Placeholder Image

字幕表 動画を再生する

  • - This video is about the ridiculous way

  • we used to calculate Pi.

  • For 2000 years the most successful method

  • was painstakingly slow and tedious,

  • but then Isaac Newton came along and changed the game.

  • You could say he speed-ran Pi

  • and I'm gonna show you how he did it.

  • But first Pi with pizzas.

  • Cut the crust off of pizza

  • and lay it across identical pizzas.

  • And you'll find that it goes across three and a bit pizzas,

  • this is Pi.

  • The circumference of a circle is roughly 3.14 times

  • its diameter but Pi is also related to a circles area,

  • area's just Pi R squared.

  • But why is it Pi R squared?

  • Well cut a pizza into really thin slices

  • and then form these slices into a rectangle.

  • Now the area of this rectangle is just length times width.

  • The length of the rectangle is half the circumference

  • because there's half the crust on one side

  • and half on the other,

  • so the length is Pi R.

  • And then the width is just the length of a piece of pizza

  • which is the radius of the original circle.

  • So area is Pi R times R,

  • area is Pi R squared.

  • So the area of a unit circle then is just Pi,

  • keep that in mind because it'll come in handy later.

  • So what was the ridiculous way we used to calculate Pi?

  • Well, it's the most obvious way.

  • It's easy to show that Pi must be between three and four,

  • take a circle and draw a hexagon inside it,

  • with sides of length one.

  • A regular hexagon can be divided

  • into six equal lateral triangles.

  • So the diameter of the circle is two.

  • Now the perimeter of the hexagon is six

  • and the circumference of the circle

  • must be larger than this,

  • so Pi must be greater than six over two.

  • So Pi is greater than three.

  • Now draw a square around the circle,

  • the perimeter of the square is eight

  • which is bigger than the circles circumference,

  • so Pi must be less than eight over two.

  • So Pi is less than four.

  • This was actually known for a thousands of years.

  • And then in 250 BC, Archimedes improved on the method.

  • - So first he starts with the hexagon, just like you did

  • and then he bisects the hexagon to dodecagon.

  • So that's a 12 sided, regular 12 sided shape.

  • And he calculates its perimeter,

  • the ratio of that perimeter to the diameter

  • will be less than Pi.

  • He does the same thing for a circumscribed 12-gon

  • and finds an upper bound for Pi.

  • The calculations now become a lot more tricky

  • because he has to extract square roots

  • and square roots of square roots

  • and turn all these into fractions,

  • but he works out the 12-gon, then the 24-gon, 48-gon

  • and by the time he gets to the 96-gon

  • he sort of had enough, but he gets,

  • in the end he gets Pi to between 3.1408 and 3.1429.

  • So for over 2000 years ago, that's not too bad.

  • - Yeah, that seems like all the precision you'd need in Pi.

  • - Right, so this goes way beyond precision

  • for any practical purpose.

  • This is now a matter of flexing your muscles.

  • This is showing off just how much mathematical power

  • you have, that you can work out a constant like Pi

  • to very high precision.

  • So for the next 2000 years, this is how everyone

  • carried on bisecting polygons to dizzying heights

  • as Pi passed through Chinese, Indian, Persian

  • and Arab mathematicians, each contributed to these bounds

  • along our committee's line.

  • And in the late 16th century, Frenchman Francois Viete

  • doubled a dozen more times than Archimedes,

  • computing the perimeter of a polygon with 393,216 sides

  • only to be out done at the turn of the 17th century

  • by the Dutch Ludolph van Ceulen.

  • He spent 25 years on the effort computing to high accuracy

  • the perimeter of a polygon with two to the 62 sides.

  • That is four quintillion, 611 quadrillion, 686 trillion,

  • 18 billion, 427 million, 387,904 sides.

  • What was the reward for all of that hard work?

  • Just 35, correct decimal, places of Pi.

  • He had these digits inscribed on his tombstone,

  • 20 years later, his record was surpassed

  • by Christoph Grienberger who got 38, correct decimal places.

  • - But he was the last to do it like this

  • - Pretty much.

  • Yeah, because shortly thereafter we get Sir Isaac Newton

  • on the scene.

  • And once Newton introduces his method

  • nobody is bisecting n-gons ever again.

  • The year was 1666 and Newton was just 23 years old.

  • He was quarantining at home

  • due to an outbreak of bubonic plague.

  • Newton was playing around with simple expressions

  • like one plus X, all squared.

  • You can multiply it out and get one plus two X

  • plus X squared.

  • Or what about one plus X all cubed?

  • Well, again, you can multiply out all the terms

  • and get one plus three X plus three X squared plus X cubed.

  • And you could do the same for one plus X to the four

  • or one plus X to the five and so on.

  • But Newton knew there was a pattern that allowed him

  • to skip all the tedious arithmetic

  • and go straight to the answer.

  • If you look at the numbers in these equations

  • the coefficients on X and X squared and so on,

  • well, they're actually just the numbers

  • in Pascal's triangle.

