Ok, so I've been learning about Pythagoras
and the dirt on him is just too good.
You've probably heard of the Pythagorean Theorem
but not the part where Pythagoras
was a crazy cult leader who thought he'd made
a deal with a god thousands of years ago
and could remember all of his past lives.
Oh, and he killed a guy.
I mean maybe it was a long time ago and he was afraid of beans.
As in beans, they just like freaked him out
or something, I don't know.
But mostly I want to talk about the murdery part.
See, Pythagoras and his cult of Pythagoreans
had this cool-kids club where they'd
talk about proportions all day.
They'd be like, "Hey, I drew a two by three
rectangle using a straight-edge and compass.
Isn't that awesome?"
And then someone would be like,
"Hey guys, I have a box that's two by three and a half?"
And the cool kids would be like,
"Three and a half? That's not a number!
Get out of our club!"
And then they'd make the units half the length
and call it four by seven, and everything was okay.
Even if your box is 2.718 by 6.28
you can just divide your units into thousandths
and you'd get a box that's a nice, even
2,718 by 6,280. It's not a simple proportion
but hey, the box still has a whole number proportion,
so Pythagoras is happy.
Unless it's a box of beans, then he freaks out.
I'd like to imagine what it would be like to
think of numbers the way they did.
Maybe you think of numbers as being on a line.
Numbers one way, zero, and negative numbers the other,
and there are numbers between them:
fractions, rationals, filling in the gaps.
But Pythagoras didn't think about numbers
like this at all.
They weren't points in a continuum
they were each their own, separate being,
which was still pretty modern because
before that people only thought of numbers as adjectives,
numbers of. In Pythagoras's world, there is
no number between seven and eight,
and there is no number three over two so much as
a relationship between three and two, a proportion.
Six to four has the same relationship
because the numbers share this evenness,
which when accounted for makes it three to two.
The universe to Pythagoras was made up
of these relationships. Mathematics wasn't numbers,
mathematics was between the numbers.
Though while people admire how much Pythagoras
loved proportions, there's a dark
flip side to that obsession.
How far was he willing to go to protect
the proportions he loved?
Would he kill for them?
Would he die for them?
And the answer was he'd go pretty far
until beans got involved.
It's time for Time Line Time.
Mathematicians like lines.
I want the context, because in school today
if you bring out the ruler and compass
and are like, "Let's do some geometry!
Let's draw two lines at 90 degree angles
using a straight-edge and compass!
Here's a happy square!"
Then you've probably had years of math class
already and think of geometry as being harder
than adding big numbers together.
You probably think that zero is a simple, easy concept
and have heard of decimals too.
Well, here's now, 2012.
Here's Einstein, Euler, Newton and Da Vinci -
- that sure was a while ago!
Now let's go all the way back to when
Arabic numerals were invented and
brought to the West by Fibbonacci.
Before that, arithmetic was nightmarishly hard,
so if you can multiply multi-digit numbers together
you can go back in time and impress
the beans out of Pythagoras.
And before that there was no concept of
zero, except in India where zero
was discovered around here.
And if you keep going back you get to the year one,
(there's no year zero, of course,
because zero hadn't been invented)
and back a bit more you get to folk like
Aristotle, Euclid, Archimedes
and then finally Pythagoras,
all the way back in 6th century BCE.
Point is, you can do some pretty cool mathematics
without having a good handle on arithmetic
and people did for a long time.
And in school when they tell you
you need to memorize your multiplication
table and graph a parabola
before you can learn real mathematics,
they are lying to you.
In Pythagoras's time there were no variables,
no equations or formulas like we see today,
Pythagoras's theorem wasn't
'a squared plus b squared equals c squared,'
it was 'The squares of the legs of a right triangle
have the same area as the square of the hypotenuse,'
all written out.
And when he said 'square' he meant 'square.'
One leg's square plus the other leg's square
equals hypotenuse's square.
Three literally squared plus four made into a square.
Those two squares have the same area as
a five by five square.
You can cut out the nine squares here
and the sixteen here and fit them together
where these 25 squares are, and in the same way,
you can cut out the 25 hypotenuse squares
and fit them into the two leg squares.
Pythagoras thought you could do this trick
with any right triangle,
that it was just a matter of figuring out
how many pieces to cut each side into.
There was a relationship between the length
of one side and the length of another
and he wanted to find it on this map.
But the trouble began with the simplest right triangle
one where both the legs are the same length,
one where both the legs' squares are equal.
If the legs are both one then the
hypotenuse is something that, when squared,
gives two. So what's the square root of two and
how do we make it into a whole number ratio?
Square root two is very close to 1.4
which would be a whole number ratio of 10:14
but 10 squared plus 10 squared is definitely
not 14 squared, and a ratio of 1,000 to
1,414 is even closer, and a ratio of
100,000,000 to 141,421,356 is very close indeed
but still not exact, so what is it?
