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.


raghu said...

now i can proudly say im good at math!
watta wonderful post :D:D

heh? ok said...

hmm, i'm guilty of laziness and laziness inspired fear. i always did wonder why i so loved grammar and couldn't do math.

Priyanka Kumar said...

i have a mathematical brain?

*clearly stupefied by the magnitude of the thought*

Polar said...


Chanced upon your blog through Mr. Munroe's at xkcd. Have to say, this is excellent writing. (most of the posts are.)

At the risk of sounding redundant, Douglas Adams phrased this argument amazingly well in one of his later novels-'Dirk Gently's Holistic Detective Agency', I think it was called. It isn't as cogent as it seems, though. Geometry, and simple rate calculus is fine (intuitive, I mean), but the brain stutters quite a bit when you get to abstract math, or logic.

(Unless you have the intuition of someone like Feynman- who pretty much imagined hexaflexagons in his mind's eye.)

ToOothlEss WOndeR! said...

ok. I guess I'm on top of the guys-with-mathematical-brains list.
I'd calculate my projected exam score with one half, and all the trouble that i'd have to go through to get to the exam on the other, and would take the logically right decission to rather-sleep-than-go-and-flunk. This was the case for every math exam in my engineering course.
It was fun. Three papers at the last sitting.
Boy, it was so much fun!

another brick in the wall said...

haha.. brilliant! the way the u've justified urself.. hehe.. yes yes.. no one shall judge u by ur marks in math(s)

Marks are only about what formulae you can memorize and remember on exam day

oh thts another misconception.. if u start deriving the formulae or understand the derivation.. u wud never feel the need 2 memorize any of them.. life wud be real easy for u!

yeah.. i've had brilliant mathematics teachers.. but my english teachers somehow refused 2 teach me :S

new age scheherazade said...

@ raghu: thanks! but then you are good at math anyway, na?
@heh- i always wondered why i could do english grammar but bengali grammar stumped me. very strange. yeah, i think it's laziness too.
@ priyanka: haha. and will you stop acting dumb, you brilliant thing.
@ polar: wow. someone who reads xkcd thought i was good. i'm overwhelmed.
and you like Feynman too. oh wow.
@toothless:haha yes-i read about that. it would be so cool if it weren't a bit scary. i think i'm inching towards that kind of sleepiness.
@brick: thanks! and yes, math(s) it is. never really understood why the difference.

raghu said...

HAHAHAHA moti saali sirf strangers ko maha bhav dethi hai!

new age scheherazade said...

abbe gadhe, i didn't know you read xkcd then.
and anyway, mamma told me, be nice to strangers.
uh...that came out wrong.i think.

Polar said...
This comment has been removed by the author.
Polar said...

The xkcd creator, and all readers (including, presumably- yourself) are normal people too- and your posts are highly articulate and lucid, by most standards. {shrug}

Commenting again is admittedly not the best thing to be doing, but will try to refrain from, err- littering your list of reprieves in the future. Didn't realize this was a semi-private blog, sorry.

Full stop. said...

That was very warm and encouraging for someone who once scored 41 in math. No kidding.
But XKCD, oh. The other day I saw a guy wearing a tshirt that said 'Raptors'. I almost went and grinned at him in XKCD siblinghood, but then I wondered if Raptors might be a band or such? So I didn't.

new age scheherazade said...

@polar: did i and raghu go and offend you in our ignorance? litter away all you want. i love comments.specially when they're nice.

@fullstop: hey- i got 50 once, back when i was scared of it. and did the tshirt have 'Raptors' struck through with a big cross? Mr.Munroe would do that, no?

Polar said...

Noted. Thanks.
That's the xkcd variant. (Never saw Jurassic Park, never could make sense of the phobia.)

Full stop. said...

Aha, that's a good point. Mr.Munroe does disapprove of raptors.

speedpost said...

see putting down on paper is really the toughest thing to do.. especially when ur a left handed person and ppl look at you with the impression that ur an escaped reticulated python.. thats when the formulae go whooshing out...

new age scheherazade said...

polar-i think i'm kindred with randall munroe. what a thought.

fullstop-that he does. violently and obsessively.

speedpost-but aren't left handed people supposed to be really clever? and are escaped retculated pythons, or captive ones, left handed? or any-handed for that matter? *scratches head*
no, really, i get the joke.

Divya said...

Well i always did wonder why i was very good at maths but despised it...

and oh god.. people who like feynman... people who know who feynman is... [sigh of contentment]


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