  • The power that one plus X raised to

  • corresponds to the row of the triangle

  • And Pascal's triangle is really easy to make,

  • it's something that's been known

  • from ancient Greeks in Indians and Chinese Persians,

  • a lot of different cultures discovered this.

  • All you do is whenever you have a row

  • you just add the two neighbors

  • and that gives you the value of the row below it.

  • So that's a really quick easy thing you can compute

  • the coefficients for one plus X to the 10 in a second

  • instead of sitting there doing all the algebra.

  • - The thing that fascinated me when I started looking

  • at those old documents was how even like,

  • I don't speak those languages,

  • I don't know those numbers systems

  • and yet it is obvious,

  • it is clear as day that they're all

  • writing down the same thing

  • which today in the Western world, we call Pascal's triangle.

  • - That's the beauty of Mathematics.

  • It transcends culture, it transcends time,

  • it transcends humanity.

  • It's gonna be around well after we're gone

  • and ancient civilizations,

  • alien civilizations we'll know Pascal's triangle.

  • Over time, people worked out a general formula

  • for the numbers in Pascal's triangle.

  • So you can calculate the numbers in any row

  • without having to calculate all the rows before it,

  • for any expression one plus X to the N

  • it is equal to one plus N times X

  • plus N times N minus one X squared on two factorial

  • plus N times N minus one times and N minus two times X cubed

  • on three factorial and so on.

  • And that's the binomial theorem.

  • So binomial, because there's only two terms,

  • one in X by is two, there's two normals and a theorem

  • is that this is a theorem that you can rigorously prove

  • that this formula is exactly what you'll see

  • as the coefficients in Pascal's triangle.

  • - [Alex] So all of this was known in Newton's day already.

  • - Yeah, exactly, everybody knew this.

  • Everybody saw this formula

  • and yet nobody thought to do with it

  • the thing that Newton did with it

  • which is to break the formula.

  • The standard binomial theorem insist that you apply it

  • only when N is a positive integer,

  • which makes sense.

  • This whole thing is about working out one plus X

  • times itself a certain number of times,

  • but Newton says, screw that just apply the theorem.

  • Math is about finding patterns and then extending them

  • and trying to find out where they break.

  • So he tries one plus X to the negative one.

  • So that's one over one plus X.

  • What happens if I just blindly plug in N equals negative one

  • for the right-hand side of the formula?

  • And what you get is the terms alternate back and forth.

  • Plus one minus one, plus one minus one, and so on forever.

  • So that's one minus X,

  • the next term will be a plus X squared,

  • the next one will be a minus X cubed

  • plus X to the fourth minus X to the fifth.

  • So that just alternating series with plus and minus signs

  • as the coefficient.

  • - [Derek] So it becomes an infinite series.

  • - Yeah, that's right.

  • If you, don't a positive integer the binomial theorem,

  • Newton's binomial theorem will give you an infinite sum.

  • - But how do you understand that?

  • Like for all positive integers

  • it was just a finite set of terms

  • and now we've got an infinite set of terms.

  • - Yeah, so what happens is if you have a positive integer

  • you remember that formula,

  • the coefficient looks like N times N minus one

  • times N minus two and so on,

  • when you get to N minus N, if N is a positive integer,

  • you will eventually get there

  • and N minus 10 is zero.

  • So that coefficient and all the coefficients after it

  • are all zero and that's why it's just a finite sum,

  • it's a finite triangle.

  • But once you get outside of the triangle

  • with positive integers, you never hit N minus N

  • because N is not a positive integer,

  • so you get this infinite series.

  • - [Derek] So I think the big question is,

  • does this actually work?

  • Does Newton's infinite series actually give you the value

  • of one over one plus X?

  • - Right, and it might be nonsense.

  • There's lots of math formulas that could break completely

  • when you do this.

  • There's, we have rules for a reason

  • but we should always know the extent to which the rules

  • have a chance of working farther.

  • If you take that whole series and you multiply it

  • by one plus X and you multiply all that out

  • you'll see all the terms cancel, except that leading one.

  • And so that big series times one plus X is one.

  • In other words, that big series is one over one plus X,

  • that's how Newton justified to himself

  • that it makes sense to apply the formula

  • where it shouldn't be applicable.

  • So Newton is convinced the binomial theorem works

  • even for negative values of N,

  • which means there's more to Pascal's triangle

  • above the zeroeth you could add a zero and a one

  • that add to make that first one.

  • And then that row would continue minus one, plus one,

  • minus one, plus one, all the way out to infinity.

  • And outside the standard triangle

  • the implied value everywhere is zero.

  • And this fits with that.

  • The alternating plus and minus ones add to make zero

  • everywhere in the row beneath them.

  • And you can extend the pattern for all negative integers

  • either using the binomial theorem

  • or just looking at what numbers would add together

  • to make the numbers underneath.

  • And here's something amazing.

  • If you ignore the negative signs for a minute

  • these are the exact same numbers arranged

  • in the same pattern as in the main triangle.

  • The whole thing has just been rotated on its side.