Welcome to another Mathologer video.
Today’s mission is to make sense of Powell’s Pi Paradox. Never heard of it?
Trust me, it’s a good one:) Also, the main tools that I’ll be using to resolve this paradox
are some amazing bits of half-forgotten medieval mathematics discovered by Indian mathematicians
more than 600 years ago. Lots to look forward to today. Okay, let me get straight into it
and introduce you to Powell’s Pi paradox. :) I’m sure most regular Mathologerers will be
familiar with this mathematical gem. A formula for Pi built from the odd numbers, without a circle
anywhere in sight. Pi over 4 =1 - 1/3 + 1/5 - 1/7, etc. Absolutely beautiful. And the Mathologer
regulars will probably also remember that I've already presented a couple of stunning proofs
of this formula, in earlier videos. Today we’ll do something totally different. Let’s start by
multiplying both sides of this identity by 4. Now we can use the infinite sum on the right to
compute arbitrarily many digits of Pi by adding more and more of the terms. Okay, so let’s
see what we get when we sum the first million terms. In particular, let’s compare Pi with that
monster sum. Here are the first couple of digits of both numbers. 3.14159 both at the top and at
the bottom, a perfect match. Great. What about the next digits. 2 at the top and 1 at the bottom.
Close but no cigar. And when you think about it, that’s actually quite disappointing. We just
added 1 million terms of our series and all we get for our efforts is a measly 6 correct digits
of Pi. Is that the paradox? Well, this ultra slow convergence is definitely a bit surprising but
it’s not the paradox. To get to the paradox, let’s look at the next pair of digits. Both 6. Must be a
coincidence, right? Well, could be. Just choosing two digits randomly there is a 1 in 10 chance that
they will coincide. What about the next digits? Both 5. Interesting? Next. Both 3. What’s
going on here? By now we’d definitely expect those digits to be all over the place. Why
do they still come out the same? Let’s keep checking. 558899779933223388 ALL the same. That’s
amazing! I guess at this point nobody watching will still believe that this is a coincidence.
Nevertheless nothing lasts forever and things actually do go off the rails again at the next
pair of digits. 4 and 7. And, surely, from now on there won’t be many more coincidences, right?
Would you bet your life on it? Well, let’s see. Yep, 1 and 6. Different. However, before
you gamble your life away, check this out. 226644338833227799 Seriously? And then. All
different okay, but there are more coincidences. And it’s not just if we add one million terms
that we get this paradoxical behaviour. A billion terms, a trillion terms, same thing. Intrigued?
I bet! This paradoxical property of Pi was first discovered in 1983 by the maths teacher Martin
Powell while he was playing around on a computer. Why had nobody noticed this earlier? That’s a bit
of a paradox in itself given the prominence of our gem and the fact that people have been obsessing
over the digits of Pi forever. Just to put things into perspective in 1983 more than 16 million
digits of Pi were already known. Okay. Now for the ancient Indian discoveries that will serve as
the key to explaining Powell’s Pi Paradox :) These discoveries consist of some beautiful
patterns hiding deep in our mathematical gem. I grew up knowing this identity
as the Leibniz formula, named after Gottfried Willhelm
Leibniz who discovered it in 1673. Well, at some point people realised that
Leibniz was actually not the first to discover this formula. The Scottish mathematician James
Gregory beat him to it a couple of years earlier. And so now some people refer to this series
as the Gregory Leibniz formula. But even that name is misleading. It turns out that this gem
was already discovered by the Indian astronomer and mathematician Madhava of Sangamagrama at
least 200 years before Leibniz and Gregory. No, seriously 200 whopping years before Leibniz
and Gregory. That’s about 1400, still in the middle ages. AND, it took until quite recently
for Western mathematicians to actually recognise and acknowledge this. Does this count as another
paradox? Those of you in the know will ask. Isn’t Leibniz’s formula usually proved using calculus?
Then how did those ancient Indian mathematicians derive it? Well they used ... calculus! Really. It
turns out that Madhava and his disciples already discovered quite a few of the crown jewels
of calculus, including some derivatives and integrals, and some really advanced calculus stuff
that you only learn about at uni. For example, when you replace the 1s in the numerators
of our series by the powers of x, like this you get the famous power series expansion
of the inverse of the tangent function. For example, I teach this stuff in first year
calculus at Monash uni here in Australia. Now, believe it or not Madhava and his colleagues
even knew this very fancy piece of calculus, hundreds of years before Leibniz and Newton
and Gregory “invented” calculus. Madhava also knew the corresponding series for sine and cosine.
