(Copyright © 2000-2001 --
this item revised 26 Nov 2000)

The greek letter pi is used for the ratio of a circle's circumference to its diameter, perhaps because pi is the first letter of the Greek word for periphery.

We usually substitute 22/7 for pi in calculations. It's not a bad
approximation, but always remember that it *is* an
approximation. The value of pi cannot be stated exactly as a decimal or a
fraction.

Pi is not just an irrational number (equal to no fraction of integers), it is a transcendental number. That means that there is no algebraic equation with integer coefficients that has pi as a solution. Therefore we can only approximate pi, never write an exact value, in base 10 or any other number base.

Nearly all ancient peoples used 3 as the everyday value of pi, and it's even enshrined in the Judeo-Christian Bible (I Kings 7:23). In 1897 one house of the Indiana legislature unanimously passed a bill defining pi as 3.2 (not 3, as often reported). The article Pi through the ages runs through a number of the historical values of pi.

Our modern approximation of 22/7 is due to the great Archimedes (287-212 BCE). Actually, he gave pi as lying inside an interval:

223/71 < pi < 22/7

22/7 is still the best small-number ratio that approximates pi; it's about 3.142857, which is correct to 0.04%.

A better approximation, good to 0.002%, was discovered by Hipparchus (late second century BCE):

pi ~ 377/120 ~ 3.14166667

The astronomer Tsu Chúng-chih (430-501 CE) came up with a much better approximation:

pi ~ 355/113 ~ 3.14159292

This is correct to 0.000008%, and is the closest existing approximation that uses a ratio of three-digit integers.

The decimal expansion of pi begins

pi ~ 3.141592653589793

which is more than sufficient for any computation of a phyhsical measurement.

Computers calculate successive approximations using convergent infinite series. The faster the series converges, the more digits of pi you can get for the same amount of work. Many sites on the Web give various series expansions of pi; one of them is Pi through the ages. Many books have been devoted to the topic. The classic is Petr Beckmann's A History of pi, but it's about 20 years out of date. Check subject heading "pi" in your library's card catalog for more recent works.

Your calculator probably has a "pi" key. If so, it almost certainly just stores a few digits of pi, and does not compute it on the fly.

- Stan Brown
- Haran Pilpel

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