Tuesday, March 12, 2013

Math is Beautiful: Euler's Identity

We live in a world where everyone and everything you come across competes for your attention span. From newspapers to IPhone apps to friends to coworkers to rascals to fine ass to driving to movies to eating to reading and (stick any word in here), we frequently miss some of the more subtle, tranquil things that have quietly removed themselves from the insanity. The heavenly lights, sunsets, architecture, cities, and ecosystems, requiring years to million of years of craft, have been infused with eternal patience. They are the silent whispers of sages which usually goes unnoticed. When an occasional voice might strike you on a ordinary day, you may be so lucky to lend a few seconds, or maybe a few minutes, to a beauty which always existed, but never appreciated. You might notice that every blade of grass and every giant of the forest pays worship to the same gust of wind. You might notice that a lake of still water creates a perfectly flat, level, and smooth surface of proportions which cannot be replicated by any technology available to mankind. You may be with a drink and friends and suddenly think that life is great the way it is, despite everything you just complained about 20 seconds ago. You might for a moment in a busy life, pause and think, "The world is beautiful".

By an age that we can babble, we come across the concept of numbers, starting with 0 and 1, and probably to 10. We don't think much of it. It is only a way to retain and transfer information from one medium to another. Eventually, we learn that we can add and multiply these things and get more numbers. All these numbers were are somehow related by the ability to add or multiply something to get something new. Somewhere along the way, maybe in 3rd grade, we learn about circles and areas. We get introduced to some weird thing called pi, or π. Although we are generally familiar with an approximation of 3.14, some of us eventually learn that pi is not a regular number. It cannot be categorized with any other number. It is unique. There is only one of them in all the universe. It has special properties, special powers. In the 3rd grade, we don't know where it came from. We're not sure how someone else figured it out. Similar to how Adam and Eve were just planted onto Earth, pi just existed, with no history, no explanation, with powers that not everyone fully understood.

If you were so fortunate to have a good math teacher for algebra, you would have learned that 0 and 1 were not as plain as you thought. 0 and 1 are the additive and multiplicative identities. More importantly, 0 and 1 would allow you to create inverse operations to addition and multiplication, which were yes, subtraction and division. A number plus its additive inverse equaled 0. A number multiplied by its multiplicative inverse equaled 1. If you were lucky enough, you would have been taught that there isn't really such a thing as subtraction and there isn't really such a thing as division in algebra. In algebra, you add additive inverses to subtract and you multiply by multiplicative inverses to divide. Addition and multiplication were associative and commutative operations. These properties did not apply to subtraction and division. In order preserve the purity of algebra, 0 and 1 had new value as operational identities.

As we proceed through junior high school, we find i, the imaginary unit. Like π, i was unique. It was special. It had powers. It opened up a world of vector math. Just like π, i just existed. It had always existed, unconcerned with whether or not you knew about it; whether or not you appreciated it, whether or not you believe in it. Studying and learning about it enabled you access to its wisdom, just like reading the bible.

After a long journey through calculus, someone wondered if there was a function which was its own derivative. As a result, a new mathematical constant was discovered, e, Euler's number. Once again, e has always existed, even before it was discovered, nonchalant about whether or not you knew or cared. Just like 0, 1, π and i, e had very unique properties. It was special. They were all special, and they were all different. They had as much in common as water and rock, night and day, heaven and hell.

In your effort to satiate your thirst for knowledge, you battle through the infinite armies of the Taylor Series'. The devastating torture devices to your brain like sin() and cos() have evolved into deadlier monsters. You are confronted with the brutal, yet magnificent, mystical, legendary beasts which guard the secrets and treasures of math and science. These are creatures which attack the mind. Physical strength is not a viable weapon. Those with the persistence, perseverance and patience to tame the Taylor Series' eventually unlock a door and find what's known as Euler's Identity.
At first glance, this is quite curious and requires a double take. It's a mess of symbols in an equation, but not a complete mess. Moments like this can suddenly amplify the voices and echos of elders, raising your alert level on why this outcome appears so daunting. A further look reveals that this equation has no variables. This means that it is not a tool to solve a problem. It's not like the Black Scholes model where the welder obtains a dominant power over those who were unable to acquire or comprehend it. It isn't there to serve a particular purpose. It isn't there to solve a problem It's simply there, representing the perfect and elegant unification of the five most important, yet seemingly unrelated mathematical constants of our universe. It just exists, to be admired. It's unjudging, irrefutable, incorruptible, unique, elegant, not open to interpretation. It does not care whether or not you are able to appreciate it, as it has existed long before the rise of humanity and will continue to exist past the death of humanity. It shares many characteristics with god. It's like Earth, Fire, Wind, Water, and Heart melding together to form Captain Planet.

Euler's Identity existed long before it was discovered, and it was there long before I learned it. The idea that seemingly unrelated things were connected in the grand scheme is very significant. Perhaps you and I are connected, and were connected, all along. Perhaps it's my current state of ignorance which does not allow me to see the full picture. Maybe every one of us and everything in the universe is also tied together by some surprisingly elegant relationship. Is it possible that the wars and crimes and poverty make it difficult to see beauty in the world? The same way that the odyssey from counting to arithmetic to algebra to calculus to Taylor Series tightly guard the sliver of overwhelming beauty on the other side? Such are the questions that I asked myself during my pause, as my admiration for this gem drowns out what is currently just noise in the rest of my life.I ponder about how there was a time that I did not know this, and I can only wonder about what else I don't know.

The most important thing that I learned isn't that some symbols can be combined into an equation, but that I know far too little and have experienced far too little to judge others. It is a rare and incredibly humbling moment to acknowledge that, my own success may be due much more to good fortune and due much more to forces beyond my understanding than I had originally thought. I've always felt that everything about me was a result of my own hard work, and therefore I need not be grateful or attribute credit to anything else. But now I feel a sense of shame after learning how foolish my viewpoint was. As I exit this temporary sanctuary whose walls have been negating the screaming distractions, I think to myself, "Math is beautiful. Life is beautiful. And if there was ever a god, he might hint at himself to us through the finer details in his work."

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