Just had a fun little tutoring session with a new student. (How could it surprise you that I tutor?) She strikes me, happily, as one of the many people who have a decent aptitude for math but, sadly, don’t like it much and don’t think they’re any good as it. How much I enjoy tutoring shocks me sometimes. The last problem got me genuinely excited: a neat geometric picture involving a stationary circle with another circle beside it shrinking away to nothing. You make a line from the top of the first circle to the place where the two circles intersect. The question, basically, is where does that line intersect the horizontal line through the center of both circles? We both looked at the picture and both of us thought, intuitively, that it should go off infinitely down the line.
But how to prove it? I suggested translating the problem from geometric to mathematical language. It involved quite a few steps, and wasn’t at all obvious how to do it at first, but she advanced from one to the next with little prompting from me, and we managed to state the whole thing as an algebra problem, where the answer would be given by a limit. Kind of astonishing that it’s possible to translate pictures into mathematics like this. We ran the limit, and with a little more work, found that the answer was: 4. A very finite number.
I have to say, this is one of the great things about math: you get surprised. Where did our intuition go wrong? What did the 4 have to do with the geometric picture (it was a graph, I should mention). I was immediately seized by a desire to start messing with the initial conditions to try to understand what made it be that number, and not another. And why not infinity? Something was wrong with our intuition, it seems. It’s a powerful thing to refine your intuition, and very satisfying.
The work of translating from one situation to another is also very key to the art of mathematics. Many people assume that all you need to do is solve problems, but before you can solve a problem you usually need to look at it from different perspectives. This algebraic/geometric correspondence is one of the richest in mathematics, in my opinion. It’s these connections between fields where the true insight lies.
A little example: have you ever noticed that as you go from one square number to another (1, 4, 9, 16, 25, 36, 49, etc.) you’re jumping an odd number each time (1, 3, 5, 7, 9, 13, etc.)? Looking at that arithmetically isn’t too enlightening: what does going from 3X3 from 4X4 have to do with the number 7?
But if you make it geometric, and think about a 3 by 3 array of dots going to a 4 by 4 array of dots, it becomes clear how to add those seven dots (in an L shape around the outside). Think about it a little more, and it becomes clear why it should be seven, and why it should always be odd.
And before you’ve known it, you have a proof of the purely arithmetic fact that the sum of the first n odd numbers is n^2. You can figure out that 1+3+5+7+9+…+91+93+95+97+99 = 2500 other ways (and I encourage it!), but the argument you get from playing with the square arrays sure is sweet.