Legal Commentary

Pi Day

By Eugene Volokh

A quibble about an item from the Pi Day site, which I?ve seen quoted in a few places:

Pi is an irrational and transcendental number meaning it will continue infinitely without repeating.

That pi is an irrational number does mean it will continue infinitely without repeating. For all rational numbers, and only rational numbers, the decimal representation of the number will at some point start repeating and keep repeating, e.g., 3.1415926926926926926.... (If you?re wondering about a number such as 2.5, it?s actually 2.50000000...., or for that matter 2.49999999....)

But that pi is a transcendental number doesn?t quite ?mean[] it will continue infinitely without repeating.? It?s true that all transcendental numbers are irrational, and therefore will indeed continue infinitely without repeating. But not all irrational numbers are transcendental, so a number?s being transcendental means something more than that it continues infinitely.

An irrational number (say, the square root of 2) is merely a number that can?t be represented as a ratio of two integers. But a transcendental number is a number that can?t be represented as a solution of any polynomial with integer coefficients: Square root of 2 is thus irrational but not transcendental, because it is a solution of the polynomial x^2?2=0. Pi is indeed transcendental, as is its soulmate e; but it would have been more precise to say,

Pi is an irrational number, meaning it will continue infinitely without repeating, and is also transcendental [possibly followed by a definition].

Full post as published by The Volokh Conspiracy on March 14, 2010 (boomark / email).