Legendre and first proved independently by Jacques Hadamard and Charles de la Vallée Poussin in 1896, using properties of the Riemann zeta function introduced by Bernhard Riemann in 1859. Elementary proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős .

If you are learning more about prime numbers, a very fascinating subject, read Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire and peruse through the treasure-chest of information at Prof. Chris Caldwell's Prime Pages.

For more about Riemann's hypothesis, read The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics by Karl Sabbagh. For a rigorous mathematical treatise, if you are so inclined, you can read Riemann's Zeta Function by Harold M. Edwards. Also read about my aborted attempt to read Stalking the Riemann Hypothesis by Dan Rockmore.

May 13, 2007

A literary walk-out

A short review of Stalking the Riemann Hypothesis by Dan Rockmore.

How (and why!) do you write a book about mathematics and numbers without using either equations or numbers? I got lost in the sea of abstract forced analogies and ended up more confused, irritated, and lost than I had when I began reading the book. I am not a mathematician by training but have a science/engineering background. Even if I did not understand all the details, I had hoped the book would at least grip my attention and make me want to learn more.

Attempting to read the book has been a stark contrast (and a frustrating one at that) to the book I just finished reading - QED - The strange theory of light and matter by the great teacher, Richard Feynman. There couldn't be two more contrasting writing styles! One enlightens and sheds light on complex topics in as simple terms as possible...the other obfuscates in verbiage that tries to be too clever for its own good.

Anyways, why spend time reading a book one is not enjoying? So, 80 pages into the book, I decided to give up (the literary equivalent of a walk-out mid-way through a bad movie!) and have decided to instead read John Derbyshire's book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, about which I have read good things.

I may try to blog more later about the Riemann hypothesis and Bernhard Riemann himself and his impact on the mathematical and science in general, vis-a-vis the impact of Riemannian geometry on Einstein's general relativity theories.

Lives cut short

Started reading two books that I got from the library last week....

The Collected Short Stories by Katherine Mansfield (13th printing, 1976, Knopf)
and
Stalking the Riemann Hypothesis by Dan Rockmore (more about this book here.)

Katherine Mansfield (14 October 1888 – 9 January 1923), about whom the famed author, Virginia Woolf is claimed to have said - her writing was 'the only writing I have ever been jealous of.'

and

Berhard Riemann (November 17, 1826 – July 20, 1866), a giant in the field of mathematics, on whose shoulders Einstein's general relativity theory and many of the advances of quantum physics rests.

Coincidentally, the lives of these two prodigious talents were cut short prematurely by the scourge that continues to plague human kind in many areas of the world - tuberculosis.

---
It is worth mentioning, for future reference, that the creative power which bubbles so pleasantly in beginning a new book quiets down after a time, and one goes on more steadily. Doubts creep in. Then one becomes resigned. Determination not to give in, and the sense of an impending shape keep one at it more than anything. - Virginia Woolf

February 15, 2007

Ramanujan Prize

The Ramanujan Prize is for Young Mathematicians from Developing Countries. The Prize, named after the famed Indian genius and mathematical whiz, Srinivasa Ramanujan, is awarded annually to a researcher under 45 years of age and from a developing country and is funded by the Niels Henrik Abel Memorial Fund

The 2006 winner was Ramdorai Sujatha of TIFR, while Marcelo Viana of Brazil won the inaugural award in 2005.

August 22, 2006

Fields Medal

Grigori Perelman, the Russian who in 2002 solved the Poincare Conjecture, one of the hardest problems in mathematics, has declined the top prize in Mathematics, the Fields Medal.

The Fields medal (Mathworld link) is awarded every four years on the occasion of the International Congress of Mathematicians to recognize outstanding mathematical achievement for existing work and for the promise of future achievement. The Field Medals, often said to be the Nobel Prize of Mathematics, were first proposed at the 1924 International Congress of Mathematicians in Toronto, where a resolution was adopted stating that at each subsequent conference, two gold medals should be awarded to recognize outstanding mathematical achievement. Professor John Charles Fields , a Canadian mathematician who was secretary of the 1924 Congress, later donated funds establishing the medals which were named in his honor, with the International Congress of Mathematicians adopting the proposal at the 1932 Zurich Congress and the first Fields Medals were awarded at the 1936 Congress in Oslo to Lars Valerian Ahlfors (Harvard University - for research on covering surfaces related to Riemann surfaces of inverse functions of entire and meromorphic functions. Opened up a new ideas in analysis) and Jesse Douglas (Massachusetts Institute of Technology - for work of the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary. ) Note: List of Fields Medallists since then.

