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take any circle
measure its circumference and its diameter
the ratio of these two numbers is a mathematical constant we call pi
while this definition is simple pi has been studied for thousands of years and
History of our understanding not just of the value of pi
But also what it means forms a history of all of mathematics it takes us from the Middle East to Europe
to China to India and even America
It's a history, which involves revolutions murder and the infinite
Maths is as old as civilization
older even
There's evidence of counting going back thirty thousand years
and two of the very earliest civilizations
the ancient Egyptians and Babylonians
both investigated pi around 4000 years ago
the Babylonians estimated PI to be 3 and 1/8 or
3.125 now that's the first of a few estimates you're gonna hear in this video so for reference
remember that the first few digits of pi are
3.1415926
There are more
that means that the Babylonian estimate of Pi is accurate to 1% of its true value
Which is kind of astonishing when you remember that this is a time in human history when?
iron was first being used
and the last mammoths went extinct
the ancient Egyptians on the other hand estimated PI slightly less accurately as
3.16 but how do you even estimate the value of pi
You have to count it by definition measure a curved surface, which is super tricky to do accurately
Well one way of doing it is to cheat and actually use a square compare a square and a circle well
It's quite a little bit like a circle
But that was much like a circle as a Pentagon which has one more side than a square
and a Pentagon doesn't look quite as much like a circle as a hexagon which has one more side again and
a hexagon doesn't look quite as much like a circle as a heptagon and
So on you can think of a circle as a regular polygon just want with an extremely large number of sides
So many sides in fact that each individual one is
Infinitesimally small meaning that the circle looks round this was exactly the thinking that legendary ancient Greek mathematician Archimedes
Used when estimating pi around 220 BC in fact
It was probably the very last thing he ever did to approximate pi
He reasoned why not measure the perimeter of a square adding up the lengths of all of its edges
And then dividing that number by the squares diameter
But what is the diameter of a square is it the length of its diagonal or the length of one of its edges?
Why not both said Archimedes draw one square with its corners just touching the perimeter of a circle
Another square with its faces just touching the perimeter of that same circle add up the lengths of the sides of each square
divided by their effective diameters
And you have two estimates for the value of pi the true value of which lie somewhere between those two numbers
But here's the really clever part
Because the difference between those two values is pretty big if you're using squares because a square isn't much like a circle
But replace those squares with Pentagon's and you shrink the difference between those two numbers
Meaning that there's a smaller range of values that PI could be your estimation just got more accurate
And if you replace those Pentagon's with hexagons you'll get an even more accurate
estimates keep increasing the number of faces on the shape that you're drawing inside and outside the circle and your estimate will get more and
More accurate as long as you have the time and patience to draw said shapes
There is a reason why this thing was called the method of exhaustion
Archimedes got up to a 96 sided shape which incidentally is called an a neocon Turki hexagon
I really hope I said that right giving an estimate of Pi between three point one four zero eight and three point one four
To nine so accurate to two decimal places as I mentioned earlier this was likely his final contribution to science
because into 1/2
BC he was killed by Roman soldiers who invaded his hometown Zaira Q's he was apparently performing this
Calculation at the time allegedly his final words were don't disturb my circles
European progress in the study of pi died with Archimedes for well over a thousand years
fortunately however there was plenty of the world which was not in Europe a
mathematicians here were also interested in PI in particular three mathematical superpowers of the first millennium ad
Were China India and Persia?
Ideas when these three nations were soon to change the world
first off Chinese mathematicians used a method of exhaustion similar to our comedians
But instead of considering the parameters of shapes they considered their areas and this dude no
I'm not going to try and pronounce his name because I'll only get it wrong used to polygon with
3072 sides to obtain pi to five decimal places
200 years later a father-and-son team used a polygon with over
12,000 sides to extend that record to six decimal places and that was a world record which stood for
800 years the problem was it was just difficult to do these calculations. They weren't especially hard to understand
it was just awkward to write down what you were doing to physically do the
Calculation and this was something that would only be resolved by the introduction of two world changing ideas
From India and Persia say that you want to do a calculation
You know that you and your friends together weigh a hundred and twenty-five kilos
And you also know that you weigh 70 kilos the question is how much does your friend weigh?
Mathematically, we'd write. This as X plus 70 equals
125 where X is your friends weight in kilos
Subtract 17 from both sides and you get the answer
55 kilos now in that simple example. I just used two ideas which were
Revolutionary to the classical world firstly I wrote large numbers like
125 and 70 using a simple notation we take it for granted these days
But the ability to write any number using just ten symbols and a place value notation
Where the position of a symbol in a number determines its size?
Massively simplifies arithmetic to see what I mean try and do that calculation
Only using Roman numerals our modern decimal notation was first developed in India some time before
400 AD and then rapidly spread to Persia where the second key idea came from the second key idea was
Representing your friend's weight using some symbol X and then manipulating both sides of the equation this of course is
algebra originally developed by Babylonian and ancient mathematicians
But truly established by Persian mathematician and all-round very influential dude Mohammed eben Musa al-khwarizmi
using decimal notation and algebra allowed for much easier calculations across all of maths and
mathematicians working on calculating PI
Used it to turbocharge their work after the Renaissance and a renewed interest in mathematics along with
Crucially new tools from the east Europe was back in the game and in 1630 the most accurate
estimate of Pi using the polygon method was achieved by Austrian astronomer Christiaan grind Berger who used a shape with 10 to the
40 sides yes really to calculate pi to
38 decimal places and then because mathematicians are sensible people with lives to lead they decided that was accurate enough
And they'd leave it there oh
wait
The adoption of algebra by European mathematicians triggered a whole new way of looking at the world a change in thinking
generally grouped under the title the Scientific Revolution
Which itself went on to inspire the Age of Enlightenment with thinkers like Rene Descartes and John Locke?
