The Discovery That Transformed Pi

• 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 circle's 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 circle's circumference, so Pi must be less than eight over two. So Pi is less than four. This was actually known for 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 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 contributing to these bounds along Archimedes' 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 outdone 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. Twenty 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 is 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, 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 number 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 will 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 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 insists 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 have 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 ten 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. 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 zeroth. 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.

But Newton doesn't stop with the integers; next he tries fractional powers like one plus X to the half. So now what does it mean? You take one plus X to the one half. Well, that's the same thing as the square root of one plus X. And he wants to understand does that have the same expansion.

Putting N equals a half into the binomial theorem, he gets an infinite series.

• That makes me think that we could actually go into Pascal's triangle, blow it up, and add fractions in between the rows that we're familiar with.

• Exactly! There's even a continuum of Pascal's triangles; between zero and one, there's this continuum of numbers that you could put in for powers.

And you can think of each fraction, like a half, a quarter, a third, as existing in its own plane where in each plane pairs of numbers add to make the number beneath them. And it doesn't have to be a positive integer anymore.

• It doesn't have to be a positive integer; it doesn't have to be a negative integer; it doesn't have to be an integer. So now we're going to take N to be a half, and he works this thing out, and then he can do all kinds of things.

For example, he could work out the square root of three very quickly and efficiently because the square root of three we can write three as four minus one. And if we pull out a four, then we get a square root of four, which is just two times the square root of one minus a quarter.

If you put in minus a quarter for X in this series, you'll get a very rapidly converging series expansion that will quickly give you the square root of three to high accuracy. Now, Newton is particularly interested in N equals a half because the equation for a unit circle is X squared plus Y squared equals one.

And if you solve for Y, well the top part of the circle is equal to one minus X squared to the half. This is basically the same expression he's been looking at; he just has to replace X by minus X squared, which adds in some minus signs and doubles the power of X on each term.

But now he's got an equation for a circle where each term is just irrational number times X raised to some power. Now we have two different ways of representing the same thing. Whenever you have something like that, magic is about to happen; fireworks is about to go off.

• But how does he use this to calculate Pi?

• Well, luckily for us, he had just invented calculus, or what he called the theory of Fluxions. He realizes that if you integrate under that curve as X goes from zero to one, you're getting the area under the curve, which is a quarter circle.

And he knows that the area of a unit circle is exactly Pi R squared, except R is one, so the area is Pi. And we want just a quarter, so the area is Pi over four. On the other side, he has this nice series and he knows how to integrate X to some power. You just increase each power of X by one and divide by the new power.

And now you have an infinite series of terms which just involve simple arithmetic with fractions. You put an X equals one, and you can calculate Pi to arbitrarily high precision. But Newton goes even further adding one final tweak.

A not good math paper has zero ideas; it's just pushing through things that everybody already knows, but nobody bothered to do. Then there are good math papers that have one new idea that's really shockingly new. Newton's on new idea number four at this point, and he's about to have new idea number five.

And new number five is instead of integrating from zero to one, he's gonna integrate just from zero to a half. When you have an infinite series, you want the terms to decrease in size as fast as possible. That way you don't have to calculate as many of them to get a pretty good answer.

And Newton sees if he integrates not from zero to one, but from zero to a half, then when he subs in a half for X, each term will shrink in size by an additional factor of X squared, which in this case is a quarter.

But if you only integrate to a half, what is the area under the curve that you're computing? Well, it is this part of a circle, which you can break into a 30-degree sector of the circle which has an area of Pi on 12 plus a right triangle with a base of a half and a height of root three on two. So that integral should come out to this expression.

And rearranging for Pi, you get the following. Now, if you evaluate only the first five terms, you get Pi equals 3.14161; that's off by just two parts in a 100,000. And to match the computational power of Van Ceulen's four quintillion-sided polygon, you would only need to compute 50 terms in Newton's series.

What, before it took years, now would take only days? So no one was bisecting polygons to find Pi ever again. Why would you? Yeah, you do all that work, and somebody comes along and beats you in a second. It's sort of like once someone builds a crane, and then somebody else is still climbing up on a ladder to put a brick on a house; that's just not how you build houses anymore.

We have new technology; are you out of your mind? We're gonna build a 100-story house; we're gonna build a five-story thing that's gonna fall over. You see it in New York City; you see literally where technology came along. There's rows and rows of five-story buildings, and all of a sudden, here's a 20-story, and here's a 30-story, and here's a 90-story. So it's all about who has the technology.

For me, this is a story about how the obvious way of doing things is not always the best way and that it's often a good idea to play around with patterns and push them beyond the bounds where you expect them to work. Because a little bit of insight and mathematics can go a very long way.

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