Fractal fractions

OK, so fractal fractions. 5 equals 5. Bear with me now. Let's explode this 5 into fifths. 25 fifths to be precise. Now I want to split it into two parts. Say, 17/5 plus 8/5. Could have been 1 plus 24, or whatever, I don't care. OK, now I'm ready for the fun part. Since this 5 down here is just as much a 5 as any 5 is 5, such as this whole thing, which is equivalent to 5, let's go ahead and replace that 5 with this messier looking but still very fively 5. This one too. Oh, and now we've got more fives, so we can do it again. And again. And again. And then you can give someone this whole thing and be like, whoa, it's 5. Making things look more confusing than they actually are is a delicate art, but it speaks to the true heart of algebra, which is that you can shuffle numbers around all day and as long as you follow the rules, it all works out. OK, here's another one. Say you want to do something with 7. Maybe you could use 7/7, which is 1. So you need six more. Why not just add it? There. 7 equals 7/7 plus 6. Now you could replace one or the other, but why not both? 7 equals 7/7, plus 6/7, over 7 plus 6, plus 6. Instead of writing it all again, why not just extend this way? There we go. This equals 7. And you can actually take this all the way to infinity. The 7's kind of disappear. But then again, it didn't really matter what they were the first place, as long as they're the same. All these 7's could have been 3, or a billion, or pi to the i, and this would still equal 7. As long as this numerator equals this denominator this fraction equals 1. And whatever else you may think about algebra, at least it has the courtesy to make 1 plus 6 be 7 every time. The fractal structure of this first fraction was like a binary tree. Each layer with twice as many terms as the one above it, growing exponentially. And this one does too, but sideways. But awesomely enough, this is obviously an ABA CABA DABA CABA pattern. That's a fractal pattern that's actually found lots of places, but I'm not going to get into that right now. Point is, if you name this innermost layer A, and the next B, and the next C, and the next D, and then try to read it from top to bottom, you get ABA CABA DABA CABA. And if your fraction was infinite, you'd get ABA CABA DABA CABA EABA CABA DABA CABA FABA CABA DABA CABA EABA CABA DABA CABA GABA CABA-- and so on. Anyway, a foolish algebra teacher would teach you that algebra is about solving equations. As if the goal of life were to get x on one side, and everything else on the other. As if every fiber of your being should cry out in protest when you see x on the left side, and yet more x on the right side. But you could replace that x with what it equals, and then you could do it again, and again, and each time your equation is still true. How's that for getting rid of the x on this side? And you can make equations even more confusing by remembering special numbers and identities. Write whatever you want, as long as you can sneak in a multiplied by 0, you don't even have to bother knowing what the rest is. Or, knowing that all you need is the top and bottom of the equation to be the same to get 1. These 6's don't need to be 6's, they could be 3's. Or 8 square root 13. Or you could even make each layer different. 7, 8, 9, 10, 11. Now look how confusing this is. Awesome. Say you wanted to actually solve one of these things. Say you started with this puzzle. What is 1/1 plus 1, but each 1 is over 1 plus 1? And so on, all the way to infinity. You could try doing it by hand, thinking maybe it'll converge on something. 1/1 plus 1 is 1/2. So the next layer, these are 1/2, add up to 1. So this is 1/1, 1. Three layers, back to 1/2. Uh oh. Any whole number of layers is going to give either 1 or 1/2. So what could this possibly be? Well, you could try doing algebra to it. Say all this equals x. Look, you've isolated x on one side, and everything else on the other, and it doesn't help one bit. Take that math teacher. OK, but if all this is x, then all this-- which is the same as all this-- is x. You can write this as 1/x plus x, which completely works. You could generate it all again by replacing x with 1/x plus x. And now that you've got those helpful x's on the wrong side of the equation, you can solve it and get the boring way to write this number if you wanted. One last fraction, this one with a caution sign. Say you want something to equal 1. Split 1 into 1/2 plus 1/2. Now these 1's could be replaced with 1/2 plus 1/2. Each time you do this, it works. What happens if you go to infinity? It's weird because if you look at any number of layers of 2's, to see if it converges to something, the result is always 2 for each fraction. Which might make you think that at infinity it's also 2 for each fraction, and therefore 1 equals 4? And just looking at this and trying to take it backwards, you might say, all this equals x, and all this equals x. So it's x plus x/2. Just try and solve that equation. The problem is, half of something plus half of something always equals that something, no matter what the x. So this could be anything, it's undefined. Or say you want to make something with all 1's like this. Now x equals x plus x/1, or x equals x plus x. You can algebra your way to a contradiction, and as far as algebra is concerned, this is undefined. Or you could think, well, there's two numbers I know that fit this description, infinity and 0. This I suppose could be either, or both at once, or nothing at all. I don't know. Why does it do that? Maybe because the numerator got lost up there, and could have been anything. Interesting though that even here, when the denominator got lost in infinity, you can still solve this back to 5. That's, to me, the cool part about algebra. Unlike the neat little problems they put in grade school textbooks, not all problems can be solved, and it's not always obvious when there's an answer and when there's not. Weird stuff happens all the time. And most importantly, algebra isn't a dead ancient thing. There are things no one's ever done before, that you can do with the simplest concepts. As simple as that x is what x is.