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 anynumber. 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 10t. 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.


  1. Pierce R. Butler says

    Even for those few writers-for-the-public who understand the meaning of “irrational numbers”, the idea of sneaking that term past their editors and to their readers without getting a flood of, ahem, unreasonable responses must look laughable at best.

  2. says

    I missed your post at the time.

    Numberphile often talks about π and how to derive it mathematically (e.g. measuring, formulae, etc.) and the tau versus pi argument is interesting.

    More interesting is Vi Hart (mathematician, musician and youtuber, mentioned in the above link) and her fun opinions on π.

    My favourite thing about π is the approximation discovered by Chinese mathematician Zǔ Chōngzhī in the 5th century CE: 1 1 3 3 5 5.

    355/113 = 3.1415929…

    It’s the best rational approximation with a denominator less than 25,000, and the only known integer ratio that with more accurate output digits (7) than input (6).

    • shallit says

      “It’s the best rational approximation with a denominator less than 25,000, and the only known integer ratio with more accurate output digits (7) than input (6).”

      False! – any approximation that immediately precedes a large partial quotient in Pi’s continued fraction will have this property. For example, the numerator


      has 217 digits while the denominator
      has 216 digits, but their quotient is good to 435 digits.

      • says

        Can I pretend I said that to see if anyone had another example? Even so, the example you gave is likely only known because of computers, like large primes. I’m fascinated by those who had to do the grunt work by hand and produced amazing results (re: my post on the 400th anniversary of Napier’s Bones).

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