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Monday, March 9, 2015

Multicultural History of Pi

Mathematicians around the world celebrate the number pi on March 14 every year. This year is a very special year and is being called the pi day of the century because besides the date the year gives more digits of accuracy and if you look at time of the day to the second you can go nine digits after the decimal point for your celebration! Various celebrations are planned around the world including in Chicago, MIT in Cambridge, Massachusetts, Arizona, New Jersey, and San Francisco (where the first Pi Day Celebration was held).

Pi is a very special number in our world. It is a number that has been explored for thousands of years in just about every culture. Why? Because people wanted to learn about their world and started to realize that the same number kept appearing when taking the circumference of a circle (distance around the circle) and divide it by the diameter (distance across the circle through the center). The Babylonians and Egyptians were the first known to start the hunt for pi about 4000 years ago. Some say they figured out the ratio of a circle's circumference to diameter is slightly bigger than three or about 3 1/8. 

Ancient Egypt and Mesopotamia c. 1450 BC
By Свифт/Svift (my work) [Public domain], via Wikimedia Commons
The first written approximation is on Rhind Papyrus in Egypt dated around 1650 BC. It indicates that pi is equal to 3.16049. However this knowledge did not spread far. The Chinese knew that the ratio was around 3 and used it for hundreds of years. 


An approximation of pi appears in the Old Testament of the Bible. In 1 Kings 7:23 it states 
23 Then Huram melted bronze and poured it into a huge mold to make a tank,[a] which was called “The Sea.” The tank was about 30 cubits around. It was 10 cubits across and 5 cubits deep. 
This implies that pi is about 3 (30 cubits around divided by 10 cubits across). However there are questions as to how the Hebrew was translated and that perhaps the translations should have actually been that the pi/3 = 111/106. That gives us pi = 3.1415094.... 
Algoritmo de Pi
Inscribed Polygons Method By User:HiTe (Own work) [Public domain], via Wikimedia Commons
The Greeks soon took on the challenge. Antiphon and Bryson of Heraclea thought of putting a polygon inside a circle. They figured finding the area of the polygon would give an approximation of the area of the circle. Then increasing the number of sides would give a more accurate estimate. Later Bryson had the polygon circumscribe the circle (polygon on outside of circle) and calculated the area. This gave a lower and upper bound of pi and is probably the first time lower and upper bounds were used to calculate a number. Anaxagoras of Clazomenae took on the problem and spent most of his time trying to square the circle. A problem that would take over 2000 years to solve. Squaring the circle is trying to construct a circle and a square with the same area using only a compass and a straightedge.
Archimedes Siracus. Line engraving by Remondini. Wellcome V0000191
See page for author [CC BY 4.0], via Wikimedia Commons
Archimedes of Syracuse was the first to have success in calculating pi.  He used inscribed and circumscribed polygons but focused on their perimeters instead of areas. He kept doubling the number of sides up to two 96-sided polygons. He found pi to be between 3 10/71 and 3 1/7. His approximation of 22/7 was used for a long time (and is still used in some math problems for ease).

In 263 AD, Chinese Liu Hui independently discovered the same method as Antiphon and Bryson and found pi to be 3.14159. Near the end of the 5th century a Chinese father and son duo, Tsu Ch'ung-chih and his son Tsu Keng-chih, used a 24,576-sided polygon to get a good range. Next it was Hindu mathematician, Aryabhata, who gave the "accurate value" as 62,832/20,000. Another Indian mathematician, Brahmagupta, incorrectly approximated it to be the square root of 10. In the 9th century Arab mathematician Mohammed ibn Musa al'Khwarizmi, used 3 1/7, 62,832/20,000 and the square root of 10 as the values of pi. The ratio 62,832/20,000 is most accurate but somehow seemed to be forgotten by the Arabs. 

Not much happened with pi until the 16th century when a French lawyer and amateur mathematician, Françle;ois Viéte, used Archimedes method of two polygons eventually using 393,216-sided polygons. His approximation was more accurate but even more importantly he was the first to write pi as an infinite product. In 1596 a German man, Ludolph Van Ceulen, used Archimedes method with polygons over 500 million sides and got 20 digit accuracy. He spent his life hunting for pi and before he died found it to 35 digits. Germans still refer to pi as the Ludolphian number in his honor. 
Ludolf van Ceulen
Ludolph Van Ceulen By Dr. Manuel at de.wikipedia 
(Transfered from de.wikipedia) [Public domain], from Wikimedia Commons
Up to this point there was no name for the ratio of the circumference of a circle to its diameter. In 1706, an English mathematician, William Jones adopted the Greek letter, π, for the ratio of the circumference of a circle to its diameter. However it did not really gain much attention until Swiss mathematician-physicist, Leonard Euler (pronounced like oiler), used π in one of his books in 1736. He had been using p for pi early in his career. (Euler is for whom the number e is named).
EulerLeonhard
Leonhard Euler by Jakob Emanuel Handmann 
[Public domain], via Wikimedia Commons
Now mathematicians throughout Europe and the world looked for equations to calculate pi. By 1750 pi was being expressed as an infinite series. In 1761, Swiss mathematician, Johann Heinrich Lambert, proved pi was an irrational number. His proofs were not completely accepted by all, but in 1794, French mathematician, Adrien Marie Legendre, provided a proof that satisfied everyone. In 1882, German mathematician, Ferdinand von Lindemann, proved that pi is a transcendental number, which means it is not of any polynomial with rational coefficients. This proved that squaring the circle was an impossible task. In the 20th century computers took over the calculation of pi. Pi has been calculated to over one trillion of digits past its decimal however only 39 are needed to accurately calculate the circumference of the known universe.

As you can see the value of pi has mattered to mathematicians throughout time and around the world. Pi has many uses throughout mathematics. Most people know it for finding the circumference of a circle or the area of a circle. For a simple activity for younger children check out Cutting Pi on Exploratorium (they have several activities for different ages there and were the first to celebrate Pi Day), Pi Day Week of Lessons, Education World has a great list including grade levels for the activities. I also listed our past posts filled with pi activities below. Pi also is used in measuring angles in radians (a more every day calculation measurement learned in precalculus). We also enjoyed looking at some books about pi.
Sir Cumference and the Dragon of Pi by Cindy Neuschwander is a fun math adventure where a knight's son must figure out a mathematical riddle to save his father who has been turned into a dragon. The riddle's answer is pi.


From the series The Biography of Numbers, Pi by Kevin Cunningham is a chapter book with different parts of the history of pi. This book was a bit above Hazel's understanding level, but she wanted me to keep reading a chapter a night.


Science in Ancient China by George Beshore has a few pages about the approximations of pi in China. 



Sources:
  • Cunningham, Kevin: The Biography of Numbers: Pi, Morgan Reynolds Publishing 2014.
  • Beshore, George: Science in Ancient China: Franklin Watts: 1998.
  • The History of Pi by David Wilson, Rutgers Univirsity
  • Wikipedia: Ferdinand von Lindemann
  • Pi Day.org 
For more posts about pi check out: