TITLE: Why is $\log_{10}\pi \simeq 0.5$?
QUESTION [5 upvotes]: I was looking at my slide rule earlier today, and happened to notice that the dedicated mark for $\pi$ is right around the middle of the scale:

Of course, this is because $\log_{10}\pi \simeq 0.4971498$, which is very close to $1\over2$.
Is there a reason or derivation for this, or is it pure coincidence?

REPLY [3 votes]: I cannot see a very deep reason why this would be so, but the continued fraction of $\sqrt{10}$ is $3 + \frac{1}{6 + \frac{1}{6 + \cdots}}$ and a continued fraction for $\pi$ is $3 + \frac{1}{6 + \frac{9}{6 + \cdots}}$ so they at least agree to $3 + \frac{1}{6}$, which after all is not too bad of an estimate, being off by only 2 in the second decimal place.
I suppose it is worth noting that a general continued fraction for square roots of the form $\sqrt{a^2 + b} = a + \frac{b}{2a + \frac{b}{2a + \cdots}}$ then there is straightforwardly no better approximation to $\pi$ by the square root of an integer since $a$ is constrained to be $3$ and $b$ is constrained to be $1$ right at the start.
I'm not sure how to consider things "coincidental" at this level of navel gazing.