“$45 \cos($ thir… I mean $\frac{\pi}{6})$,” said the student, catching himself just before the axe reached his shoulder.”

“Thirty-nine,” said the Mathematical Ninja, without a pause. “A tiny bit less.”

The student raised an eyebrow as a request to check on the calculator, and the Mathematical Ninja nodded in assent. “$38.97$,” said the student. “Not bad, sensei, not bad at all. I suppose you’re going to tell me how you did it?”

“Of course,” said the Mathematical Ninja. “For a long while, I was perfectly happy with the approximation $\sqrt{3} \simeq \frac 74$.”

“It’s about 1% too high,” said the student. “Good enough for government work.”

“Indeed,” said the Mathematical Ninja, wistfully. “But not good enough for Ninja work.”

“Naturally. So you found a better approximation?”

“I did,” said the Mathematical Ninja. “$\sqrt{3} \simeq \frac{26}{15}$.”

Tappety tap. “That’s only off by 0.074%”, said the student. “So pretty much always three significant figures.”

The Mathematical Ninja cocked his head. “Is that so? I’ll leave that as an exercise.”

“It’s also got the advantage of having an even top, so you can say $\frac{\sqrt3}{2} \simeq \frac{13}{15}$ without getting your head in a muddle.”

“My head,” said the Mathematical Ninja, “is never in a muddle.”

“Of course, sensei. But that means $45 \cos \left( \frac \pi 6\right)$ is just about $45 \times \frac{13}{15}$, or 39. Cunning!”

“I’ll confess that I sometimes use $\sqrt 3 \simeq \frac{52}{30}$ or $\frac{104}{60}$ to make the division easier.”

“So you could even do $\frac 53$ and add 4%?”

The Mathematical Ninja nodded. “If you were so inclined.”

“So, if I had $20 \sqrt 3$, I could call that $20 \times \frac{5}{3}$, or $\frac{100}{3} = 33.33$, and add 4%… which is… erk… $1.33$, making $34.66$?”

“$34.64$, said the Mathematical Ninja. “It goes down by a hundredth in every 13 or so.”

The student threw down his pen. “You win, sensei.”

A smug little grin: he always did.

Edited 2015-02-21 to fix a typo - thanks to Mark Ritchings for pointing it out!