On reddit, an interesting question:

Given one hour, an unlimited amount of chalk, and an unlimited amount of blackboard space, how many (correct) digits of $\sqrt{10}$ could you find?

(without any calculation aids, obviously).

At the moment, the idea of having a free hour to do anything, let alone work out huge bits of chalkboard arithmetic, seems like a crazy luxury. However, I can grab a few minutes here and there to try out a few different methods. I have four morally different ones I might attempt.

Method 1: the long division method

There are few things in maths, especially mental maths, that I dislike intensely. Despite @colinthemathmo’s excellent write-up on the method, the long division way of finding square roots sadly leaves me cold. I could probably do it, in a pinch, but it’s not my idea of a fun hour. Let’s put that to one side.

(I’m going to treat “Figure out $n$ so that $n^2 \approx 10$, then use that to find $n^2 \approx 1000$, etc” as a variation on this.)

Method 2: Taylor series / binomial expansion

This is probably the method most of my mathematically-minded friends would try: use $(1+x)^{1/2} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \dots$. Stick in $x=\frac{1}{9}$ and out pops $\frac{\sqrt{10}}{3}$.

It’s a pain, though. The numbers get big, quickly. And it’s also a pain because the Mathematical Ninja is likely to pay a visit to anyone who uses that form rather than $(a^2+b)^{1/2}= a + \frac{1}{2a}x - \frac{1}{8a^3}x^2 + \dots$. My first thought was $a=3$ and $b=1$, but I’ve since realised that $a=19$ and $b=-1$ converges much more quickly – two terms of that gives $\frac{721}{228}$, which is correct to better than one part in a million.

It may also be worth trying $(1-0.1)^{1/2}$, which gives $0.3\sqrt{10}$ and numbers that aren’t too horrible.

Method 3: Babylonian!

I’d forgotten that the Babylonians had a method for this: let your initial guess be $x_0$; then $x_1 = \frac{x_0 + \frac{10}{x_0}}{2}$ is a better guess. This doesn’t converge super-quickly, and the fractions get ugly fast, but I still prefer it to long division.

Method 4: Continued fractions

Of course I’d do it with continued fractions. $\sqrt{10}= [3; 6,6, 6,\dots]$, so it’s fairly easy to generate the convergents using the following process:

  • $x_0 = 1$; $x_1 = 3$; $x_{n+2} = 6x_{n+1} + x_n$
  • $y_0 = 0$; $y_1 = 1$; $y_{n+2} = 6y_{n+1} + y_n$

And $\frac{x_n}{y_n}$ is a good estimate for $\sqrt{10}$.

A few minutes’ work got me to $\frac{168,717}{53,353}$, which is good to ten or eleven decimal places.

What about you? Do you have any better methods, either more efficient for chalkboard work or less likely to go wrong? I’d love to hear about them!