Pi Day

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].

UPDATE: The same quibble applies to the CNN story, which says “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 ….” The “That means” is precise as to the “irrational” part, but I think it doesn’t adequately capture the meaning of “transcendental, meaning it is not algebraic.”