## Monday, December 22, 2014

### How to prove Ramanujan's q-Continued Fractions

 The photograph of a page in Ramanujan's Lost Notebook where he expanded a ratio of two series in terms of three continued fractions. These three are among the continued fractions explained in this paper.

Its the 125th year of Ramanujan's birth, but how many of his formulas do you know? Here is an opportunity to get familiar with 9 of Ramanujan's continued fraction formulas. These include the three continued fractions that appear in the Lost Notebook in the above photograph.

Abstract:
By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.

A Preprint of this expository paper is now available from arXiv. The latest version fixes a typo. The final version appears in this book. You may wish to buy/access the entire volume from the AMS, its really an amazing piece of work.

Update (Sept 7, 2018): I presented this topic in IISER, Mohali, after adding a few ideas from the recent joint work with Mourad Ismail. Here is the presentation.

Update (December 20, 2014): Published in Contemporary Mathematics: Ramanujan 125, K. Alladi, F. Garvan, A. J. Yee (eds.)  627, 49-68 (2014)

Update (July 17, 2013): Accepted for publication in the proceedings of Ramanujan 125, where I presented this paper.

## Monday, March 31, 2014

### How to discover 22/7 and other rational approximations to $\pi$

A short article written for a magazine that caters to high-school students. The basic idea is to use continued fractions found using Euclid's algorithm, and then to chop off the continued fraction to get rational approximations. Written at a high-school level. Some of the material was already present in my book Maths Concepts.

## Thursday, February 06, 2014

### Interview in Annulus - Hindu College math department magazine

This is an email interview with Annulus, a magazine taken out by Acuity, the mathematics society run by mathematics students of Hindu College, Delhi University. Most of it was published in the magazine. I thought it was meant as a tribute to Saroj Bala Malik, who taught me 4 classes when I attended Hindu, from 1984-87, but they edited out the question about her in the final article.

I wanted to become a mathematician. I joined Hindu college because it was the best college where I managed to get admission to study Math. It was the best! And I had the most wonderful time.

Hindu was a most liberal place, where a lot of leeway and freedom was given to students to figure out their own approach to life. Teachers did not impose upon us. The students came from all kinds of economic and social backgrounds, which was great for me, because I had gone to a somewhat elitist school. The cafeteria those days was really a park in front of the hostel, with a few chairs, but plenty of sunshine. A lot of people (even from other colleges) just hung around. A lot of time was spent in ‘Café Hons’.

The key highlights was the annual ‘trip’ which was 4-5 days of concentrated fun, followed by discussions of what all happened there for the rest of the year. Plus, of course, Mecca. One year in Mecca, we took out a daily magazine called ‘The Quark that Quakes’, consisting of mathematical puzzles, limericks and bad jokes. It was a big hit.

What career option were you looking for when you decided to take up Mathematics?

I wanted to be a research mathematician. I knew that all that I had studied in school was essentially stuff known to Newton and Archimedes many centuries ago. I wanted to reach the frontiers, whatever that meant. I had no idea what it takes to discover and prove your own theorem. I just thought it would be cool to have one I can call my own!

Questions on whether I could become rich, or even survive financially, didn’t really enter my head. Perhaps the practice—prevalent in Hindu—of treating our friends’ money and possessions as our own, contributed to this attitude.

How hard was it to make it to IIT Delhi back then? Any tips that you would like to share with the students?

I don’t recall having studied at all for the IIT entrance. The exam was so tricky that it was fair game for anyone. About 250 students took the test, and 20 were selected. My rank was 2, so I suppose I did quite well! The test required understanding the basic ideas/definitions rather than extensive knowledge of the subject. In fact, I recall that one of the questions was to state and prove your favorite theorem, so they were looking to see if you liked math and what you liked in it.

An idea that works for me is to find one book that gives a historical overview of a particular subject. After going through it in a week or two, I am able to understand what’s happening for the entire semester. There are books like this for algebra, analysis, complex analysis, number theory—you just have to find one that you like.

If you do this, then you can begin to appreciate the beauty of the subject, and are able to understand why you are doing what you are doing. The subject becomes easy, and you will be able to answer the kind of questions that examiners are looking for.  You will also be able to slog through the difficult theorems and proofs, because you have a sense of where you are going.

Any Dr. Saroj Bala Malik memories?

SBM has been one of the most influential teachers in my life. In our first class, she asked questions and I was one of the two or three students who answered her. The same day I met her at the bus stop, and she recognized me. I told her I want to do research in mathematics. And from that day on, she took it upon herself to help me in whichever way she can.

