Math geeks can probably tell that my old screen name “exipolar” is a tribute to Euler’s equation: . If you are somewhat of a math geek, then I probably don’t need to relate to you how beautiful of a thing this was for little 17-year-old Steve when he figured it out in high school.

But if you’re not a math geek, keep reading.

I didn’t do very well in 8th grade algebra, so my parents bought me my TI-83 graphing calculator to aid me when I took the course again in 9th grade. There were plenty to go around in the classroom, but I could take mine home to play and use it for my homework. At least that’s what my parents had intended it for.

Teenagers are just childish enough to harbor a naïve curiosity for the world but just grown up enough to share that curiosity with some adult freedoms. It’s here where they do most of their damage. They go out with friends late at night, take illicit substances, and generally do whatever it takes to climb their social ladders. I wasn’t like that; middle school taught me well that it simply wasn’t worth being at the top with the peers fate had given me. It would have been a meaningless struggle for a hollow victory, but that doesn’t mean that I didn’t rebel.

Instead of social norms, I directed my rebellion towards my education, particularly towards math. Math is pretty much the only subject you’re allowed to do that in as it’s so easy to be absolutely wrong instead of being partly right like you can with History or Language arts. I tried wherever I could, but math was most rewarding. You can try out an idea and it only takes a little work to show if it’s the wrong way, especially always having a graphing calculator on hand. It was an experimental approach to education: the right approaches survived, the wrong ones fell aside.

This way, I formed a sort of mathematical vocabulary. It allowed me to “see” the behavior of any particular equation as long as I was familiar with the pattern it was put in. This ability earned me straight A’s throughout my freshman year in algebra and geometry. I decided to learn higher math on the computer that summer through programs my father had bought for earlier that year. Much to my surprise, I made it all the way through calculus in time for my sophomore year.

It was during this time that I was first introduced to imaginary numbers. An imaginary number is simply the square root of a negative number; no real number can be squared (multiplied by itself) to produce a negative result, but philosophically speaking, one ought to have a meaningful result when any operation is applied to any number, so one ought to be able to get a result by taking the square root of a negative value. Thus, imaginary numbers. Imaginary numbers are pretty much like normal numbers except that it’s value is a real value multiplied by : the imaginary identity . So if you needed something like the square root of -25, your answer would be . Another property of imaginary numbers is that they do not mix with real numbers; a number that has a real part and an imaginary part is called a complex number. If you wanted to perform you would only by able to combine imaginary with the imaginary and the real with the real yielding . To make matters more complicated, you can multiply real and imaginary numbers (even complex numbers) with one another, but since then complicating multiplication a bit. It suffices to say, complex numbers aren’t your daddy’s math, and they really weren’t in the textbooks I was given either.

Ultimately, I became curious about how equations like would behave. Now normally, looks like this

The TI-83 has a complex number mode, and when I tried to graph , it gave me this

I had apparently left Kansas sometime earlier and entered a Louis Carol novel. (Which isn’t too far from the truth)

No, it’s not the opening to the outer limits, nor is it a DNA double helix. What it is, though, is what could be called a metric wave function. Waves can be produced with trigonometric functions like sin or cos but they did so for a reason. Though, I had no clue why it looked this orderly for , but I didn’t know what to expect in the first place. What I found truly bizarre was that the real part of matched up exactly to . To make matters worse, there was no algebraic, trigonometric, or even calculus method I could use to figure out why it turned out like this.

It took me until the beginning of my Junior year to figure it out.  I flipped to a page on Taylor series polynomials 2 weeks into actually taking Calculus. Fairly quickly, I realized there was an intimate relationship between and trigonometric functions like and . After another week of wrestling with the algebra of Taylor series polynomials, I had found my answers.

This is Euler’s equation. Simply, this equation illustrates that there is a natural mechanism in mathematics that ties two very desperate things in a simple and elegant way. That there is a transcendental link between rectangular coordination and polar coordination. More generally, it can be viewed as .

To me, it was the most beautiful thing in the world. No matter how wrong you tried to make something in math, there was always an answer, and the harder you tried to break it, the more beautiful it became. From Euler’s equation, one can make this identity:

Illustrating the link between 4 fundamental values of math: e (Euler’s constant, used for evaluating continuously compounding interest), , i and -1.

Don’t worry if it doesn’t make sense to you. That’s a part of the beauty though, one can rationally access and use a vocabulary of knowledge that is beyond our ability to directly abstract. Even in my mind, I have to remember that the i in the exponents position connotes an entirely different pattern than that associated with real numbers. While it’s common in english, you almost never have mathematical equivalents to “slaughter” and “laughter”.

So anyways, It was about this time that I had gotten involved in ISTF at school. They wanted me to have an aim account to support communication online among the other team members, if you’ve made it this far, then you understand how obvious and stupid it was. But I typed it, exipolar, and it’s been stuck with me ever since.