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Mathematical Annihilation

Last year, while sitting in Algebra II and being profoundly uninterested in the subject matter, I found myself thinking up some of the strangest and most absurd ways to pass the time. Occasionally, these manifested themselves as actually useful trajectories of thought. And one time, with the help of my seatmate, I came up with a little trick of math that I still use from time to time, and may serve you well.

Coined the “Annihilator Method” by Thomas, the smart-ass sidekick, it’s a simple concept. Essentially, it’s a method of simplifying algebraic equations — nothing fabulously new, I’m not going to win any prizes for it, but it’s a way of looking at things you might not otherwise see.

Consider the following arbitrary equation:

Pain in the ass, frankly. Can we simplify it? Sure, lots of ways — but here’s one you might not have thought of.

What are we doing? Merely dividing one of the ugly terms by itself, removing it. On the other side, of course, the same thing is done… but the other side is zero. Zero divided by anything is still zero, so the term just disappears. It’s gone. You’ve removed it from the equation entirely, no muss, no fuss. (The zero division has not been shown, to save space.)

Moving along:

That was so fun, we did it again. The other stupid, complex term goes away, vanishing into the zero, which happily consumes it like a hungry black hole. Finally:

Here we go. Now we’re talking.

And all we’ve done here is simply to divide. This is an exaggerated example, but the basic concepts can be applicable in many instances. Remember that you need to have zero on one side, and things will fall into place.

Now, it’s important to understand the limitations of this method. When we first cooked it up, I found myself trying to use it all the time, even in cases where it’s not much help. As a general rule of thumb, it’s best used when you have only multiplied or divided terms to deal with — if there’s any addition or subtraction of terms going on, it can still be done, but may end up taking more, rather than less time.

What we did above is to destroy the terms by division, but it can be done just as well with multiplication — say, if you have several terms over several others, in a massive division problem. Just multiply to destroy the bottom (denominator) terms, then divide out the top (numerator) terms, and be done with it.

Also, be careful not to go too far. In the illustrated example, you may notice that we could have just kept going, dividing both sides by (x – 5), and thus proving with epic finality that 0=0. However, unless you particularly want to prove that zero is, in fact, zero, this may not be too useful. Don’t erase any terms you need — only the ones you don’t.

There are actually two variations on the Annihilator method that we came up with — in one, you can prove that absolutely anything equals absolutely anything, but unfortunately the math is not valid. Fun, but not much use. In the other, the true annihilator method, you simply start by multiplying both sides of the equation by zero, killing everything, and going to sleep. That’s the one you use when you’re tired of doing your math homework and want to go out with a bang.

This entire system may seem impossible, but it actually makes total sense, if you think about it. Imagine: If one side of an equation is zero, that means the other side is also zero. No matter what’s over there, it equals zero overall. So if it’s zero, then isn’t anything times zero or divided into zero still zero? You’re merely removing those superfluous terms; but it will be zero no matter what, unless you add or subtract something.

There’s also one subtle flaw in the Annihilator method, which probably nobody but a mathematician would notice. Fortunately, it doesn’t often become a problem. Bonus points if you notice it.

UPDATE: The aforementioned flaw has been discovered, much more quickly than I would have expected. Kudos to the BHS community. Read the comments for this post to get the details (a good idea whether or not you’re mathematically inclined, since it reveals the limitations of this concept).

— Brandon