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 is $\sqrt{\frac{1}{3}}$, which comes up fairly frequently, as it's $\tan\left(\frac{\pi}{6}\right)$ .

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.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.