The Fields Medal is made of gold, and shows the head of Archimedes together with a quotation attributed to him: "Transire suum pectus mundoque potiri" ("Rise above oneself and grasp the world"). The reverse side bears the inscription: "Congregati ex toto orbe mathematici ob scripta insignia tribuere" ("the mathematicians assembled here from all over the world pay tribute for outstanding work"). - adapted from the Mathworld link mentioned above.

Other References (via Wolfram Research)

Albers, D. J.; Alexanderson, G. L.; and Reid, C. International Mathematical Congresses, An Illustrated History 1893-1986, rev. ed., incl. 1986. New York: Springer Verlag, 1987.

Fields Institute. "The Fields Medal." http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html.

International Mathematical Union. "Fields Medals and Rolf Nevanlinna Prize." http://elib.zib.de/IMU/medals/.

Joyce, D. "History of Mathematics: Fields Medals." http://aleph0.clarku.edu/~djoyce/mathhist/fieldsmedal.html.

Lopez-Ortiz, A. "Fields Medal: Historical Introduction." http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/node19.html.

Lopez-Ortiz, A. "Why Is There No Nobel in Mathematics?"

MacTutor History of Mathematics Archives - The Fields Medal.

Monastyrsky, M. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: A. K. Peters, 1997.

Technische Universität Berlin. "The Four Fields Medallists and the Nevanlinna Prize Winner of The International Congress of Mathematicians, Berlin 1998." http://www.tu-berlin.de/presse/pi/1998/pi182e.htm.

Other Links to Perleman (from the wiki entry)

Perelman's eprints on the arXiv
Grigori Perelman at the Mathematics Genealogy Project
Staff listing for Perelman at Petersburg Department of Steklov Institute of Mathematics
Mathematics & Mechanics Faculty of St. Petersburg State University
Petersburg Department of Steklov Institute of Mathematics
Notes and commentary on Perelman's Ricci flow papers
International Mathematical Olympiad 1982 (Budapest, Hungary) Individual Scores
Maths genius living in poverty

June 27, 2006

Coincidences, Chaos, and All That Math Jazz - Book Review

I am 70% of my way through Coincidences, Chaos, and All That Math Jazz by Edward B. Burger & Michael Starbird but have to return it to the public library soon and so I figured I might as well compile and blog about what I have read so far... since I did glean some more interesting tidbits about numbers, learned about a poll debacle during the 1936 Presidential election due to survey of an un-representative sample of the population, and so on..

Here is a "review"...subject to change and modification as and when I read the un-read chapters and finish the book...

Overall a very good read. Here is a quick review of the book, divided into 4 parts, with about 2-3 chapters in each part.

First a preface:

“When most of us think of math, we first think of numbers. And when we think of numbers, we think of counting. While at first blush we may not view the act of enumeration as profound, it is certainly something we may count on. (Sorry.)”…(Chapter 5, p79).

The book is full of such punning and corny jokes but while they are cause for a pause, they are a welcome distraction for someone like me reading the book in a very technical mood and are probably a necessary diversion to lighten the load for people who are of a mathematical bent of mind - we are after all talking about math and numbers, a much dreaded subject for many in their high school days – although I doubt that despite the mighty goals of such books to reach out to people without a mathematical propensity, do such people really read these books? That said, while the book is definitely void of any mighty equations that may scare some people away, it is not dumbed down and did make for an enjoyable read.)

