Tuesday, February 17, 2009

My Anti-hero, Zero

This is the first in a series about a bunch of numbers. We're starting at zero, because it's easily the beginning or center of it all-- whole numbers, integers, reals, complex numbers. Well, all of it except for the natural or "counting" numbers, but I think we can blame the Greeks for that, frankly. (see Seife. More on his book below).

Zero, in a more innocent time:

The number 0 is a weird number to claim as heroic. Sure, it makes multiplication dead easy-- anything times zero is zero. It also makes addition and subtraction a bit of a snap, but you have to remember that, say, 4 - 0 =/= 0 - 4. The first is 4, the second is -4. But that's as hard as it gets.

Until you get to division, anyways. Watching an elementary school math teacher who doesn't have a degree in a math subject explain why division by zero is wrong is like asking your priest why premarital sex is wrong-- the answer is always "it just IS", followed by some very uncomfortable squirming. It's actually fun to watch, especially since kids love the word "why?" and will repeat it in a high-pitched whine: "WHYYYYYYYYYY?"

Well, there is no real reason why, except that any number divided by zero is, well, NOT a number the same way infinity isn't a number. Although that doesn't make any sense, either.

Well, here's a basic reason:

If n is any number, we know n * 0 = 0. Don't take my word on this-- intuitively, it makes sense. Zero n's means there are no n's to count-- nothing. Nada. Zip. Zero.

Yet notice that if n is any number, and n * 0 = 0, then n = 0/0. Meaning that 0 divided by itself can be anything you want it to be, which is impossible. Nothing divided by nothing can't be any something... can it? This leads to n * 0 = m * 0 -> n * 0/0 = m * 0/0 therefore n = m. Try it with n=1 and m=2 and you come up with 1 = 2.

Things get weirder. Think of 1/n where you try to get n as close to zero as possible. This is the notion of "limits", calculus' best friend. As n gets smaller and smaller, 1/n gets closer and closer to infinity. It'll never reach it because, well, infinity. It isn't a "real" number and it can't be reached. There's been a few attempts to try to force 1/0 = a conceivable infinity, but alas, it turns out this can't be consistently worked into our number systems.

I won't even get into 0 raised to the 0 power (which isn't 1), or the 0th root of 0 (which isn't 1 either), or L'Hospital's rules (which can resolve functions of 0/0 or 0^0 into actual living, breathing functions), or for that matter, 0! (zero factorial) which DOES equal 1. That last one confounds me, but it works mathematically. And it's critical in probability. The proof just hurts my head, though.

This is but a taste of the paradox of zero. I highly recommend Charles Seife's Zero, the Biography of a Dangerous Idea, which is a great read no matter how much you like or dislike math. It's almost as much about the cultural idea of zero as it is about the mathematical idea of nothingness.

Next up: Something between 0 and 1, and one of the least celebrated but most important concepts in the number system.

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