Category Archives: Pi

A Brief History of π

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People have been studying pi for almost 4,000 years.  Some believe that ancient Egyptians had some knowledge of pi, because the Great Pyramid at Giza, constructed around 2500 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832.

The earliest written approximations of pi (both within 1 percent of the true value) are from ancient Babylon and Egypt from around 1900-1600 BC. Babylonians are known to have used a value of 3 for pi, but one Babylonian tablet indicates a value of 3.125. The ancient Egyptians used a value of (16/9)2 or 3.1605.

Pi was first rigorously calculated by Archimedes of Syracuse around 250 BC Archimedes approximated pi by inscribing (inside) and circumscribing (outside) polygons around circles. Knowing that the area of the circle must be between the area of the 2 polygons allowed him to find an upper and lower bound for pi.  He was able to narrow down this range by increasing the number of sides on the polygons. Archimedes did this up to a 96-sided polygon, allowing him to approximate pi as being between 223/71(3.1408) and 22/7 (3.1429).

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[Steven Strogatz has a great article about Archimedes and this “method of exhaustion”. Also, I found a nice worksheet for older students here.]

An interesting side note, legend has it that when the Roman army invaded the Greek city of Syracuse, Archimedes was so engrossed in his work that he failed to respond to a Roman soldier who was questioning him. His last words are said to have been “Do not touch my circles!”, before being beheaded by the soldier.

An approach similar to that of Archimedes was developed in ancient China by Lin Hui around 1 AD.  Zu Chongzhi calculated the value of pi to be 355/113  (around 480 AD) by applying Lin Hiu’s algorithm to a 12,288-sided polygon. This value remained the most accurate approximation of pi available for the next 800 years.

The calculation of pi was revolutionized by the development of infinite series techniques in the 16th and 17th centuries and again, by the advent of computers, in the mid-20th century . The decimal representation of pi has now, as of late 2011, reached over 10 trillion digits!

Magnificent Pi

Pi (π) is the ratio of the circumference of a circle to the diameter of the circle. No matter the size of the circle, pi is always the same.

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Pi is important to geometry and trigonometry, of course, because of its relation to circles, ellipses, and spheres. But it is also found in other fields of study, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. Pi seems to be everywhere. That is why it is one of the most widely known mathematical constants.

Pi is sometimes approximated as 3.14 or as the fraction 22/7, but pi is actually an irrational number, meaning it cannot be expressed as a fraction and its decimal places continue infinitely with no pattern. If you wrote it out in full (which is impossible), its decimal places would continue forever.

Pi is also believed to be a normal number, which means simply that no digit, or combination of digits, occurs more frequently than any other. Being infinitely long and completely random, means that any string of numbers can be found somewhere in the digits of pi. Pi contains every phone number in the world, and if you converted numbers to letters, you’d find every book that has ever been written. You can search for your own phone number (or any other number) in the first 200 millions digits of pi at The Pi-Search Page. (If you don’t find a number it doesn’t mean it isn’t there, just that it is not in the first 200 million digits. Remember that pi is infinitely long.)

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