Bad Mathematics on Π Day

Every March 14 you can expect a flood of stories about the famous number π = 3.14159... . And you can also expect that most of them are bad, written by people who know very little mathematics and don't bother to get their facts checked by a mathematician.

Here's an example: the writer, Tia Ghose, says that π is "an infinitely long, nonrepeating number". Later she says "Because pi is an infinite number...".

This is not correct. More than that, it illustrates a very, very common misunderstanding: confusing a real number with its representation in some base.

Real numbers do not have "lengths" and are not "infinitely long". π is not an "infinite number". (All real numbers are finite.) Instead, one could speak about the base-10 representation of a number (or, more generally, about the base-k representation of the number for some integer k ≥ 2). This representation is "infinitely long", but so is the representation of every real number! For example the base-10 representation of 1/3 is 0.3333.... and the base-10 representation of 1/2 is 0.5000.... . The thing that makes π different from these examples is not that π's representation is "infinitely long", but rather that it is not ultimately periodic. "Ultimate periodicity" means that, after some point, the representation consists of a single block of digits that repeats forever, so the representation looks like x.uvvvvvv.... for some blocks x, u, v.

That's what the author apparently means by "nonrepeating", but "nonrepeating" is such a vague term that it should be avoided. Some kinds of repetitions are inevitable in the base-10 representation of any number. For example, in the base-10 representation of π we quickly find the repetitions "33" and "88" and "99". The first group of three consecutive identical digits to appear is 111; the first group of four consecutive identical digits to appear is 9999. At the same position we even get 999999 ! Now, not every real number will have a base-10 representation with blocks of consecutive identical digits, but there are beautiful theorems, like Dejean's theorem whose proof was recently completed, that say specifically what kinds of repetitions cannot be avoided.

More bad math follows in the same article. The writer claims that "normal numbers" are those "numbers that have the same frequency of all the digits". This is not correct. A normal number to base 10 satisfies a much stronger property: namely, that every block of digits occurs with the expected frequency: if the block has length t then it must occur with limiting frequency 10^-t. An example of a number that "has the same frequency of all the digits", but is not normal, is .01234567890123456789... = 13717421/1111111111. No rational number can be normal, because the number of distinct blocks of length t that appear in a rational number is eventually constant.

What happens when bad mathematics journalism like this gets reposted by the World's Worst Journalist™, Denyse O'Leary? Why, she just quotes it verbatim with no understanding. Despite the fact that nobody knows for sure whether π is normal, and despite the fact that nearly all mathematicians suspect it probably is normal, she insists that it is not! Even when commenters try to set her straight, she still doesn't seem to understand that "normal" is a term of art in mathematics, and does not correspond to the ordinary English meaning.

Bad mathematics all around.