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Fractal Fractions: How to make snazzy-lookin' fractal equations using simple algebra. For more abacabadabacaba: http://www.abacaba.org/ and http://books.google.com/books?id=QpPlxwSa8akC&pg=PA60&lpg=PA60&dq=abacabadabacaba+%22martin+gardner%22&source=bl&ots=92eAyvrZGV&sig=J2uvF2DAyn9kY8nSarSy-XIXW74&hl=en&sa=X&ei=Np45T4K9GYixiQK92NG2Bg&ved=0CCEQ6AEwAA#v=onepage&q&f=false
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- Okay so fractal fractions.
- 5 equals 5 okay, bear with me, now lets explode this 5 into 5ths: 25 5ths to be percise
- Now I want to split it into 2 parts, 17 fifths and 8 fifths.
- Could have been 1 plus 24, I don't care. Ok now I'm ready for the fun part
- since this five down here is just as much a five as any five is five, such as this whole thing,
- which is equivalent to five, lets go ahead and replace with that five,
- with this messier looking but still very fivily five
- 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, so you wanna do something with 7, maybe you could use 7 over 7, which is 1,
- so you need 6 more, eeeehhh- why not just add it? There.
- 7 equals 7 over 7 plus 6 now you could replace one, or the other but why not both?
- 7 equals 7 over 7 plus 6
- over 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 sevens kind of disappear but then again it didn't really matter
- what they were in 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 als it's numerator equals it's denominator,
- this fraction equals 1 and whatever else you may think of algebra,
- at least it has the courtesy to make one plus six be seven every time.
- The fractal structure of this first fraction was like a binary tree,
- each layer with twice as many turns as the one above it, growing exponentially.
- And this one does too, but sideways. Bus awsomely enough, this is obviously
- an ABACABADABACABA pattern. That's a fractal pattern that's actually found
- in lots of places, but I'm not going into that right now
- My 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 ABACABADABACABA, and, if your fraction was infinite,
- you'd get ABACABADABACABA-EABACABADABACABA-FABACABADABACABA-EABACABADABACABA-GABACABADABACABA
- and so on. Anyway, a foolish algebra teacher would teach you that algebra
- is about Solving Equations, as if the goal of life is to get X on one side and the rest 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 would still be true
- How's THAT for getting rid of the X on this side?
- And you could 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 equal 1,
- these sixes don't need to be sixes. They could be threes, 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 puzzel:
- What is 1/(1 + 1) but each 1 is 1 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+1) is a half. So the next layer we'll use our half added up to one so this is one over one: One.
- Three layers and you're back to 1/2. Uh-oh. Any whole number of layers is going to give either 1 or 1/2.
- What could this possibly be? Well, you could try doing algebra to it
- Say, all this equals X. Look, you isolated X on one side, and everything else on the other
- and it doesn't help a bit, so take that, math teacher!
- OK but if all this is X, than all this, which is the same as this, equals X
- You can write this as 1 over x plus x which completely works. You can generate it all again
- by replacing X with 1 over 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 want.
- One last fraction, this one with a caution sign. Say you want something to equal one
- Split 1 into 1/2 + 1/2. These ones could be replaced with 1/2+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 layer of two's
- to see if it converges to something, the result is always two for each fraction
- which might make you think that at infinity, it's also 2 for each fraction and therefore, 1 = 4?
- And just looking at this, and trying to think it backwards, you might see 'all this equals x'
- 'and all this equals x' so it's x+x over 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 is. So this could be anything, it's undefined.
- Or, say you want to make something with all ones, like this. Now, x=x+x over 1, or: x = x+x
- You can algebra your way into a contradiction, and as far as algebra's concerned, it's undefined.
- Or you could think "well, there's 2 numbers I know that fit this description: zero and infinity...
- ...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 where the denominator got lost in infinity, you can still solve it back to 5
- That, to me, is the cool part of algebra. Unlike the neat little problems they put in gradeschool 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 has ever done before that you can do with the simplest concepts
- as simple as that x is, what x is
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