Dear Uncle Colin,

If I didn’t have a calculator and wanted to know the decimal expansion of 2, how would I be best to go about it?

Roots As Decimals - Irrational Constant At Length

Hi, RADICAL, and thanks for your message!

There are several options for finding 2 as a decimal (although why you’d want to know it beyond about three places, I have no idea).

One is the ‘long division’ method that @colinthemathmo has explained in more detail than I would ever care to, so I’ll just send you there if you’re interested in that.

But there are two other methods I would consider.

The binomial expansion

If you have a reasonable guess for the value of Rn, you can find an improved guess by working out R=R+nR22R, which comes from the binomial expansion. For example, if you guessed that 21, your second guess would be 1+212=1.5, which is significantly closer.

You can repeat this process as often as you like: with R=1.5, the next guess is 1.5+2943=32112. That’s 17121.4167, which is still closer to the true value.

You can either work with fractions and do a great big long division at the end, or you can say (for example) “15 is my estimate for 200”, or “142 is my estimate for 20000” - which will converge more slowly, but may be less annoying.

Continued fractions.

With a litte bit of surd work, you can show that:

2=1+(21)=1+11+2

But, you know that the 2 on the bottom of the last term can be written as 1+11+2, which would make the whole thing 1+12+11+2. That can be repeated again, and it turns out that:

2=12+12+12+, following the same pattern forever. The more levels you expand the continued fraction to, the more accurate your result - the first few terms in the sequence are 1, 32, 75, 1712, …

In fact, if you know that the kth term is ab, the (k+1)th term is a+2ba+b, and a better approximation to 2.

Again, you can use this to generate an approximation as close to the answer as you would like!

Hope that helps,

- Uncle Colin

* Updated 2019-02-06 to correct LaTeX in title.