Saturday, July 14, 2007

Millisecond Math

[This was written for flamebird' online magazine, and is reproduced here because priyanka said she needed another post(which makes me feel wanted, a bit), and i already had this written. Exams sneaked up on me again, and I will probably be failing this time, so please, no best of lucks.]

The one subject most people have real trouble with in school is math. Whether it's the way it's taught, with no emphasis on understanding and application, or whether it's the confidence-demolishing teachers and exams, a lot of people I know are afraid of the subject to the extent that they let it affect their choice of later studies. But the fact remains that math is just another language with its own grammar and a surfeit of logic; and as Feynman said, it's the language in which nature speaks.
It's also the one language which the human mind is programmed to understand from birth. Every sense organ of our bodies is a delicately tuned receiver of mathematical data for the brain; which in turn does an unbelievable amount of math to present us with, say, the correct picture of Seurat's Sunday Afternoon on the Island of Le Grande Jatte with its innumerable dots, or to show us this printed page.
And everyone's brain does it, not just Einstein's or that belonging to the smartest kid in the class.
So then. Here’s a wow problem I came across in Halliday Resnick the other day. Very interesting.
What happens when you hear something? The sound goes in your ears; the brain interprets it, blah blah, yes. But how does your brain know which direction it’s coming from?
One clue it uses is the time delay, which we’ll call Δt, between the arrival of the sound at the ear closer to the source, and the arrival at the farther ear.
Here, when the wavefront (the wave) reaches your right ear(very originally named R here) it still has the distance d to travel before it reaches your left ear (yes, that is the one named L).

Some basic high school trig tells us that this distance is Dsinθ, where D is the separation between your ears (directly proportional to your fatheadedness). Using the basic equation, time=distance/speed, we get:
Δt= d/v = Dsinθ/v (where v is the speed of sound, in air)
And then, the brain does some millisecond algebra and comes out with
θ=sin-1 (vΔt/D). And, based on the measured value of Δt, a lifetime of experience which gives v, and the obvious knowledge of D, the brain computes the angle θ and tells you instantly which corner of the room your friend’s screeching at you from. And all this in a few milliseconds, to enable you to duck the textbook she throws at you.
(There’s some interesting extra info in Halliday about what happens in water, where the brain’s confused about v, but you can look that up yourself.)

The same thing happens when you see something, walk, talk, or do anything at all. The brain’s very good at math. And so are you. After all, it’s your brain. And if you’re having any problems, it’s either the way you were taught, or the fact that unlike the brain, you don’t practice enough. Whichever it is, there is no such thing as a ‘mathematical brain’ unless that refers to every brain in the world. Each of us is good at math, and we can all understand the language in which nature speaks. Marks are only about what formulae you can memorize and remember on exam day. The real test is when you hear a car radio two cars away in the traffic jam and can instantly tell where it’s coming from, and that it’s unmistakably Elvis singing. The rest is just putting down on paper what you can already do in your head.