Pi (Math Lair)

"The primary purpose of the DATA statement is to give names to constants; instead of referring to pi as 3.141592653589793 at every appearance, the variable PI can be given that value with a DATA statement and used instead of the longer form of the constant. This also simplifies modifying the program, should the value of pi change."
—FORTRAN manual for Xerox Computers

If you were to measure the diameter of circles of many different sizes, you would notice that the ratio between the circle's circumference and its diameter remains constant no matter what its diameter is. This ratio is represented by π (that's the Greek letter pi).

The earliest-known record of this ratio was written by the Egyptian scribe Ahmes around 1650 B.C. and has been preserved in the Rhind Papyrus. In one passage, Ahmes implies that this ratio is (¹⁶/₉)², or 3.16049... . To five decimal places, the actual value of π is 3.14159. Ahmes' value was within one percent of π's true value.

What kind of a number is π, though? It turns out that π is not rational; that is, it cannot be written as the ratio between two numbers. Furthermore, π is transcendental; that is, π cannot satisfy any algebraic equation. Since this number is irrational and transcendental, the digits of the number are inherently unpredictable.

Currently, over 5 trillion digits of π have been calculated. You can view the first 10,000 digits here. You might be wondering: why? Certainly nowhere near that many digits are required in calculations to provide results that are very close to the "exact" value. There are several reasons. One reason is that calculating π to so many digits shows how powerful a supercomputer is. Another reason is that mathematicians are interested in whether the sequence of numbers in the decimal representation of π is as random as it looks. It is possible that we may discover some sort of pattern within the decimal places of π if we have enough data.

One interesting recreation is to find fractions that are close approximations of π. A well known one is ²²/₇, which is 3.142857, which gives π to two decimal places. Another is ³⁵⁵/₁₁₃, which gives 3.141592..., which is π to 6 decimal places. Yet another curious one is ^21053343141/_6701487259, which is 3.141592653589793238462381742774869013595...; the first 20 decimal places are the same as the first 20 decimal places of π!

One book about π that I particularly enjoyed is The Joy of π by David Blatner, which was released about ten years ago (for more information on this book see my bibliography page).