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# Fractal fractions

## Video transcript

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.