Amongst other ideas the Enlightenment movement emphasized the value of Reason over
Tradition and new mathematical ideas were held up as Paragons of this
they were pure reason the change in how 17th century European mathematicians calculated PI is arguably a
Perfect example of the shift from following what the ancients did to new rational?
Theoretical approaches because while the ancients like Archimedes may have measured the perimeters of shapes increasingly similar to circles
Now European mathematicians were using a method based entirely on reason a method based on
Infinite series an infinite series is just an expression
Made up of things added together one after the other after the other after the other and so on until
Forever if those contributions keep getting smaller as you go on then the series converges to a particular value
Sometimes you can work out what that value will be using logical arguments?
but sometimes you just have to keep calculating term after term after term until you reach an accuracy that you're happy with the
Method of using infinite series to calculate pi was first used not in Europe
But again in India you could kind of argue that what Archimedes did was an infinite series?
But the first person to write a mathematical function as an infinite series was Indian
mathematician math hava of Sangamo grammar in the 14th century
He wrote down expressions for the sine cosine and tangent of an angle as well as the inverse tangent
quick refresher if you write the expression y equals tan of X the
Expansion for the tangent would tell you what y equals?
If you already know what X is while the expansion of the inverse tangent would tell you what X is?
If you already know what Y is by its definition
the function tan of X precisely equals 1 when x equals 1/4 pi
That means that if you have an expression for the inverse tangent
Then if you plug 1 into that expression and keep calculating terms
You'll end up with an increasingly accurate estimate of 1/4 pi madhava did this and calculated PI to
11 digits, but then his method seems to have been forgotten only to be apparently
independently rediscovered in 17th century Europe by Scott James Gregory and German, Gottfried Wilhelm
lightness and at this point
everything kicked off
the new decimal notation and algebraic technique allowed for record calculations of Pi in
1699 it was calculated to 271 digits by abraham sharp who was beaten in
1706 when John machen reached a hundred digits who was in turn beaten by thomas von tete de l'année
I hope that's how you say his name in
1719 with 112 digits it wasn't just the case that each of those mathematicians had more spare time than the previous one
They were competing with each other using different infinite series, which converged on PI
faster instead of just using the inverse tangent infinite series
They might use a combination of different inverse tangent values or something completely different
the competition then became less about which mathematician had done the most calculations and instead which
mathematician had the fastest converging infinite series
Development of increasingly efficient infinite series continued well into the 20th century
With the technique kind of coming full circle as the current infinite series of choice was developed by Indian prodigy mathematician
Srinivasa Ramanujan of course by the 20th century
Mechanical computers had been invented making it much easier to calculate pi
You basically just used one until he got bored in 1949 Americans D. F, Ferguson and John wrench calculated PI to
1120 digits
But they were bringing a knife to a gunfight because that very same year
the first calculation of Pi by an electronic computer was done
nearly doubling their record with two thousand and thirty seven digits from here the history of Pi is basically a list of
increasingly powerful computers running for a long time and spitting out
Increasingly absurd numbers of digits at the time of recording the world record for digits of pi
Calculated is held by peter trib with a shade under twenty two and a half trillion digits
Calculated the question of course is if we know that pi is going to keep going on forever. It's a transcendental number
Why should anybody bother calculating anymore dishes?
Well for one thing calculating pi is actually a really good way of making sure that your brand new shiny computer is
working properly
Calculating pi uses up a lot of mental brainpower for the computer you have an answer that you can check yours
Against and also if you keep going just a little bit longer than the previous person you can have a casual world record
Secondly pi is actually a really good random number generator
If you look at the first two hundred billion digits of pi. You'll find the number zero occurs almost precisely
20 billion times and the same goes for the other digits 1 through 9
That means that if you were to pick a random digit in those 200 billion
There's an almost exactly 10 percent chance of it being one under almost exactly 10% chance being to and so on
This makes calculating PI to a large number of digits very
Valuable to people that want to generate random numbers
people working in cryptography for example
But lastly and arguably most
importantly
People keep calculating
More digits of pi for the same reason that why people memorize tens of thousands of digits of pi
And the same reason why people climb mountains and swim oceans and invent the double luge
because they can
Humans are weird. We like to understand the world around us and as our civilization has developed
We've built increasingly complex tools to help us understand the world
It wasn't essential for our survival that we did that we just did it because of the way we're wired
because we could pie is a thread that's gone through all of human history because it's a microcosm of how we
interact with the natural world from the ancients to present day through
Revolutions in Thor and across the world as long as there are people. There's always going to be somebody who just wanders
What's the next digit long may that continue?
I'd like to close out this video with two quick announcements announcement the first I
Finally launched a website
Go to the very shiny and new Simon ox fist calm for a hub for everything
I do online including this YouTube channel
Sponge in electric and the wiki cast there's a page there
Detailing the gear that I used to make these videos and coming soon
there's also going to be a page detailing all the advice that I give out to people who are
Interested in applying to the universities of Oxford and Cambridge and then announcement number two if you'd like a look at my website
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