Kind: captions
Language: en
this video is about the ridiculous way
we used to calculate pi
for 2 000 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 pie and i'm going to show you
how he did it but first pie with pizzas
cut the crust off a 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 is 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 sort of 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 equilateral 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 uh 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-gun 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 and turn all these into fractions
but uh he works out the 12 gone and then
the 24 gone uh 48 gone and and by the
time he gets to the 96 con 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 need in pi right so this goes way
beyond precision for any practical
purpose this is now a matter of uh
flexing your muscles you know this is
this is uh 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 2 000 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
archimedes line
and in the late 16th century frenchman
francois viet 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 ludov van
kerlen 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 greenberger 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 was raised to
corresponds to the row of the triangle
right and pascal's triangle is really
easy to make it's something that's been
known from ancient greeks and 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 you know 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 are 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 going to be
around well after we're gone and 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 1 plus x to the n it
is equal to 1 plus n times x plus
n times n minus 1 x squared on 2
factorial plus n times n minus 1 times n
minus 2 times x cubed on 3 factorial and
so on and that's the binomial theorem so
binomial because there's only two terms
one in x bi is two there's two nomials
and uh theorem is that this is a theorem
that you can rigorously prove that this
formula is exactly what you'll what
you'll see
as the coefficients in pascal's triangle
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 right
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 i mean
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 1 for the right hand
side of the formula and what you get is
the terms alternate back and forth plus
1 minus 1 plus 1 minus 1 and so on
forever so that's one minus x the next
term will be a plus x squared the next
term will be a minus x cubed plus x to
the fourth minus x to the fifth so it's
just alternating series with plus plus
and minus signs as the coefficients 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 1 times n minus 2 and so
on when you get to n minus n if n is a
positive integer you will eventually get
there and n minus n is 0. 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 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 break
completely when you do this right
there's we have rules for a reason but
uh 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 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 row 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 you know 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 1
plus x to the half
so now what does it mean you take 1 plus
x to the one half well that's the same
thing as square root of 1 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 you know a continuum of
numbers that you could put in for powers
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 n doesn't have to be a
positive integer anymore 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
could do all kinds of things for example
he could work out the square root of 3
very quickly and efficiently because the
square root of 3 we can write 3 as 4
minus 1
and if we pull out a 4 then we get a
square root of 4 which is just 2 times
the square root of 1 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
1 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 a rational
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 about 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
flexions 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
x2 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 in 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 like one new
idea that's like really shockingly new
newton's on new idea number four at this
point and he's about to have new idea
number five a new idea number five is
instead of integrating from zero to one
he's going to integrate just from zero
to a half you know 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 0 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
twelve
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
hundred thousand
and to match the computational power of
van curlin'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
pie ever again
why would you yeah do you do all that
work and somebody comes along and beats
you in a second it's sort of like uh you
know once 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 you're gonna you know we're
gonna build a hundred story house you'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 like 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 in
mathematics can go a very long way
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