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A brief history of pi

International love for the most irrational of numbers
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Pi Day is celebrated on March 14. CORINNE LANGMUIR/THE VARSITY
Pi Day is celebrated on March 14. CORINNE LANGMUIR/THE VARSITY

Pi — the ratio between the circumference and diameter of a circle — is perhaps one of the most famous numbers in all of mathematics. Irrational, transcendental, and never-ending, it has frustrated and fascinated mathematicians across cultures for centuries, and it has become a universally recognized symbol for the field in popular culture.

The concept of pi is ancient — it’s been known to humanity for at least four thousand years. In 2000 BCE, Babylonian mathematicians performed the first known calculation of the area of a circle using the circumference of an inscribed hexagon, and derived an approximate value for pi of 3.125. A little further south, in ancient Egypt, the Rhind papyrus, dated circa 1650 BCE, reveals that a value of 256/81 — around 3.16045 — was used to approximate pi.

Around 250 BCE, Archimedes made a major breakthrough in calculating a more accurate value of pi by using the Pythagorean theorem. He approximated the area of a circle by comparing the areas of a polygon inscribed inside the circle and one that surrounded it. By doing so, he proved that there was a constant ratio between the area of a circle and the square of its radius, and he was able to obtain an upper bound of 22/7 and a lower bound of 223/71 for the value of pi.

In the centuries that followed, mathematicians around the world were able to extend the number of known decimal places of pi by performing extensive calculations. One notable contributor to these calculations was Chinese mathematician and astronomer Zu Chongzhi, who approximated pi to be around 355/113. 

Unfortunately, very few specifics are known of his work, as his book has been lost to history. However, based on his results, it is hypothesized that he would have used a method similar to that of Archimedes, starting with a regular polygon with 24,576 sides inscribed inside a circle and performing lengthy calculations involving hundreds of square roots carried out to nine decimal places in order to to arrive at his answer. 

By 1600, pi had been calculated up to 35 digits using the method of inscribed polygons, although with little theoretical progress made beyond Archimedes’ work. In the seventeenth century, however, new analytical techniques were developed in the field of mathematics which allowed for improved calculations of pi using infinite series. 

One of the earliest formulae for calculating pi was proposed by English mathematician John Wallis in 1656. While calculating an integral in an attempt to find the area of a circle with a radius of one, he established a formula involving the multiplication of an infinite series of fractions that was based on the value of one-half of pi. Later that century, Sir Isaac Newton — in one of his many achievements — used his binomial theorem to quickly calculate the value of pi up to 16 decimal places.

Next came the Gregory-Leibniz series, which made use of both infinite series and trigonometric functions to develop formulae for values of pi divided by four and six. By the end of the eighteenth century, over one hundred digits of pi had been calculated using this method and others derived from it. 

In this same century, two major developments also occurred concerning the nature of pi: in 1761, Johann Lambert proved that pi was irrational — that is, that it cannot be expressed as the ratio of two numbers — and Ferdinand von Lindemann concluded that it was transcendental soon after. It was also during this period that the use of the Greek letter pi was introduced by William Jones, in 1706, and that it was later popularized by Leonhard Euler, beginning in 1737.

Progress slowed once again for the next few hundred years, as more calculations were made but with little conceptual development. Eventually, though, in the early twentieth century, Indian mathematician Srinivasa Ramanujan developed an incredibly efficient formula for calculating pi based on its reciprocal fraction, which was later incorporated into computer algorithms. 

The subsequent development of more advanced computers meant that pi could be calculated to higher and higher decimal places. By 1949, the ENIAC computer could calculate 2,037 digits of pi, and the IBM 704 was able to calculate 16,167 digits a decade later. The IBM 7090 broke the 100,000 digit mark in 1961, and the CDC 7600 was able to reach pi’s millionth decimal place in 1973. 

Even now, the value of pi continues to be further unravelled up to and beyond its 50 trillionth digit. Mathematicians are still searching for possible patterns within pi — if its digits should ever begin to repeat. However, even if no pattern is ever found, calculating the value of pi is still valuable for the development of computer programs.

In modern times, the number pi has become ubiquitous in popular culture, in ways that reach far beyond its roots in ancient geometry. Prime-time television shows such as The Simpsons and Twin Peaks, as well as films including the Oscar-winning movie Life of Pi and the Hitchcock classic Torn Curtain, have made use of the number for humorous one-liners, mysterious codes, and everything in between. 

There is even a holiday dedicated to pi, appropriately known as Pi Day, which is observed annually on March 14 — or 3/14. In 1989, physicist Larry Shaw was the first to hold a large-scale organized celebration of Pi Day, when he and his colleagues at the San Francisco Exploratorium marked the occasion by doing laps around a circular area of the building and then eating fruit pies. The holiday has since been recognized by the U.S. House of Representatives, honoured in two Google Doodles, and celebrated by people all over the world. Each year on March 14, math lovers gather for pie eating, pi recitation contests, and discussions on the significance of pi. 

The constant, which developed from a simple fascination with circles and numbers, has altered the course of science and mathematics, charmed the most brilliant minds of human history, and captivated millions around the world. Pi Day is a lighthearted way of recognizing the importance of the number itself, but it’s also a celebration — not only of mathematical and scientific achievement, but of the dedication and passion we give to what intrigues us and what we love.