You don't know anything about algebra. Nothing, nada, zilch, absolutely none is what you know. Ok, now that I got that obligatory math teacher berating out of my system, let me backtrack a bit and let you in on one of my big pet peeves. Like the boring goofball I am, I like to peruse the various math related internets. Most of what I find is enlightening, fun (to me at least), or at the very least correct. Every once in a while I'll stumble on a page or facebook group (*massive groan*) that touts something about paradoxes and the weak foundations of mathematics. "Oh really?" I think, sounding interesting I continue reading, thinking they must be referring to all the problems and paradoxes found in set theory (the actual foundations of math). And here I read something like: "Oh math is so crazy, through logical foundations crazy paradoxes result. Here follows a list of paradoxes proving that the foundations of math are shaky at best." Then they list some paradoxes like they promised (they're not jerks after all, just dumb), and most of them are your typical logical paradoxes, you know the ones like: "All bandits are liars," said the bandit. So on and so forth, which are all fine and dandy until they insist on adding this one:
No, math isn't crazy you're just dumb and you know nothing about algebra. Are there paradoxes in math? You bet. Are the foundations of math shaky at best? Possibly, but not because you don't know that you can't divide by zero. That's right, to get line 5 we divide by (a-b), but a = b, thus we were dividing by zero. So this isn't actually a paradox, it's just a case of bad algebra skills. Someone who didn't look at this carefully might have been fooled. Which is a fun trick and all, but what's more disturbing to me is that someone who believes they are bad at math might look at that and surrender "God! Math really doesn't make any damn sense at all, I'm doomed." And that's the problem with these "paradoxes," they imply that math is confusing, when really it's just some moron playing with algebra who hasn't read the instruction manual. And if you're interested there are a ton of these invalid proofs floating around out there, and each one can easily (usually) be debunked if you know the rules.
Therein lies the problem actually. Math does make sense when you know all the rules, the problem is people think they're the shit and plow through algebra without carefully understanding how it all works. Here I imagine a big lummox who's trying to play chess but doesn't know how to move the bishop. So he just plows the piece across the board knocking over everything before declaring "King Me!" I'm not talking about the big picture stuff here, the general how algebra works stuff. I'm talking about the nitpicky stuff, the little things you might not pick up on, but can cause your understanding to completely fall apart in an instant. Take, for example, one of my favorite invalid proofs.
But we also know that
What does this mean? Is 1 = -1? Obviously the answer is no, so which one is right? This one is a little harder to pinpoint, but again it's a case of playing with a tool when you haven't read how it works. Those of you who are nerds like me go get your algebra book. Seriously, go get your algebra book, I can wait. If you don't have an algebra book, then what the hell are you doing here? Ok, got it? Good, now look up the square root. There might be a chapter on it or a section somewhere. You might have to look up something called the product rule for radicals. If you have Cockswold *cough* ... er I mean Rockswold's algebra book then look on page R-55, you'll see the following:
That text at the beginning is just as important as the rule itself. The text tells you when you can use the rule. Think of it as chess, you can only castle your king if you haven't moved your king or your rook at all during the game so far. You can't multiply radicals like in the above rule unless they are defined*. Now, when is the square root defined*? When what's inside it is a real number that's non-negative. So we could actually rewrite this rule to say:
There we go, so this move can only be played if a and b are positive or zero. Therefore the above statement with the square roots of -1 wont work because you're trying to play the move even though you don't satisfy the requirements to do so. You're trying to castle your king when both your rooks have been captured, you're moving your pawn and screaming, "connect four," with confidence, you just don't get it. But you get it enough to confuse a lot of people, you might know how the pieces move, but you don't realize there's no "king me" in chess, or that the pieces must be moved on a board and not your kitchen floor (although kitchen floor chess is pretty damn fun).
So many students see things like this and lament that math is just too confusing and too obtuse. What they're really saying is that there are lots of rules to remember and each rule comes with lots of conditions for its use and they have trouble keeping up with it all. On one hand I can sympathize, but on the other hand, when I think of how many useless things people keep in their brains (baseball statistics, american idol winners since season 1, pin codes) I don't think there's much excuse for not at least trying to get these down in the brain box. All it takes is practice and a suave genius like me to guide you through it all, seriously. That and a little damn to give.
*You know, Rockswold really should have specified that the radicals must be defined Real Numbers, not just defined. Square roots of negative numbers are defined in the Complex numbers, but still the rule doesn't hold. It's little inconsistencies like this that make me hate his stupid book (almost as much as Hughes-Hallet's craptacular book).
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