Part I - Undersanding Uncertainty – Coincidences, Chaos, and Confusion

Some of the initial stuff regarding the role uncertainty, luck, and sheer statistics plays in explaining “coincidences” was elementary, in the sense that it was too obvious to someone with my background and interests. Although I personally would have liked the first two chapters to be shorter, it is likely that for a non-science non-mathematical person, these chapters on unbridled coincidences we encounter in our lives and the origin of chaos and its role in preventing us from knowing the future were perhaps necessary introductions that helped set the mood for what is to follow. The third chapter, titled ‘Digesting Life’s Data – Statistical Surprises’, was a interesting read even for someone like me who has a decent background in statistics… enlightening me about the bias in polling that led to a historic doomed poll conducted by the Literary digest for the 1936 US Presidency poll and also does a good job explaining basic statistical concepts for readers without a statistics background through interesting examples like SAT scores, HIV-AIDS testing, Air vs. road safety statistics - pointing out how statistics an help us understand the world by highlighting random or unknown features of the data but at the same time can also be used to manipulate data and lie with the right (or wrong) interpretation and presentation. (Related aside: Also see my blogpost about nonsensical surveys.)

Part II – Embracing Figures: Sensing Secrecy, Magnificient Magnitudes, and Nature’s Numbers

Skipped Chapter 4 on Cryptography for later reading and also skipped over Chapter 6 on the Synergy between Nature and Numbers (skipped the latter after giving it a quick scan as I plan to read The Constants of Nature : From Alpha to Omega--the Numbers That Encode the Deepest Secrets of the Universe by John Barrow & Just Six Numbers: The Deep Forces That Shape the Universe by Martin J. Ree, which should give more detailson these synergies.)

Chapter 5 titled, ‘Sizing up numbers’ was a great read and succeeds in its goal to put a ‘face to a number’, using common examples that arise in daily life that help us get a more intuitive feel for what thousands, millions, billions, and trillions actually mean….with some funny and interesting tidbits and examples like “The number of hours a student spends in class during a college education is one or two thousand; that number is also approximately the number of hours we sleep in a year. Coincidence? We think not…”; that it is possible for all 6.4 billion people in the world to fit into one cubic mile; and asking interesting questions like “How much will a million dollars weigh” (answer is estimated to be about 1600lbs); and if Bill Gates would be better off going off the clock to pick up a hundred dollar bill lying on the floor, if we assume his annual pay is the twenty billion dollar personal wealth increase a few years ago (the answer is NO.. at 20billion a year, he earns 2800$ a second and so earns 100$ every 1/28^th of a second…so, “he should not only not clock out to pick up the 100$ bill, he shouldn’t even stop to look at it”.) But the best and most interesting example to me was the one used for explaining what a quadrillion is… where the authors use an example of folding a paper repeatedly into halves… showing that after 10 folds, we have a thousand layered stack which is 4 inches thick, after 20 folds, it is one million layers and 350 feet thick, …after 25, it is 2 miles thick, after 30, it is 1 billion layers thick and 64 miles thick…and by 50 folds it has reached 1 quadrillion layers and goes for 64 million miles…with the thickness going beyond 128 million miles (well past the sun) at the 51^st fold. Ofcourse, in practice, we cannot go beyond 7 folds with any piece of paper.. (try it!)… but this example boggles the mind as even I didn’t anticipate or think of how quickly explosive repeated doubling gets. The next example in the chapter is also an interesting one on arranging a deck of card on top of each other without any glue but such that they do not topple over…and apparently (see page 90-91) it is possible to arrange a deck of 52 cards such that the top card is more than a mile beyond the end of the table so that we can actually sit on that top card without collapsing the leaning pile. Again, this may be impossible to do but the authors provide a sound mathematical and physical argument for why this and other interesting card arrangement tricks should be possible.

Part III - Exploring Aesthetics: Sexy Rectangles, Fiery Fractals, and Contortions of Space

Even though I have read quite a bit of technical literature in the area of chaos (by no means am an expert or even claim I understand chaos theory though), the chapters that lead into the discussions on chaos made for new and exciting reading. New & exciting because, based on my previous readings, I already understood how chaos arises mathematically from Bifurcation Theory but to see the patterns and ordered chaos of something like paper-folding was an amazing revelation. Chapter 7 gives us a good introduction to the aesthetics of the golden ratio and its role in art & nature and the “beauty” in golden rectangles whose base to height is the Golden Ratio, 1.618 and a delightful method to construct one with a straightedge which is unmarked and a compass. However, there are other detailed expositions of the subject (eg: The Golden Ratio : The Story of PHI, the World's Most Astonishing Number by Mario Livio) that are on my to-read list (if I ever get to them) and probably will make for even better interesting reads on the subject.