Pythagoras wanted to find the perfect ratio
he knew it must exist, but meanwhile
someone from his very own Pythagorean brotherhood
proved there wasn't a ratio, the square root
of two is irrational, that in decimal notation
(once decimal notation was invented)
the digits go on forever. Usually this proof
is given algebraically, something like
this, which is pretty simple
and beautiful if you know algebra,
but the Pythagoreans didn't.
So I like to imagine how they thought
of this proof, no algebra required.
Okay, so Pythagoras is all like,
"There's totally a ratio, you can
make this with whole numbers."
And this guy's like,"Is not!" "Is too!"
"Is not!" "Is too!" "Fine have it your way.
So, there's a whole number ratio in simplest
form, where this square plus this square equals
this square." "Yeah, that's the Pythagorean
theorem, I made it." "Yeah, though for this triangle
you don't even need the full theorem.
it's easy to see that it's the same area
by cutting each part into four triangles."
"But I don't want to divide the squares up
into triangles, I want unit squares."
"So kind of like this, where this square is
divided into units and so is this one and
they all fit perfectly into this one and vice versa,
but not like this. It almost works, but you
start dividing this square evenly to fill up
the two equal other squares, and you've got
this one odd one out. There's an odd number
of squares to begin with, so you can't
divide them evenly between the two squares."
"That's not even a right triangle, what's your point?"
"Just that you know an odd number like seven
isn't gonna be it without even trying.
An odd number times itself gives an odd number
of squares, so whatever this number is,
it can't be seven, it has to be even."
"Ok, so the hypotenuse is even, that's fine."
"So what if I proved the leg is even too?"
"Then it's not in simplest form."
"Any ratio where both are even you divide by two until
you can't divide anymore, because one of them is odd
and then that ratio is the best. I thought
we assumed we were talking about the simplest form ratio."
"We are. If there's a ratio in simlest form at least
one of the numbers is odd and since the hypotenuse has to
literally be divisible by two, then the leg must be the odd one.
So what if I proved the leg had to be even?"
"You just proved it's not. It can't be both."
"Unless it doesn't exist!
What you forget Pythagoras is that if this is a square
then the two sides are the same. Just as this is divsible
right down the center so too is it divisible the other way!
And the number squares on this side, which are
the number of squares in just one leg is an even number.
And for a number of squares to be even what
does the number have to be, Pythagoras, oh my brother?"
"If leg squared is even then the leg is even.
But it can't be even, because it's already odd."
"Unless it doesn't exist."
"But if they're both even you can divide both
by two and start again, but this still has to be even
which means this still has to be even, which means
you can divide by two again, but then it has to be even
so everything is even forever and you never find
the perfect ratio. Aww, beans"
He had a vision, a beautiful vision of a world
made up of relationships between numbers.
If this wasn't a whole number ratio, then what was it?
The Pythagoreans still believed, wanted to believe
that irrationality was somehow false and the world
was as they wanted it. So this proof stayed secret.
Until someone spilled the beans.
According to some, it was all a guy named Hippasus
and Pythagoras threw him of a boat to drown him
as punishment for ruining what had been perfect.
Or maybe it was someone else who discovered it
or Hippasus or someone else who was killed
by the Pythagoreans long after Pythagoras was
dead or maybe they just got exiled, who knows?
And how did Pythagoras die?
Well, according to one guy some guys got mad
because they didn't get into the cool kids' club.
So they set Pythagoras' house on fire.
And Pythagoras was running away and they
were chasing him, but then they came upon a field
and not just any field, but a field of beans.
And Pythagoras turned around to face his pursuers
and proclaimed: "Better to be slaughetered
by enemies than to trample on beans!"
And he was. Others say he ran off and starved
himself to death. Or just got caught by his enemies
because he ran around the bean field instead of through it
or who knows what happened. People claim Pythagoras
didn't like beans because he thought they were
bad for digestion, or gave you bad dreams
or reminded him of male genitalia or
because he didn't want a clubhouse full of
flatulating mathematicians or he just didn't
like them metaphorically.
He and his followers were or weren't vegetarian
did or didn't sacrifice animals
possibly were only allowed to eat certain colors of birds
I mean he definitely had a lot of rules to follow
but just what they were and what they meant
is lost to history.
I'd like to give you a colorful story about exactly
what happened with Pythagoras, but somehow
that kind of truth doesn't last.
What I do know is that the square root of two is
irrational, that there's no way to have
the length of a side of a square and of the
square's diagonal both be whole numbers.
Mathematical truth is truth that indures.
This proof is just as good now as it was
2500 years ago, I mean it's awesome and
it shows that there's more to the
world than whole numbers and
shame on the Pythagoreans who didn't have
the beans to admit it.