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# 畢格拉斯怎麼了? (英): 畢格拉斯出了一個問題，跟豆子還有無理數有關。當時是怎麼回事，我不知道。但是根號2是無理數，還有豆子很好吃。

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- 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.

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