I was a tolerably good student at that time, especially good at high school level math. But while I could easily and frequently score 100 marks in calculus and algebra tests, I just could not understand how one can cancel some terms involving delta-x because it is close to zero but not others; or how one can divide by delta-x that is almost close to zero.
I really had trouble during my +1 year. My math teacher was one K.V. Srinivasan or KVS, an old school teacher (He had a B.A. in math, not B.Sc.) at St. Joseph’s, Chengalpattu. I repeatedly asked him about the delta-x issue and he couldn’t give me a satisfactory explanation. His explanations looked very weak to me – I am sure he was as tired of my questions as I was tired of his answers.
Similarly, just wtf is an imaginary number? I mean, what number, when squared, yields a negative number? I was able to use them well, without really understanding it. Or even the negative numbers – why in the world would multiplying two negative numbers yield a positive number? I was used to thinking about negative numbers as debt – but why would multiplying one debt by another give me a profit? I didn’t have any teacher, any book, any resource that could answer such fundamental questions. I also couldn’t see any use for trigonometry. Algebra – which was my favorite branch of math at that time – felt like just symbol manipulation. Geometry was one branch I never truly liked – primarily because my drawing skills were very poor – I couldn’t draw a circle even with a compass, my circles would never close. But also, I just couldn’t see the point of opposite angles of intersecting lines being equal or the artificial distinction between but equilateral and isosceles triangles and so on. I remember asking in 6th or 7th grade where all the theorems about isosceles angles are all used in real life and getting a huge scolding from my teacher.
I can keep going on. Basically, higher level math – anything beyond simple arithmetic – felt like just a load of useless, impractical, theoretical garbage.
Aside: Algebra was my favorite branch because I had discovered formulae like a^2-b^2 = (a-b) * (a+b) by myself independently. That was my first intellectual achievement ever. My only clue was the observation that the difference between successive squares differ by 2. I was writing up squares of all numbers upto 32 as part of my classwork and noticed that 24^2=576, 25^2=625, 26^2=676; i.e. 25^2-24^2 = 49 and 26^2-25^2=51 and noticed that the pattern can be extended forever i.e. 27^2-26^2 = 53, 28^2-27^2 = 55 and so on. I was 11 or 12 around that time. And it took me around 4-5 months of thinking and thinking about this pattern before I came up with the formula. I didn’t know algebra at that time, I actually expressed it in terms of words, Tamil words at that 🙂 Something like:
ஒரு வர்க்கத்திலிருந்து இன்னொரு வர்க்கத்தை கழித்தால் கிடைக்கும் எண் அவற்றின் வர்க்கமூலங்களின் வித்தியாசத்தை வர்க்கமூலங்களின் கூட்டலால பெருக்கினால் கிடைக்கும் எண்ணுக்கு சமம்
When I came across algebra in 9th standard (I think), it felt great! I suddenly had a language to think about these kind of things. But 2-3 years later, I had come to think of algebra as rote manipulation of symbols according to certain predefined rules, just useless theory.
Then I came across Bell’s book. Bell made math come alive for me. Suddenly, I understood limits after reading the chapters on Fermat and Newton. In fact, it made perfect sense. Though I didn’t expect to use it myself, I suddenly saw why calculus is useful; not just useful, but necessary to understand the laws of physics. I suddenly saw imaginary numbers in terms of a co-ordinate plane and things just clicked. Multiplication of one negative number by another? Just a directional change along the number line.
When I read about Descartes, suddenly geometry became a discipline that I could master, by coming from a different angle. I got pretty decent at co-ordinate geometry, and I attribute that to entirely Bell’s effect. In fact, I had a voice at the back of my mind going that I could have done this, I could have invented co-ordinate geometry myself, Decartes’ achievements aren’t that great. That feeling that I could have created analytical geometry myself was enough to hook me onto math forever.
I read about Poncelet and the romantic account of his discovery of Projective geometry from prison, it made great sense. I desperately wanted a textbook on projective geometry and couldn’t get one. When I joined engineering, that one chapter gave me a fundamental understanding of engineering drawing class. I was still pretty bad at drawing, but I could clearly understand and in fact even go beyond what was taught in class. One chapter – just 10 pages or so – made a discipline real for me.
Of course, there were things I didn’t quite understand as well e.g. Fourier series, Analysis in general, Ellipitic integrals, Group theory, Riemann Hypothesis, chapters on Hermite and Poincare, but it just didn’t matter. I still don’t understand many of these topics. But I found enough to fascinate me. Cantor’s infinities, the Dedekind cut, Zeno’s paradoxes, Weirstrauss’s continuous but non-differentiable curve, Non-Euclidean geometries (The instant I was told to imagine a sphere, I could see how the fifth axiom is not valid any more), Fermat’s Last Theorem, the list goes on.
I didn’t see the point of of several of these topics as well e.g. I didn’t think Hamilton’s quarternions would ever find any use. But now I was confident that somewhere there must be a use for it. I read that number theory didn’t have much practical use, but I found it the most fascinating of all branches of math. Now of course, primes are crucial in cryptography…
Bell had a habit of combining the biography with tantalizing bits of math. For instance, he just mentioned Wilson’s Theorem in passing and would make a remark like a high school student can prove it. I think I spent 3-4 months thinking about it before I came up with a proof of my own. I can still vividly remember the thrill when I realized that every number has an inverse in modulo arithmetic. Wilson’s theorem was one of my rare successes, though. I probably failed in trying to prove 90% of what I attempted. No matter. I don’t think I had that much fun learning, just trying to figure out things before and after those 12-15 months in my teens.
I also spent hours ranking the mathematicians in that book in my mind. Gauss was always the #1. Descartes never got a high ranking from me because I felt that if a high school student like me can derive analytical geometry, then Descartes doesn’t deserve lot of credit. But I remember ranking Archimedes, Fermat, Newton, Leibniz, Euler, Abel, Galois, Riemann and Cantor as the top mathematicians.
One of my great regrets is that I couldn’t convey my enthusiasm for the book to my daughters – or for that matter, anybody else. I even bought a copy for Jeyamohan, but I doubt whether he actually opened the book any time. 🙂
Later I came to know that Bell would now and then exaggerate to make the biography more interesting. I couldn’t care a rat’s ass. He had done his job – kindled a life long fascination with math in a govt school kid brought up mostly in villages. I just wish I had gone beyond the dilettante stage and actually learned math. I would have enjoyed working on number theory!
Aside: When my daughter went for a familiarization tour to her college (UC Santa Barbara), Yitang Zhang – one of the romantic stories of math today – was giving an introductory lecture on the prime gap. I think I was the only one who was listening; I was actually thrilled with that lecture! My wife and daughters still make fun of my asking questions at the end of the lecture. I could see how his proof that the prime gap is less than 70 million ties in with Goldbach conjecture, and I was remembering Bell gratefully…