- 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.
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 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 a 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 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 gonna take N to be a half
and he works this thing out
and then he could do all kinds of things.
For example, he could work out the square root of three
very quickly and efficiently 'cause the square root of three
we can write three is four minus one.
And if we pull out a four, then we get a squire root 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 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.
And 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 an 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
four quintillion sided polygon,
you would only need to compute 50 terms in Newton 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 a, once someone builds a crane
and then somebody else is still climbing up on a ladder
to put a brick on a house,
like 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 we're 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 the 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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