Our entire batch was her favorite. It wasn’t that she was all mushy or soft on us. She practiced what is called ‘tough love’. She worked hard at her teaching and demanded we work hard at our learning. She asked a lot of questions. She praised us when we could answer, and, well, took our trip, when we couldn’t. She went out of her way to fund our activities, and covered for us in case we got into trouble with other faculty members!

However, there were a few rocky moments too. I used to organize a weekly puzzle contest. Every week, I would post a new puzzle, along with the answer of the previous puzzle, and the names of those who got it right. All went well for a few weeks. Until one day, when SBM got (in my view) the wrong answer! Her view was that the question was wrongly worded. She demanded that I correct my mistake. We fought long and hard. It wasn’t pretty—but it was interesting, and kind of fun!

The question above was not included in the printed interview.

I spent only a year in IIT Delhi. IIT was mostly about very brilliant lecturers and a fun hostel life. But I did not learn much there, because I did not work very hard. Most of the time I was busy applying abroad. I got a scholarship, and left without finishing my MSc. But there was one important aspect of my year at IIT. I met the person whom I eventually married. So all-in-all it turned out to be a good year!

What was the experience at Ohio State University like?

Ohio State was truly the best educational experience possible. There were many famous mathematicians who taught me, among the best people in their area. My Ph.D. advisor was Steve Milne, who had given the first combinatorial proof of the Rogers-Ramanujan identities, thereby solving a long-standing problem. My story with him was similar to SBMs. He gave a talk about his area, and showed how he had extended a famous result of Ramanujan. Right after his talk I went and told him I wanted to work with him for my Ph.D.

The biggest truth I learnt at Ohio State was that mathematics is learnt by doing mathematics. Your professor can be the most brilliant lecturer in the world (or not), but you will learn only if you do all the problems of the textbook on your own.

In our department, there were people from all over the world; plus, I interacted with hundreds of American students as their Teaching Assistant. Living in the US, with enough money to have some fun, and hanging out with many people of many different countries—I think that was the most amazing and enjoyable part of doing a Ph.D. in the US.

From Modern School to Ohio State University, how has Mathematics shaped your life?

When I was in class 11, I took a Math Olympiad exam, where I happened to crack a problem I had never seen before. And I felt wonderful! I had got an exhilarating high, and it happened because I got a creative idea in mathematics. I figured that I want to have this feeling again and again, for the rest of my life. So I decided to become a mathematician.

From Modern to Hindu and IIT, and on to Ohio State, I stayed with this for nearly 15 years.

But I forgot about this after returning to India after my Ph.D. After a year in ISI, Delhi, I took up a job in the industry and thought I cannot pursue math any more. This went on for a few years, and I was totally miserable, and didn’t know why. Then one fine day I got a project to write a math book, and got reminded about this exhilarating feeling again!

That is when I realized that math is what keeps me happy. Now, despite a full time job, I look to do something mathematical, whether it is research, teaching, writing books, articles or papers, or even reading math books.  The thrill that comes from solving a math problem—especially a tough math problem—has never gone away. That is what keeps me happy.

The generation of today is somewhat reluctant to pursue Mathematics as a subject. What will be your advice to the students who are looking to or currently pursuing Mathematics?

My advice would be to do as much as you can handle, and then a little more. If you cannot do math just for the love of it, then consider the following 5 things that a math education does:

#1: It teaches you to question.

Why prove theorems, when they have been proved a million times before? Because, as our teachers tell us, you need to see for yourself that the theorem is true. This is so unlike the real world, where often people tend to prove things to you by intimidation, or by asserting their authority. However, unless you question things, you will not get creative ideas. And in math, we question everything!

#2: It teaches you to reason.

We learn to apply logic to prove theorems. In the real world, people frequently confuse a statement with its converse, and don’t believe that if ‘A or B’ is true, then both ‘A and B’ could also be true! Your capability to reason correctly and think clearly will quickly get you noticed.

#3: It teaches you to communicate clearly.

The practice of understanding mathematical definitions and proving theorems teaches us that words have a precise meaning. Being able to communicate clearly is perhaps the most important requirement for success.

#4: It teaches you to think abstractly.

As you grow in responsibility in an organization, you need to deal with a large number of facts. However, the time to deal with them is finite. At this time the ability to think abstractly becomes hugely important. Abstraction is a key requirement of any leadership position whether it is in academia, industry or the government!

#5. It gives you confidence.

If you have done well in mathematics, or even reasonably well, you should take a huge amount of confidence from this. For someone who is a master of epsilon-delta proofs, point-set topology, or abstract algebra, most management or technical problems at the workplace are a piece of cake!

In short, a good mathematical education gives you an unfair advantage in the real world. So if you can handle it, go for it!