Archimedes’ Last Words and the History of π
π is probably the most famous or at least the number that is recognized by most out there. Almost all of us know that if we take a circle and divide its circumference with its diameter, we get π.
It happens not just for one particular circle but all the circles out there can be considered in a plane. It is important to note that this definition would not hold true if the surface on which the circle was drawn was not a plane, for example, if we were to consider a circle drawn on a ball.
However, it is astonishing how old this number is. Understanding how π was first calculated and what it means forms the very backbone of the fundamentals of mathematics. Its history takes us from the middle east to China, to India, to Persia, and even Italy!
Counting, the very basis of mathematics dates back to 20,000 BC. The evidence of which was found in the Ishango bone, and maybe even before that! While the knowledge of counting spread and mathematics as a whole developed, both the ancient Babylonians and the Egyptians attempted to find the value of π. The Babylonians found the value of π to be around 3.125, which happens to be accurate to 1%! and given these calculations date back to when iron was first used by the human civilization, this is astonishingly accurate!
The way π was initially calculated by a method known as the method of exhaustion, you will soon enough know why it was called the method of “exhaustion.”
The ancient mathematicians were struck by the problem of calculating the length of a curved surface, though they could have used easier methods like using a string to measure the length and then find the value by comparing it with a linear scale, they stuck to the method of exhaustion. Think about this, a square is a little like a circle, but when a side is added to form a pentagon, it comes closer to resembling one, and when another side is added it gets more closer, and so on. A circle can be thought of as a polygon with infinite sides, which was exactly the thought process used by Archimedes to calculate π!
Archimedes first thought of calculating π by fitting a square inside and outside a circle so they are all touching each other and then dividing the perimeter of the squares by the diameter. It is interesting how Archimedes in his papers used the term “diameter.” So, what is the diameter of a square, is it the diagonal or the edge, turns out it is both! Archimedes considered the effective lengths (the edge of the outer square and the diagonal of the inner square) for each square,
So he calculated, 4. (side of the outer square)/ side of the outer square and 4. (side of the inner square)/ diagonal of the inner square, and π had to lie somewhere in between these two numbers. Turns out he was close, but still far from the actual answer with an average value of 2.828.. < π < 4.000; which is a very large gap! This is because the square is not a very good representative of the circle. This is why the square was later replaced by a pentagon and later again by a hexagon and so on to reach higher precession.
And you can keep increasing the number of shapes to reach higher and higher accuracy until you get tired. This is precisely why it is called the method of exhaustion, however, Archimedes was able to reach a 96 sided shape which is now called the ennecontakaihexagon bringing the accuracy down by a LOT, to 3.1408 < π < 3.1429, accurate to two decimal places! However, this was maximum accuracy he could reach as soon later, his home town, Syracuse was invaded by Roman Soldiers who along with killing almost everyone along with Archimedes closed the book of π in Europe for the next millennium. It is interesting to know, Archimedes’ last words were, “Do not disturb my circles,” which was a reference to his, then, ongoing attempt on increasing the accuracy even further!
Meanwhile, the Chinese mathematicians used a similar method of exhaustion except they considered areas instead of perimeters and were able to get very close to the value of π. A Chinese mathematician, Liu Hui conducted the same procedure with areas, with a shape of 3072 sides, to obtain π to four decimal places, 3.1416! Soon later, Zu Chongzhi and Zu Gengzhi, who were a father and son pair, used a 12288 sided figure to 3.14159! Though these concepts were not very difficult to understand, the calculations were terrible to do physically, but mathematicians participated anyway as it was more like a competition among them.
However, things were about to get easier. With the invention of digits consequently in India and Persia and the establishment of the Babylonian discovered/ invented Algebra by Muhammad Ibn Musa al-Khwarizmi, the calculation of π became much easier! As the renaissance passed, with these new tools like digits and algebra, European mathematicians were back in competitions, and soon enough π was calculated to an accuracy of 38 decimal places by an Austrian mathematician, Berger, who used a 10⁴⁶ side shape to achieve it! I say that was accurate enough, but we shouldn’t stop should we?
With the adoption of the, then, newly discovered algebra into the European culture and mathematics, a new perspective of looking at the world was born, which is now famously known as the “The Scientific Revolution,” which inspired the age of enlightenment with, then, thinkers and, now, well-known mathematicians like Descartes who were anti-romantics, or valued reason over tradition. This spurred a new rational and theoretical approach in mathematical history, which is very evident from the shift in the ways of calculating π, from ancient times when mathematicians of the likes of Archimedes calculated π by the lengths of geometrical shapes and structures to now, when something purely theoretical, infinite series was used to discover the unknown digits of π. Infinite series are usually sums of numbers or generalised expressions that follow a particular pattern and go on forever, an example could be something like 1+1/2+1/4+1/8+… to ∞! However, it was not Europe where infinite series were discovered and first used but India, where Madhava, first found the infinite series expressions for sine, cosine, and tangent functions! Madhava in the 14th century was the first person in the world to use infinite series to calculate π, let’s see how.
Madhava discovered the infinite series expression of tangent and the inverse tangent of x, as:
And the expression for the inverse tangent of x,
As we know, tan π/4 precisely equals 1, if we substitute x = π/4 and y = 1, in the second equation, we get an infinite series that predicts the value of π very accurately:
This very accurate method of calculating π was in fact used to calculate π by Madhava till 11 accurate digits! However, it was soon forgotten until it was rediscovered by European mathematicians like Gregory and Leibniz, and at this point, the use of infinite series and calculation of π allowed for record-breaking calculations of the digits of π. The first 100 digits of π were calculated by John Machin in 1706, who was soon beaten in 1719 by Fantet de Lagny who calculated π till 112 digits, and soon turned into a heated competition among 18th through 20th-century European mathematicians. And soon mathematicians started coming up with their own infinite series that converged to π faster than the ones used before in order to excel in this competition, which went on till 20th century when Indian self-taught genius, Srinivasa Ramanujan developed one of the most fastest converging infinite series ever discovered!
However, this competition soon faded away with the invention of mechanical computers, when 2037 digits of π was calculated by the first mechanical computer invented by John von Neumann and chums: ENIAC (Electronic Numerical Integrator And Computer)
From then on, the calculation of π no more dealt with interesting infinite series on pen and paper but turned into a list of more and more efficient computers. At the time of writing this, the world record is 50,000,000,000,000 digits and was achieved by Timothy Mullican (USA) in Huntsville, Alabama, USA, on 29 January 2020.
But why did we spend so much time and effort into calculating trillions of digits of a number that we don’t really need? Well, there are benefits, the calculation of digits of π are used to check the efficiency and accuracy of newly developed computers as they can directly be checked with the already found digits of π with are in trillions! Moreover, π is also used in random number generations for various statistical and nonstatistical studies. If you look at the first 200 billion digits of π, the number of 0,1,2,..,9 in there are:
Which leads to an ~10% chance of having any number from 0 to 9, which means these can be used to generate random numbers and even happens to be useful in fields like cryptography! But, finding more and more digits of π are also a cool way of having a world record and people will never stop doing it, humans are curious, weird, un-understandable yet they are us!
Homer says, “Thank you for reading”
References:
- https://datatorch.com/life/Archimedes_faced_death_with_courage
- https://www.economist.com/science-and-technology/2018/04/07/an-ambitious-african-science-project-is-getting-into-its-stride
- http://www.hellenicaworld.com/Greece/Science/en/Archimedes.html
- http://computerscience.chemeketa.edu/cs160Reader/HistoryOfComputers/Generation1.html
