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• - 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.