Saturday, November 19, 2005

Book Review: A Course of Pure Mathematics

This is one of the two great books which influenced my life deeply. And reading it for the first time I came to know "what does a masterpiece look like?". I will never forget the excitement with which I read "Pure Mathematics" for the first time and then over and again for quite a number of times.

So what's this book all about? Well, it is an introduction to mathematical analysis (in crude terms: calculus). If you ever had any doubts or confusion in elementary calculus then this book is for you. And believe me, this book gives the best introduction to calculus.

The author, G. H. Hardy, was one of the greatest British mathematicians of his time and did significant work in number theory. He was also a man of somewhat eccentric character. But above all, we Indians should be thankful to him for his discovery of Ramanujan. He brought Ramanujan to limelight and together they collaborated for 5 years on a variety of mathematical researches.

The book begins with the theory of real numbers in the first chapter. This is the distinguishing feature of the book. The theory of Dedekind cuts is explained very nicely using English words instead of mathematical symbols. And thus Hardy creates reals out of the rationals and demystifies the nature of irrational numbers. This beginning chapter is like a gem and is the foundation on which he develops the entire calculus. Do not skip this chapter while reading the book (as I had done the first time).

Then the book goes on describe functions and complex numbers. After that we meet "Limits" and the concept of limits is explained in so great a detail and so nicely in prose that you will feel as if a teacher is sitting beside you and explaining you all the concepts face to face. The conceptual framework of limits, continuity, derivative and integral as explained in this book is the best I have ever found. After reading these chapters you will never ever have confusion regarding these topics. I owe my own understanding of calculus to this book.

Some of the most difficult theorems (the ones usually stated without proof in textbooks) like "continuous functions are integrable", "every polynomial equation has a root", "binomial theorem for non-integral exponent" and many others are proved in an elegant style which is unmatched. This book satisfied all my mathematical curiosities (except irrationality of Pi) at that time. The book also contains material on infinite series, logarithm and exponential functions, and elementary analytic functions.

The book is unique in all respects. The material presented in this book is not to be found anywhere in any other book on calculus. The best thing is the prosaic style in which the author communicates his ideas to you. You will never feel bored by mathematical symbols. Instead you will find nice English paragraphs explaining all the concepts to you. I really wonder why don't other authors also follow the same style. And I also feel disgusted by the lack of proofs in many books on calculus. It is the proofs which clarify your concepts and not the problems at the end of the chapter.

As an aside let me tell you the interesting story about how I got hold of this book in the first place. Some time around 1996 when I was 16 years of age, my father happened to be the member of a technical library in his workplace. And this library was open only to the employees and not to their children. My father wasn't any good in technical subjects but he had become a member only to get good books for me. Since he didn't know much about technical books, I told him to get books at random from each of the sections dedicated to physics, chemistry and mathematics. By my great luck, one of these random selections led me to this great classic "A Course of Pure Mathematics".

At that time I was in 11th grade and I did not know about the author G. H. Hardy (there was no internet access available and Google was yet to come, so books and teachers were the only source of any knowledge). Moreover the book did not have any introduction about the author. Even his qualification was not mentioned. But when I started reading the book I felt that the author must be a knowledgeable guy. Only much later (in my college) I came to know that this author turned out to be a great mathematician of the century and I also came to know about his Indian connection (with Ramanujan). I now feel that there was no need of mentioning his qualification in the book. The name of the author itself was enough.

If at some point in your life you had a liking for mathematics, then you must grab this book and devour it. If you were ever entangled in the mysteries of calculus, then this book is for you. However, I must mention that this book is primarily for those whose interests have a mathematical bent. If you're not one of those then you might prefer to be content with this review.

3 Comments:

Anonymous Captain Jack Sparrow said...

Another great review.

The following text was most inspirational - inspiring me to buy this book.
"Some of the most difficult theorems (the ones usually stated without proof in textbooks) like "continuous functions are integrable", "every polynomial equation has a root", "binomial theorem for non-integral exponent" and many others are proved in an elegant style which is unmatched."

Many a times we see in text books the statement "Outside the scope of this book", and that is most irritating. And I must add that writing a complete book on mathematics is impossible, but if a book like this tries to explain some of these proofs, then it should really be recommended as reference text in schools and colleges.

10:57 AM  
Anonymous Robert said...

"This is one of the two great books which influenced my life deeply."

What is the other book you leave a mystery in this post?

7:07 AM  
Blogger paramanand said...

Hello Robert,
The other book is "Atlas Shrugged" by Ayn Rand which gave justification to the philosophy of my life.

1:41 PM  

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