There, that’s the series for sine and that’s the one for cosine. Now, back to our gem. Madhava
also discovered some beautiful hidden patterns in this series that were largely overlooked in
the West until fairly recently. These patterns allowed him to speed up the slow convergence of
the series on the right in a spectacular way, and it’s these patterns that we’ll use
to explain the Pi paradox. Okay, what are those mysterious patterns hiding in our series?
Well, let’s focus on the obvious patterns first. What we’ve got there on the right, is a so-called
alternating series, plus, minus, plus, minus, the plus and the minus are alternating, they
take turns. That’s one easy pattern. And then, of course, there is the fact that the
denominators are just the odd numbers. Now, let’s really go for it and use the sum
on the right to approximate Pi. Hands on. Okay, so the whole thing times 4 again. Now, roll
in the number line. There is Pi. Perfect. Now, what does the first term of our series tell us
about Pi? Pi is approximately equal to 4. Cool. Throw in the second term -
4/3 third term and so on. Okay, so adding the first five terms gives
3.34. It's better than 4, but 3.34 is still pretty pathetic. In fact, as I already mentioned
earlier, although the infinite series on the right side of our Pi formula definitely sums to Pi, it
does so ridiculously slowly and is effectively useless for obtaining decent approximations. So,
Madhava’s formula is beautiful but useless? Well, let’s have another close look at what
happens when we sum this series. There, first term. Here we overshot Pi. Second term.
Here we undershot Pi. Next overshoot, and so on. Stopping the adding and subtracting at any point
gives an approximation of Pi. Yeees, but have a closer look. Let’s say we are here. And the plan
is to stop after adding one more term. Okay. Well, doesn’t it look like we would end up much, much
closer to Pi if we only added HALF of that final term? Like so. Really much closer to the Pi line
than going all the way. Or what about here? Go halfway. Again, better than all the way. Okay,
so if we are after a really good approximation, it seems like we can do much better by choosing
the last number we add in a smart way. And adding or subtracting half the final term seems pretty
smart. But is there a way to even be smarter? Yep, there is. To chase it down, we’ll look for
a pattern in the difference between those partial sums of our series and the value of Pi. That’s
these differences here. There. There. Got it? It’s one of these differences that we're trying
to adjust as best as possible in that last step. But HOW can we possibly go pattern spotting in
these differences between the partial sums and Pi? After all we don’t know Pi, Pi is what we want
to calculate. Well, here's an idea. Remember, we entered the story in India, around 1400. And by
that time the local mathematical hotshots already knew some damn good approximations for Pi. Of
course, there is Archimedes’s super famous 22/7, you all know that one from school. 3.14 pretty
good. But those Indian mathematicians already knew much better approximations of Pi, like for example
the fantastic 355/113. 7 digits coincidence with Pi. Not bad, hmm? And now here’s the key
idea. Given one of our partial sums, forget about comparing it to Pi, and instead look at the
difference between our partial sum and 355/113. We then hope that whatever patterns there are, these
patterns will still show up when we use 355/113 instead of Pi. Okay, so let’s calculate the
differences between our partial sums and 355/113 Here's the first difference in numbers. This first
difference pans out to be this fraction. Second difference Third difference and so on. On to the
pattern spotting. Can you see any pattern in those fractions? No? Not easy, but don’t be too fixated
on spotting an exact number pattern. Anything? Still no? Well, here's an easy one: in all those
fractions the number at the bottom is larger than the number at the top. There, 113 at the bottom is
larger than the 97 at the top, 339 is greater than 161, and so on. Not impressed? No, I didn’t think
you would be. But there is more. How much larger are the bottom numbers than the top numbers? Let’s
see. Well, roughly, the number at the bottom is equal to the one at the top plus a little bit.
Here the bottom number is about 2 times the number at the top. Next. Here the bottom number is about
3 times the number at the top. Next. 4 times. Looks like there may be some pattern here. So
the bottom is equal to the top number and a bit. To figure out how bitty that extra bit is, divide
both the top and bottom by the top, like this. Overall still the same number of course but on top
there is now a 1. and at the bottom we get 1 and the bitty bit we are after. There. Repeat for the
next fraction. So divide the top and the bottom by the top. 1 on top and 2 and a bit at the bottom.
