When reading deeper about Aryabhata’s and his work on the approximation of Pi I ran into the his description of it in these words,
Chaturnik shartmast gundam duasti stayam sahsranam Ayutathya vithkambh sya sanna vritahaprinaha
– Aryabhatta epic
The meaning of this sentence is:
(4+100) x 8 + 62000 = 62832 units of circumference for that circle whose diameter is 20,000 units. Then π=62832/20000 which results in π =3.1416
This sparked my curiosity about the value of pi and how much the approximation of finding it has changed throughout the history of time, The Life of Pi. I wanted to investigate what differed on how the value was derived.
First we start in ancient Egypt, 1650 BC. On an antique document called the Rhind Papyrus pi was first formulated using the area of a circle. A problem on the Papyrus asks the area of a circle with diameter of 9khet. How they solved this problem will help us understand their value of pi. First they multiplied the diameter of the circle by (1/9)th, this determined the area to be (d-d/9)^2 =64/81 * d^2. From this we deduce (since A = π d^2 /4) that the ancient Egyptians used the value 256/81 = 3.16049…. What is so surprising is that that left only an error of 0.0189!
We then jump past a few other mathematicians to Archimedes, 250 BC, and perspective of pi. “Archimedes’ approach is to circumscribe and inscribe regular n-gons around a unit circle. He begins with a hexagon and repeatedly subdivides the side to get 12, 24, 48 and 96-gons.” (Rosenberg, 1) The more polygon sides the more accurate of an approximation, the better “circle”. Hence the formula: n * sin(180/n). For example: a 12-gon would result in 3.10582 where-as a 96-gon comes to 3.14157.
The Chinese had to enter the ring in the fight for reaching a value of π. With the help of Zu Chongzhi (Tsu Ch’ung-chih) (430 -501) the Chinese were able to hold an accurate value that wasn’t matched in accuracy till the 1500’s. Zu came to this value by using the Ptolemaic value of 377/120 and subtracting its numerator and denominator from the Archimedean value of 22/7 which comes to 355/113. This resulted in = 3.1415927. It is really hard to pin-point exactly how he came up with this value but it was very accurate compared to previous century.
Now I want to jump even further in time to 1424 AD where Persian astronomer Amshid Al-Kashi took on a new way to approximate the value of π. Amshid took on this task by using a polygon with 6 sides, taking it to the 28th power. Amshid used what is now known as the Lambert Identity to find α ° = 180° −360°/ 3(2n) (where n ≥ 0).
“From here he found the lengths of the sides of inscribed and circumscribed regular polygons each with 3(2^n) sides (n ≥ 1), in a given circle. Then he determined the number of sides of the inscribed regular polygon in a circle whose radius is six hundred times the radius of the Earth in a such a way that the difference between the circumference of the circle and the perimeter of the inscribed regular polygon in this circle will become less than the width of a horse’s hair.” (Azarian, 8)
Al-Klashi calculated the value of π correct to 16 decimal places, using inscribed and circumscribed polygons, each with 805,306,368 sides!
Today pi has a current decimal approximation of 5 trillion digits! During my adventure through the life of pi I have come to realize that this value is what it is today because of the melding of many cultures and ways of thinking. Each culture added their own unique flavor, math traditions and way of solving problems. It took thousands of years of building on top of the works of another to get an approximation this accurate, one value of π. That, to me, is what makes math so cool!
Sources – MLA Format
Azarian, Mohammad. “AL-KASHI’S FUNDAMENTAL THEOREM.” International Journal of Pure and Applied Mathematics 14.4 (2004): 500. Web. 7 Sept. 2014. http://www.ijpam.eu/contents/2004-14-4/5/5.pdf
Berggren, Lennart. Pi, a Source Book. 3rd ed. New York: Springer, 2004. Print.
Boyer, Carl B., and Uta C. Merzbach. “Values of Pi.” A History of Mathematics. 3rd ed. New York: Wiley, 1989. 181. Print.
Dayal, Shiv. “Possible to Square a Circle with PI.” IJPCMF International Journal of Physics, Chemistry and Mathematical Fundamentals 2.01 (2012): 11. Web. 6 Sept. 2014. http://www.ijpcmf.com
Pi Through the years: http://www.preceden.com/timelines/43012-pi-through-the-years
Rosenberg, Burton. “Archimedes and Pi.” Www.phaser.com. NSF, 7 Sept. 2003. Web. 7 Sept. 2014. http://www.phaser.com/modules/historic/archimedes/meas_circle.pdf