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Pi vs Infinity

I was watching a TV show the other night when one of the characters repeated one of my pet-peeve clichés, “Pi is infinite”. I realize it's petty of me to care, but I find this irksome. You can parse this sentence in ways that make it technically true, but in most of those senses, it's true of other things as well. In addition, it's false in the most natural sense.

The most natural way to interpret “Pi is infinite” is as meaning, “Pi is not bounded above”. It is obviously false in this sense, since pi is less than four.

I think the sense that most people are thinking of when they say “Pi is infinite” is, “The digits in the decimal representation of Pi just keep going forever”. This is entirely true. It's also completely unimpressive, since this is true of the digits in every real number, including zero -- you just get an unbounded string of zeros.

Of course, if you were to represent zero by a finite string of digits, you could do it exactly. In fact, this is true for every fraction whose denominator in lowest-terms has only factors of two and five (since ten is divisible by both of these). So, they could mean “Pi is infinite” in the sense, “You can't represent Pi exactly as a finite decimal”. The problem there is, this is also true of every fraction whose denominator in lowest-terms has a factor besides 2 or 5, such as 1/3 or 1/7; no matter how long you make a finite string of decimal digits, it won't be exactly one third of one.

From here, they could point out that every infinite string of decimal digits derived from a fraction will eventually repeat itself, whereas Pi doesn't repeat. So, they might say “Pi is infinite” to mean, “Pi can't be represented by a simple pattern of decimal digits”. This is true, but it is still true of more mundane numbers, such as the square root of 2. Still, I can accept this, since many people find square roots to be exotic enough that their properties are still exciting.

The funny thing is, I could actually go on. (Well, OK, I find it funny. Although if you've read this far, you probably find it funny too.) As it happens, there's another set of numbers between the Rational Numbers (fractions) and the Real Numbers called the Algebraic Numbers. The Algebraic Numbers consist of those numbers that are the zeros of polynomials with rational coefficients. The square root of two is an algebraic number, because if you plug the square root of 2 in for x in the polynomial “x² - 2”, you get 0. Every rational number is an algebraic number, as is every number you could make by adding together roots, or nesting roots, or other clever tricks with fractional exponents. However, not every real number is an algebraic number, since Pi is real, but not algebraic.

They might try to say that they meant “Pi is infinite” in the sense that “The decimal digits of Pi don't follow any predictable pattern”, but this is actually false. One can predict the digits of Pi using any of the many, many algorithms used to generate the digits of Pi. There is a way this can be made technically true, as I understand that most commonly used statistical tests for randomness fail to detect patterns in the digits of Pi. However, this is at best saying that “The decimal digits of Pi don't follow any *of these* predictable patterns”, which doesn't have the majesty of the statement “Pi is infinite”. Besides, I don't follow statistics research, so for all I know they've come up with a new test that does detect a pattern in Pi.

There are a few other directions one could take this. One could look at the methods for calculating the digits of Pi, and state that they are all of a certain complexity class. One could also restrict one's number-generation language to certain functions, and note that Pi is not among the numbers generated by this language. Ultimately, though, the reason the statement “Pi is infinite” bothers me is because it's so often used in a place where a reference to transfinite ordinals or large cardinals could work just as well, but instead the writers took the easy way out.

Yeah, I'm a bit of a snob that way.