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 P: 4 Have you ever noticed that when you get 0 involved in even the simplest multiplicative operations, it manages to screw them up? For instance, take the functions f(x)=1/x and g(x)=xx. As for f(x)=1/x, f(0)=1/0, an expression that, from one side of the equation's graph, appears to represent positive infinity, but from the other side appears to represent negative infinity. And what about g(x)=xx? That seems to be a pretty straightforward operation. Yet, g(0)=00; conventionally x0=x/x, but 0/0's value is context-dependent, and no context was set by use of g(0); hence, g(0) could mean anything depending on its own context, but by using it in two different contexts, we could get, say, g(0)=2 and g(0)=3, and since 2$$\neq$$3, one could conclude by substitution that 0/0$$\neq$$0/0. But what about just using 2 or just using 3? 2=2 and 3=3, so substituting from either of these cases, we could find that 0/0=0/0. So there you have it: an expression that looks like two completely different terms depending on what side you approach it from and an expression that both does and does not equal itself. Weird, right? Why is it that all the operations in math that are illegal, that have no answer at all, all seem to trace back somehow to li'l ol' harmless 0? If so many things related to 0 are illegal operations, might it then be possible that in mathematics that run completely parallel to the functioning of the universe, 0 itself should not exist? Back when 0 was first invented, I'm certain it was during the early years of mathematics, where it had much more to do with the real world, back when it was used for tallying; 0 was probably used to represent, for example, the complete absence of oxen to pull carts, or chickens to lay eggs, or maybe just bananas. But is there ever a complete absence of bananas? Isn't it possible that no matter how many bananas you take away, there's always just a little tiny bit of banana left over? You can't really get rid of bananas completely; they're dropping atoms and molecules all over the place with each step you take while carrying them. Parts of themselves. Fractions of bananas. Maybe even if you try to take away parts of those atoms and molecules, there will always be little bits of them left. Maybe everything trails infinitesimal pieces of itself wherever it goes. Why does there have to be a 0, a point where is absolutely nothing? What if there's never absolutely none of something? If that's the case, 0 shouldn't even exist, and if it does, it's a mathematical concept with no real-world parallel. Of course, all of this is hypothetical; if I knew any of this for sure, I wouldn't even be talking about it here. You have 6 dollars. You want to split the money amongst no friends. How much does each friend get? Well, obviously, 6/0 dollars. But let's just talk about division by things less than 1 in general: You have 6 dollars. You want to split the money amongst half a friend. Yeah, I know, it's gross, but let's just say you're a creepy serial killer guy. How much does the half-friend get? 6/0.5 is 12, but the half-friend can't possibly get 12 dollars; you don't even have that much. Analyzing things this way, I see that dividing money up amongst friends when you have less than 1 friend isn't really what it seems; sure, it's giving an equal amount of money to each friend, but when we have less than one friend, we must consider the definition of "each friend." If you have a half a friend and you're giving the half-friend some amount of money, then if you had 1 friend, that friend would be two half-friends, so you'd be giving that friend twice the amount of money as you'd give a half-friend (to be fair to the other two half-friends that make up the 1 friend). If you only have 6 dollars, of course you're only going to give the half-friend 6 dollars; that's all you've got, and the half-friend is your only half-friend that's there. That means that if you had 1 friend, you'd give that 1 friend 12 dollars. But even though you don't have 12 dollars, you don't have 1 friend. So you don't have to worry. So now that I've demonstrated that if you have a half-friend and 6 dollars, he doesn't get 12 dollars, he gets 6, let's say you have no friends. You're a sad little fellow, aren't you? Each of your 0 friends gets 6 dollars. There you go, 0 friends. They get 6 dollars. You lay the 6 dollars down at the location of your 0 friends. Nothing happens. No-one takes it. As you stare into the money, it seems to stare back, asking you to take it back. But you don't. Now it belongs to your 0 friends. If you ever have 1 friend, it's only right to give every instance of 0 friends in your 1 friend 6 dollars (to be fair). How many 0 friends are in your 1 friend? 1/0. Multiply that value by 6 and find 6/0. This means that if every group of 0 friends in your 1 friend gets 6 dollars, your friend should get 6/0 dollars. The problem is, no matter how many groups of 0 friends you have, you still don't have a single friend. You poor soul. You must be so depressed constantly putting together groups of 0 friends. You know, Einstein once said that insanity is defined as doing the same thing over and over again and expecting a different result. Like you're doing. You poor, insane soul. But wait! Have you ever been near people? Hey, at least you've been near yourself. To not be near yourself is impossible. Everywhere you go, you drop cells, little tiny portions of friends. And even if nothing drops cells, things might drop atoms. And even if nothing drops an atom, who says things can't drop electrons? Or quarks? Or really, really, REALLY freaking small things that you can't see? So maybe you're never alone. Perhaps there's a faint glimmer of hope for you after all. See what I mean? Maybe we're never alone. Maybe there's never no bananas to eat, no chickens to lay eggs, no oxen to pull the cart, no friends to give out money to. Maybe 0 is nothing but a concept with no real-world truth behind it. ... Of course, that means that you can never take everything away from something. And if you can never take everything away from something, that must mean that subtracting a number from itself is also nothing but a concept with no real-world truth behind it. That's... kind of lame if you ask me. Come to think of it, I guess even if 0 is just a concept with no actual real-world counterpart, I bet nobody will care. Even if there can't ever be none of anything, pretending that there can seems to be working well enough for us all.