The history of pi

What is pi? How do we define it and who first thought of it? We explore the history of this quintessential mathematical constant.

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Pi is one of the most common mathematical symbols in use today. It appears when we calculate the area of a circle, the volume of a sphere, or the surface area of a cone. But, what do we know about pi? We all probably know that it is approximately 3.14, but what else do we know? Well, there is an endless list of interesting facts about pi that you should know.

The Egyptian method

The Egyptian method of estimating pi.

A definition of pi is: the circumference divided by the diameter of a circle. In other words, pi is the ratio of the circumference of a circle to its diameter. You can find the value of pi yourself when you measure around the edge of a circle and divide it by the measurement across the circle.

The existence of pi was known long ago–almost 4000 years ago. The Egyptians’ approximations of pi were indicated through the Rhind Papyrus: they had an approximation of about 3.160. A translation of a part of the Rhind Papyrus states how the approximation was derived. They hypothesised that the area of a square with a side length that is 8/9 of a circle’s diameter would be equal to the area of the circle. Within a given circle in which the diameter is 2$r$, Egyptians took away 1/9 of its diameter, which leaves 16$r$/9. They squared the measurement to get the area of the square, which is 256$r^2$/81. Comparing this measurement with the formula of a circle’s area, Egyptians concluded that pi
was 256/81, which is about 3.160.

The Babylonian method

The Babylonian method of estimating pi.

The Babylonians, on the other hand, had an estimate of about 3.125. Many Babylonian tablets were found in Susa, and one of them shows a list of mathematical constants. 24/25 is one of the constants, and it conforms with the ratio of a perimeter of a hexagon, with side length $r$, inscribed in a circle of radius $r$, to the circumference of the circle. Equating 24/25 to the ratio of lengths 6$r$/(2 $\pi r$) and rearranging gives pi as 25/8 or 3.125.

The Archimedian method

Archimedes of Syracuse was the first to attempt calculating pi. Through observation, he hypothesised that perimeters of polygons drawn inside and outside of a circle would be close to the circumference of the circle. Subsequently, he used Pythagoras Theorem to find the areas of a polygon inscribed in a circle and a polygon within which the circle was circumscribed. Archimedes could successfully find the limits of the circle’s area as it is bigger than the area of the polygon inside the circle, but smaller than the area of the polygon outside the circle. He started this rigorous calculation by drawing hexagons. He first drew a circle with a diameter of 1 and used the diameter and the fact that an angle of a hexagon is 60 degrees to find the measurements for each of the sides of the inner and outer hexagons. He figured out that the perimeter of the inscribed hexagon is 3, and the perimeter of the circumscribed hexagon is about 3.46. He concluded that pi would be bigger than 3 but smaller than 3.46. Archimedes used polygons with bigger numbers of sides so that he could get a better approximation for pi. He ultimately used polygons with 96 sides and utilised these limits driven from calculation to support that pi is between 3.1408 and 3.14285.

The Archimedean method of … you can probably guess what.

Modern approaches

William Jones in 1706 first used the modern symbol for pi, $\pi$. The reason why $\pi$ was chosen instead of other Greek letters was that, in Greek, is pronounced like the alphabet ‘p,’ which stands for the perimeter. The symbol was later popularised in 1737 by Leonhard Euler. Since then, pi has been widely used in diverse fields by engineers, physicists, architects, and designers… anywhere that mathematics is involved! Some people suggest that the Great Pyramid at Giza was built based on pi because the ratio of the perimeter to the height of the pyramid is strikingly close to pi. In fact, the Pyramid has an elevation of 280 cubits and a perimeter of 1760 cubits, and the ratio of the perimeter to the height is 6.285714, which is approximately two times pi.

Since pi is a non-repeating decimal, attempts to calculate pi up to as many digits as possible also have some history. The most accurate calculation of pi before computers was one done in 1945, by D.F. Ferguson. He calculated pi up to 620 digits. This was the most accurate and lengthy calculation since, before Ferguson, William Shanks was known to have calculated pi up to 707 digits–in which only 527 digits were correct. Later on, pi was calculated up to trillions of digits with the help of digital programs. To list a couple of people who made world records by carrying out the most extended calculation of pi, Takahashi Kanada calculated pi up to 206,158,430,000 digits with a Hitachi SR8000 in 1999, and Shigeru Kondo calculated up to 10 trillion digits with Alexander Yee’s y-cruncher program in 2011.

Pi pie. The best representation of pi.
Image:Wikimedia commons (CC BY-SA 3.0) .

In the past, pi was only used by specialists such as mathematicians, scientists, and architects. However, nowadays, pi is commonly used by non-specialists and is even used in literature, movies, and museums. For instance, Pi is the nickname of the main character in the award-winning novel Life of Pi. An episode of StarTrek: The Original Series included a scene in which Spock orders a computer to calculate pi up to the last digit. What’s more, ‘3.14159’ is included in cheers of Georgia Institute of Technology and MIT.

Today, perhaps the most popular cultural phenomenon regarding pi is Pi day. It is celebrated on 14 March all around the world. On Pi day, people who are enthusiastic about mathematics write and talk about pi, eat pie, and make bad jokes. Now that you have learned more about pi, maybe you can tell people around you about all the exciting things about pi on Pi Day!

Doyeon Jang is a grade 10 student studying at Branksome Hall Asia, an international school located in Korea. She is interested in discovering mathematics in everyday life and thinking about various applications of maths.
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