While I’m no Mathematical Ninja, it does amuse me to come up with mental approximations to numbers, largely to convince my students I know what I’m doing. One number I’ve not looked at much ((In fairness, the vast bulk of numbers have never had any attention at all from me, and never will)) is $\sqrt{\frac{1}{3}}$, which comes up fairly frequently, as it’s $\tan\left(\frac{\pi}{6}\right)$ ((that’s 30º in silly money.)) .

Ninja-chops taught me all about the square root of three – $\frac{52}{30}$ is good to about one part in 1,350 – and that gives two different ways to estimate $\sqrt{\frac{1}{3}}$.

The first is to simply use the reciprocal: $\sqrt{\frac{1}{3}} \approx \frac{30}{52} = \frac{15}{26}$. Coincidentally, the Ninja has also recently revealed his secrets about twenty-sixths, so we can say that $\frac{15}{26} = \frac{1}{13} + \frac{1}{2} = 0.5\dot 769 23\dot 0$.

Another way is to multiply the approximation for $\sqrt{3}$ by a third to get $\frac{52}{90}$, which isn’t hard to work out: $5.2 \div 9 = 0.5\dot7$.

So, the square root of a third is somewhere between 0.577 and 0.578. In fact, it’s pretty close to 0.57735.

Wolfram|Alpha lists $\frac{15}{26}$ as one of the convergents of the square root of a third; the best approximation with a denominator smaller than 100 is $\frac{56}{97}$, which is 0.57732 or so, correct to one part in nearly 19,000.