However, what I quite enjoyed was the transition made in copying over the golden triangles repeatedly, leading us from the precise beauty of the triangle to reproductive chaos. And even though I didn’t quite understand this section perfectly, it left me wanting more… the last sentence of the chapter summarizes it well.. “We now see that aesthetics and mathematics are deeply related. There’s beauty in mathematics and mathematics in beauty”. Well said.

And more delightful discussions of organized chaos did come in Chapter 8, which challenges us to go back to the ideas of paper folding and leads us through a series of very interesting pattern recognition exercises involving the valley and ridges in the folds of the paper. The authors show that results are quite mind-boggling – flitting between “sheer chaos” and “complete regularity” and in fact go on to show that the paper-folding sequence for arbitrarily many folds is actually the output of an extremely simple five-line Turing machine program!! (Some of this may be elementary for someone familiar with Turing machines and computer science basics but again… I am not sure I followed this section perfectly well ..not that I couldn’t but I gave it a quick read and did not bother to wait and study the details. The same can be said of the next section on folded swans and leading up to the Dragon curve and the common fractal structure of the self-similar Sierpinski gasket (which you can create in Excel, btw and then measure the resistance in!)

Still to read Chapter 9.. which takes us on an exploration of an “elasticized world” followed by Part IV – Transcending Reality: The Fourth Dimension & Infinity.

April 30, 2006

The Joy of Numbers

This blog post was started on reading the books, Imagining Numbers by Barry Mazur and The Joy of Pi by David Blatner. I hope to add to it by including books, interesting snippets, and sometimes my own comments as I read and research the joy of numbers and mathematics in this thread.

HISTORY OF ALGEBRA & NUMBERS

Etymology of Algebra: Via Italian, Spanish or mediaeval Latin, from Arabic 'al-jabr' or the ‘reunion, resetting of broken parts’, used in the title of al-Khwarizmi’s influential work, ‘ilm al-jabr wa’l-muqābala, or 'the science of restoration and equating like with like’.

Pre-Modern Algebra (pdf)

Bertrand Russell's Definition of a Number

BOOKS

Some books shortlisted to read
Number by Tobias Dantzig
The Constants of Nature : From Alpha to Omega--the Numbers That Encode the Deepest Secrets of the Universe by John Barrow

A related book is Just Six Numbers: The Deep Forces That Shape the Universe by Martin J. Rees, which discusses the recurring six numbers that come up in explaining the universe around us – which are:

nu (a ratio of the strength of electrical forces that hold atoms together compared to the force of gravity which is 10 to the 37th power)
epsilon (how firmly the atomic nuclei bind together which is 0.004)
omega (amount of material in the universe)
lambda (force of cosmic "antigravity" discovered in 1998, which is a very small number)
Q (ratio of two fundamental energies, which is 1/100,000)
delta (number of spatial dimensions in our universe)

Other related books of interest on numbers:

1. A History of Pi by Petr Beckmann

2. Pi: A Biography of the World's Most Mysterious Number by Alfred Posamentier & Ingmar Lehmann

3. Squaring the Circle by Tom Stoppard

4. The Golden Ratio : The Story of PHI, the World's Most Astonishing Number by Mario Livio

5. Zero: The Biography of a Dangerous Idea by Charles Seife

6. To Infinity and Beyond by Eli Maor

7. The Infinite Book by John D. Barrow

8. An Imaginary Tale Hardcover by Paul J. Nahin

9. Fortune's Formula by William Poundstone

10. e by Eli Maor

11. The Heart of Mathematics : An invitation to effective thinking by Edward Burger

12. Coincidences, Chaos, and All That Math Jazz by Edward B. Burger & Michael Starbird

13. Beyond Coincidence : Amazing Stories of Coincidence and the Mystery and Mathematics Behind Them by Martin Plimmer

14. Trigonometric Delights by Eli Maor

15. Gamma : Exploring Euler's Constant by Julian Havil

16. Prime Obsession : Berhhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire

17. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics by Karl Sabbagh (On the subject, for a mathematical treatise, read: Riemann's Zeta Function by Harold M. Edwards)