Etc. 1, 2, 3, 4, 5, does this pattern continue? Yep, this pattern continues forever and ever
after. What does this mean? Well, take, for example, the sum of the first four terms of our
series then to improve this pathetic approximation of Pi we can simply add 1 fourth. In the case
of five terms we minus a fifth In the case of six terms we add a sixth. Let’s check how much
better we are doing by using this correction term. There, the first six terms add to 2.97.
Now, add 1/6th 3.14, much better. Now, the first time 3.14 comes up in the partial sum of
the original series is as we add the 121st term. Let’s compare this to the adjusted sum. 7 digits
compared to 3, much better. But the ancient Indian mathematicians didn't stop there. They then
had a closer look at the plus-a-bit fractions. 16/97. 17/161 and so on. Let’s zoom in on
the first fraction. Go on autopilot repeat for everything in sight. Now, pattern spotting.
Let’s have close look at the new integer parts. 6, 9, 13, 16, 20, 24. There's definitely
something there. Right? These numbers are a little wonky at the start but then they’re all
multiples of 4. In fact, it turns out that after the first three blue numbers, the multiples of 4
pattern continues perfectly, forever. So after 16, 20 and 24 it’s 28, then 32, 36, 40, and so on.
And this suggests that we’ll get even better approximations for Pi if we choose the correction
terms to include these multiples of 4. Like this. There and there. And here are the three
different approximations we’ve got so far On top just the sum of the first six terms, in the
middle using 1/6th as the correction term, and at the bottom using our new refined correction term.
Not bad, hmm :) And here the same for 121 terms. Again, the differences in the numbers of correct
digits is very impressive. Now, in general, if we are adding N terms the correction term is this.
1 over (N+ 1/(4N)). What’s next for our Indian mathematicians and us? Is there a next? Of course!
Look at the remaining bitty fractions and try to discern a pattern. I'm going to skip the details.
Let me just say that the Indian mathematicians were able to discover one more refinement
of the correction term. It looks like this. And here is the approximation of Pi that we
get using the super-duper correction term. Again a very impressive improvement. In total,
we’ve gone up from a 3-digit agreement with Pi to 15-digits. Anyway, beyond this the Indian
mathematicians couldn't discern any more patterns and so their part of the story ends here.
Well almost, but before I talk about the almost, let me now use these correction
terms to explain Powell’s Pi paradox. Remember this picture here? These are the
first 69 digits of Pi at the top and at the bottom the corresponding 69 digits of the sum of
the first 1 million terms of the Madhava series. To start with let me explain where
the first discrepancy comes from. The approximation that corresponds to the first
correction term 1/N is much better than the millionth partial sum at the bottom and actually
coincides in all the visible digits with Pi at the top. And what’s the difference between these
two numbers? Well, that’s just the correction term 1/N. And since N is 1 million the difference
is one millionth. And what’s the difference between these two numbers? Well, that’s just
the correction term 1/N. And since N is 1 million the difference is 1 millionth. 0.000001,
five zeros and then the 1. And this explains the first discrepancy between the partial sum and Pi
in the highlighted digits. In other words, for those first couple of digits visible up there the
partial sum coincides with the smack on corrected sum except for this one highlighted digit. Not
bad. What about the other discrepancies. Well, the digits at the top are those of Pi. At the same
time they are also the digits of our very best approximation. The second number at the bottom
is still our monstrous millionth partial sum. Now let’s calculate the difference by plugging
N=1,000,000 into our correction term. There that’s the correction term, On closer inspection
you see a bunch of strings of 0s and 9s separated by some other strings of digits. Roughly,
those strings of 0s and 9s coincide with the regions of coincidence between Pi and
our millionth partial sum. And, of course, these regions come from our super approximation
of Pi that is identical to Pi within this range. I am not going to dwell on the details. I
think it’s clear enough what is going on. But here are a couple of observations. First, for
many different types of numbers like the powers of 10 the associated corrections will all have
this 0, 9 structure. Second, it is clear that, as before, strings of 0s should mostly correspond to
regions of coincidence. Also strings of 9s are not that different from strings of 0s since they are
really just strings of 0s in disguise. What do I mean by this? Well, our 0, 9 correction difference
is equal to the difference of these two numbers Right, just focussing on the first string of 9s.
This string is really just the difference of this red blue string on top minus the 1 at the bottom.
