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Pi

by Liliana Usvat

π Day

The mathematical constant Pi, sometimes written as Pi, is approximately equal to 3.14159... Each year, Pi Day is celebrated on March 14 by math enthusiasts around the world.

Definition of π

Pi (Greek letter “π”) is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to its diameter — which is approximately 3.14159.

Pi has been calculated to over one trillion digits beyond its decimal point. As an irrational and transcendental number, it will continue infinitely without repetition or pattern. While only a handful of digits are needed for typical calculations, Pi’s infinite nature makes it a fun challenge to memorize, and to computationally calculate more and more digits.

History

The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer approximation.

The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for pi.

The first calculation of pi was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed.

Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71.

A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for pi, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.

An Eighteenth century French mathematician named Georges Buffon devised a way to calculate pi based on probability.

Despite efforts to calculate pi by everyone from Archimedes to Sir Isaac Newton to present-day mathematicians with supercomputers, there is still no formula that would allow you to figure out, in base 10, any digit of pi without having to calculate everything that came before it. In other words, if you wanted to know the 24,000th digit, there's no way of figuring that out without putting down all 23,999 numbers before it. Such calculations can be done in binary, but it's not so interesting to know whether it's just a 0 or a 1.

Mathematicians know that pi is irrational -- it cannot be represented as one number divided by another -- and transcendental, meaning it is not algebraic. That means, theoretically, that its digits will continue on indefinitely without ending in repetition -- in other words, the digits won't suddenly continue infinitely as 5s after 3 trillion digits (Pi's digits were calculated out to a record 2.7 trillion places in December by French computer scientist Fabrice Bellard).

March 14 is also Albert Einstein's birthday.

The music of π

Who named π?

William Jones (1675 – 1749) » First mathematician to use the Greek letter p to represent the important ratio While talk of the ratio has been around for about 4,000 years, and the number itself probably a bit longer, the symbol p is just reaching the big 3-0-0. A fellow from the Welsh island of Anglesey, who grew up to be a rather well-connected but unmemorable mathematician, decided to use the Greek symbol for “p” in a math-for-beginners guide he published in 1706. He figured it would look nicer than the “p” used in prior years, which stood for “periphery.”

It probably wouldn’t have sunken in to the math community, had it only been used by a guy who spent most of his mathematical prime (young adulthood) teaching math onboard naval battleships and in London coffeehouses. But when a much bigger fish named Leonhard Euler used Jones’ new notation in his works about 30 years later, the symbol was here to stay.

Jones attempted to secure a teaching job at an actual math school, but even with references written by his friends, Sirs Isaac Newton and Edmund Halley, he was rejected and stuck to his coffeehouse crowd. He spent his later adult life serving in a variety of cushy political jobs, which he got through some key connections after losing all his money in a bank collapse. It’s always nice to have friends in high places.

Philosophy of Pi

The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine.

The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511…) It cannot be expressed in terms of the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a dimension that transcends the linear rationality that it contains. Our holistic perspectives, feelings and intuitions encompass the finite elements of the ideas that are within them, yet have a greater wisdom than can be expressed by those ideas alone.

"Chance favors the prepared mind." - Louis Pasteur