18. The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry by Mario Livio

19. Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem by Simon Singh

20. The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time by Keith J. Devlin

21. The Equation That Couldn't Be Solved by Mario Livio

22. The Mathematical Universe : An Alphabetical Journey Through the Great Proofs, Problems, and Personalities by William Dunham

23. God Created the Integers: The Mathematical Breakthroughs That Changed History by Stephen W. Hawking

24. The Triumph of Numbers: How Counting Shaped Modern Life by I. B. Cohen

25. Wonders of Numbers: Adventures in Math, Mind, and Meaning by Clifford A. Pickover

26. Meta Math! : The Quest for Omega by Gregory Chaitin, renowned for finding another proof of Kurt Godel's incompleteness theorem and another for Alan Turing's "halting problem" in computation and discoverer of the Omega number, which is an exquisitely complex representation of unknowability in mathematics.

Also see my compilation of links on Mathematics.

Indian Mathematicians

Famous Indian Mathematicians.. - not talking about ancient Indian math stalwarts from centuries ago like Brahamagupta, Bhaskara or Aryabhatta nor THE famous Indian math wizard of the last couple centuries - Srinivasa Ramanujan ..

But want to bring to your attention two mathematicians making the headlines/news in the last 5-6 years..

1. Chandrakant Khare - Read my blog post about 'Chandrakant Khare and Fermat's Last Theorem' for further details..

2. Also another mathematician making the news recently.. - Divakar Viswanath of Univ. of Michigan-Ann Arbor (PhD at Cornell). In 1999, he wrote a paper about "Divakar's constant" which is the 'limit of the ratio of consecutive random fibonacci numbers'. Lot of discussion around this since with some people calling this new constant almost as important as the "golden ratio" (I need to find a reference/citation for this bold statement - for someone calling the constant as important as the much vaunted golden ratio - it was quoted to me by a friend).

Chandrakant Khare and Fermat's Last Theorem

I had seen Simon Singh on Book-TV some years back talk about this book, Fermat's Enigma regarding the quest to solve Fermat’s Last Theorem (FMT) but though its been at the back of my mind, I never read up more about it till yesterday when I found a great website, Larry Freeman‘s blog on the subject.

The blog is a great site for amateur readers like me to read about ‘the story behind FMT and Wiles' proof in a way accessible to the mathematical amateur.’ You can also read about the intrigue and excitement that caught even the fancy of the media when Andrew Wiles, with the help of Richard Taylor, proposed that he had a solution to FMT in 1995. (If you insist on reading it, here are the papers, all 129 pages of it – most mathematicians also do not follow it – so, do not say you were not forewarned!)

In 2006, another mathematician, Chandrashekhar Khare (previously at TIFR and now Associate Prof at University of Utah), has provided a very significant result (here is the paper) that builds on the work done by Wiles.

The Slashtdot entry on this says:"An Indian mathematician, Chandrashekhar Khare, is poised to make a significant breakthrough in the field of number theory with his solution of part of a major outstanding problem in algebraic number theory. He is currently an associate professor in Mathematics Department of University of Utah. "

Actually, Khare does not provide another proof of the FMT but proved what is known to experts as the ‘level-one Serre conjecture’. This conjecture was posed in 1972 by the Fields medallist Jean-Pierre Serre, and belongs to the field of Arithmetic Algebraic Geometry.

From: http://plus.maths.org/latestnews/jan-apr05/serre/

Chandrashekhar Khare, a mathematician from the University of Utah, has announced that he has Serre's conjecture is in a sense a parent of Fermat's last theorem: mathematicians have known for some time that if the first is true then so is the second. In fact, it is a certain part of the conjecture which implies Fermat's last theorem, and this part was proved by Khare and his collaborator J.P. Wintenberger, and independently by the mathematician Dieulefait.

Fermat's last theorem as well as the conjecture by Serre, are ingredients of a wider program to unify various areas of mathematics, known as Langlands philosophy, (conceived by the mathematician Robert Langland in the 1960's and consists of a set of conjectures concerning the intimate relationship between number theory, geometry and algebra). The idea behind such unifying theories is that it should be possible to directly translate every concept in a given area of maths into all the other areas of maths……and the relationship between the objects should be the same in both areas.

See more on other recent work on Serre’s conjectures

Tasty Morsels