Okay. So when we see a string of 9s we also expect about as many digits of coincidence plus some
possible differences at the beginning and end of this string. Anyway, overall I hope that’s
clear enough for you to see what’s going on. I should also mention that if you actually
zoom further out, it turns out that Madhava’s third correction term FAILS to give a complete
explanation of the remaining few coincidences of the millionth sum and Pi. However, it turns out
that there is a fourth correction term, which Madhava missed. And this fourth correction term
takes care of these remaining coincidences for the millionth sum. Here is that fourth correction
term. In fact, there is an infinite sequence of these correction term refinements. Anyway, overall
clear enough what’s going on. Don’t you think? How did Madhava miss these other correction terms?
Well, remember, we were working with 355/113, that’s just an approximation of Pi, instead of
with Pi itself and so it’s not surprising that we will no longer pick up further patterns from
some point on. Of course, Madhava could have used his super correction term to come up with a
better approximation than 355/113 and then used that refined approximation instead of 355/113
in everything he did, in the hope that more of the true patterns will become apparent. And
that would have worked, but of course it would have been very hard to put into action in 1400,
without a computer, or much of anything except his brilliant brain. Anyway, if you're interested
in more details, I’ve put a bit of a discussion of the nitty gritty bits plus links to relevant
papers in the description of this video. A very nice explanation, don’t you think, and a
good place to stop. But before I call it a day, I should say a little bit more about those medieval
Indian mathematicians and their achievements. As I said, these guys definitely did discover a lot
of deep calculus related mathematics, ages before their Western counterparts and they deserve a lot
more credit for what they did and how they did it. Of course, as with all things dating that
far back, quite a few of the original records have been lost. In particular, there are no
surviving mathematical texts by Madhava himself, only some texts on astronomy. What we do know
about his mathematical discoveries is from later commentaries, written by other members of his
school in the two centuries following Madhava. What’s particularly interesting for mathematicians
like myself are not just the results but also the proofs. And Madhava and his disciples
definitely cared about proving things properly and essentially complete proofs even by today’s
standards can be found in their manuscripts. For example, the known Indian proof of Madhava’s
formula up there anticipates the modern slick integral based proof. Truly amazing. In terms
of how exactly Madhava discovered his three correction terms we are not absolutely sure.
In fact, I’ve based the line of reasoning that I presented to you today on some reconstructions
by math historians. Also, we do not know to what extent Madhava had proofs that his correction
terms are valid forever and ever after. The first documented fairly involved proofs are part
of western post- Newton-and-Leibniz mathematics. Again, check out the description of this video
for a lot more details of what is known today. Here is one final thing worth noting. Madhava
also used his correction terms to come up with some new beautiful Pi formulas featuring
very quickly converging series. For example, the first correction term translates
into this spectacular formula for Pi. and the second correction term corresponds to
this pretty formula. How does one translate one of these correction terms into a new series? The
explanation is actually quite simple. So to finish off, let me quickly show you how the translation
works in the case of the first correction term. This one here. Okay, so we start with our original
infinite series. Now calculate again the sequence of partial sums. Important observation: Notice
that the terms of the infinite series below can be recovered from the sequence above. For
example, the fourth term here is just the difference between the fourth and third elements
of the sequence above. Right? Blue minus green, the first three terms cancel. There. There There,
what remains is -4/7 the fourth term of our series at the bottom. Similarly with all the other
terms. There blue minus green equals orange. Blue minus green equals orange. Blue equals
orange. Now let’s add the correction terms. This gives a new sequence of numbers. Now what’s
the series that corresponds to this new sequence? Easy. As before its terms are just the differences
between consecutive elements of the sequence. Like the fourth term here is just the difference
between the fourth element and the third element. Blue minus green, well the same
terms as before cancel. There there and there. And so the fourth term of the
new series is this. Now, when an expert ponders the general formula for the nth difference a bit,
they will notice that this term can be rewritten in this very beautiful concise form. Not obvious
but at the same time really just a bit of algebra. And then the other terms are calculated in the
same way. There, there and there. In the same way, all correction terms can be turned into new
infinite series formulas for Pi. However, only the first two appear in those old Indian
manuscripts. Why? Possibly because only in those first two the terms of the new series can can be
written in surprisingly simple and beautiful ways. Also, in those old Indian palm leave manuscripts
all these formulas were not written in the concise mathematical language that we use today.
There they were all, believe it or not, they were all expressed in verse! And I guess
you think twice before you note down anything ugly in verse :) Anyway that’s it for today.
I leave you with a detailed look at a page from one of those old manuscripts.
See whether you can spot any x's :)
