My favorite math moments - vanilla geekery

Joey asked me if I have a favorite math moment. Truthfully, after spending nearly a decade tutoring math to college students ranging from the most basic remedial stuff to calculus and trigonometry, I have several.

Euler's identity:

e^iπ+1=0

Oh. my. God.

When I first discovered this I was astounded. I'll let you read wikipedia's explanation of the mathematical beauty if you want. I was convinced that somehow, locked within this equation, was the secret of the universe.

How. HOW did the irrational base of the natural logarithms, raised to the power of the square root of negative one times the (also irrational) ratio of a circle's circumference to its diameter... equal negative one? HOW? It astounds me even now. I've worked it out, and it comes out right and proper as it should. It's astounding.

I loved introducing this concept to a student who was just being introduced to imaginary numbers, and watching his or her eyes grow wide. Well, when the student was intelligent enough to grasp the awesomeness. They weren't always. Sometimes they'd just go "Huh?"

Why do I need to learn this?

I don't know if this is really a favorite or if it's just one of those things I loved to hate - but if I had a dime for every time a student said this to me, I'd have a sizable sum of money. My favorite answer, after years of it, was from a book I once read titled Everything Bad is Good for You. You don't learn algebra to use algebra any more than you go to the gym to learn how to use a treadmill. You do both because they build a certain kind of muscle - in the case of algebra it's a problem solving muscle. Algebra gives you the tools to solve problems you may not otherwise know how.

Another thing I'd always tell them is that you never know where you're going to wind up. I certainly never thought I'd wind up being a (mostly math) tutor for a third of my life.

Holy crap I've got it!

The lightbulb moment. When any student gets it. You've been working with the person for hours, possibly days, and the light in his or her eyes has dulled. He's resigned himself that he's never going to get it. She will plod through the exercises as you guide her, but she's never going to understand and she will fail her test.

And then, suddenly, the light returns. He's got it. She's got it. He flies through the next page worth of exercises with no difficulty at all. They can't even be called problems anymore, because they don't present any.

Guiding a person to that moment is fantastic. It happened far less than I would have liked - many students had just resigned themselves to achieve low scores and be happy just to pass, but the ones who struggled at first and then suddenly had it? Oh, yeah, that's a high.