sqrt of e to the i pi

My earliest memory of learning math is the frustration of trying to learn how to subtract two three digit numbers. I still remember this girl Preethi das who would subtract two numbers quite effortlessly while it looked like black magic to me.

The next shred of memory that I can recall clearly is memorizing multiplication tables. I would pace up and down the hall of our one bedroom flat in Calcutta reciting the tables while my dad sat in the light of the flickering kerosene lamp listening and helping me out when my memory failed me. In those days (early 1980s) load-shedding was very common in Calcutta. My dad insisted that I memorize multiplication tables from 2 to 20 while our teacher thought 2 to 10 was sufficient. My dad would test me asking "how much is 14 times 7" and I would be expected to answer from memory.

My next significant memory is solving algebraic equations. Two years ago Ram was thrice as old as Shyam. Next year Shyam's age will be half of that of Ram. How old are Ram and Shyam? The problem seemed quite daunting to me. Simultaneous equations had not been taught to us then. After trying for sometime, I asked my dad how to solve it and my dad taught me simultaneous equations. Then I solved my entire homework using simultanous equations. Next day my math teacher, Jana Sir asks me,
"Where did you learn to solve like this?"
"My dad taught me."
"Did you understand or did he solve everything for you"
"No sir I did"
"You will be able to solve a new problem this way on your own?"
"Yes sir, I think so." He believed me.
He was nice person and a sincere teacher and I liked him very much. When I heard of his sudden death about twelve years ago, I wished I could have met him one last time. I think that was the first time I ever felt a sense of loss and the permanence of someone's death.

Over the years I have learnt about negative and positive numbers, rational and irrational numbers, transcendental numbers and even complex numbers! The more I learnt howver, the harder it became to relate math with physical reality. Addition, subtraction, multiplication, division and even finding square roots, all had a definite physical meaning to them. If two people gave seven people gave me ten apples each I would have seventy apples. If a floor had 25 square tiles then each side was five tiles. My first transcendental number 'pi' seemed quite physical and tangible - it was just the ratio of circle's area to its diameter!! Then came along 'e' - the Euler number and challenged my understanding significantly. No matter how much I tried, I found it very hard to understand 'e'. However, soon I learnt that 'e' was the maximum possible money you could make at the end of the year if you started with a principle of one rupee compunded it at an interest rate of 100% per annum as often as possible during the year. I found that quite fascinating. Soon some interesting things came by like no one knows whether pi plus e or pi times e or pi to the power e is transcendetal or not and that 2 raised to the sqrt of 2 is transcendental.

Even before I was recovering from the transcendental shock came along 'i' - the square root of minus one. It seemed arbitrary and absurd. No matter how much I tried, I had to contend myself with the explanation "i is defined that way - it is the square root of minus one." This meant that I could potentially conjure anything I wanted and create rules up and call it my own math without any relation to physical reality!! I think that was when math ceased to be something physical for me. I was taught that multiplication with the square root of minus one was equivalent to a 90 degree anti-clockwise rotation. Then followed the relation between trigonometric functions and complex numbers and 'e'. I felt that these explanation were just tricks to make things easy to work with, easy to remember the rules of complex numbers and had nothing to do with reality. I had to contend myself with the fact that square root of minus one really did not exist and it just a fancy tool.

Calculas interested me a lot, there was something physical once again. I struggled to understand what Eigen values and vectors of matrices physically meant and partly found some physical explanation that satisfied me. It turns out that one can conjure a physical way of explaining almost all math if he or she tries hard enough. After all every idea that we get has its foundation in some physical phenomenon we have experienced. Somehow finding a physical equivalent seemed to help my understanding of the underlying math. It almost became a burden for me.
Math as a bunch of symbols was somehow not acceptable.

Nevertheless many a times finding the physical meaning of some math can be very hard. For example, the Laplace transform, what is that if not a mere tool? Perhaps one can find a physical explanation for the Laplace transform as well but it would seem a bit too unnatural. Something that was created just for the sake of doing so. I started to wonder if my approach of trying to find a physical meaning about every piece of math that I learnt was in itself flawed.
Even today my dillema remains unanswered. Even today I find peace when I can look at math and understand how it is something physical.

What is math? Today, to me, math is something unreal yet true. It borders on the fantasy and reality carefully avoiding taking sides either way. Its a merely a collection of beautiful thoughts, ideas and mnemonic rules. That brings me to the title of my blog. The first time I heard of e raised to the i times pi is from a friend. Somehow it fascinated me but I never really understood what that really means. To me 'e' raised to the 'i' times 'pi' is an esoteric and obscure way of saying minus one. In fact its perhaps merely a very inefficient way of representing minus one and yet it fascinates many. Whats fascinating is the fact that you do something strange with a bunch of transcendental numbers and rules of complex numbers and end up with a minus one!! The complexity of the expression and the simplicity of the result shows what an inextricable mesh math is.

Now let us examine the equation square root of e raised to the i times pi equals i. How fascinating is this equation? Somehow to me this expression does not seem nearly as fascinating as the previous one. How can it be that by simply taking a square root on both sides of the equation of such a fascinating expression, I have destroyed the beauty and wonder inherent to the original equation? The answer is that 'i' is as incomprehensible as the expression sqrt of e raised to the i times pi - that is not considered as surprising. Somehow math seems to become beautiful only when in the end something real comes out of it - like minus one. No wonder all the big financial firms hire mathematicians today.

Is minus one indeed a tangible number? Can you touch something made of minus one objects?
What does it mean to multiply two negative numbers and get a positive number physically?
Aren't negative numbers negative because of where the zero lies. If I move my zero arbitrarily I can turn any negative numbers to positive. Why then do I need negative numbers at all? Yes, minus one is also a concept - theres nothing physical about it. Why is it then so surprising that its square root 'i' is also a concept? How come then e raised to the i times pi equals minus one is more facsinating then square root of e raised to the i times pi equals i, when everything was only a mere concept to begin with?

Its all in the mind. We have all been hypnotized into believing whats real and whats not. We have been programmed since our childhood whats beautiful and whats not. This hypnosis makes sense to us and our society thrives on it.

When I read some english text, while it may mean something simple, the play of words makes it very hard to understand and yet some find pleasure in the same. This is perhaps not very different from today's law system or bureaucracy where at one point all that remains is rules that need to be followed blindly and all meaning is lost somewhere in the abstractions. Some use this to their advantage.

I must confess that I still find the fact that e raised to the i pi equals minus one, quite fascinating. Perhaps I too have been hypnotized and beyond repair.