A Brief History of Pi

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 the 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 see in this article 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 diameter of the square

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 to try and do that calculation

Only using Roman numerals our modern decimal notation was first developed in India sometime 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 ben 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 east Europe was back in the game and in 1630 the most accurate

an 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 the pure reason the change in how 17th-century European mathematicians calculated PI is arguably a

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 the reason a method based on

Infinite series an infinite series is just an expression

Made up of things added together with one 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 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 the 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 launched a website The Turks where you can read more similar articles.