Continued fractions and the square root of 3.
I’m a Big Fan of both @standupmaths and @sparksmaths, two mathematicians who fight the good fight.
I was interested to see Ben tackling the square root of 3 using the ‘long division’ method. It’s a method I’ve tried hard to love. It’s a method I just can’t bring myself to do or recommend. @colinthemathmo has an explainer here, probably as nice an explainer as the method permits.
I have two alternative methods for finding the square root of three.
The first, Ben alludes to in the video: use the binomial expansion. My instinct is to find a number
Now you can apply the binomial expansion:
… use continued fractions. Or a matrix version thereof.
Like Ben, I’m going to skip the derivation and jump straight to the method: the key matrix is
Applying this matrix to
And we can be clever about calculating
The approximation from
From
Squaring
Squaring
The operations here are not trivial 1, but they’re more tedious than difficult. Multiplying and adding large numbers is typically easier than long division – and in fact, multiplying repeatedly by
You’re left with one big division to do at the end, which is a bit more difficult than the other sums – but it’s just one division!
Now, Sir Isaac may not have had continued fractions available to him - but the maths involved here is certainly achievable on quill and parchment. I’m definitely not saying this is the method Ben should have used in the video (I mean, the whole point was to do it an absurd and 17th-century way), but figured it was a nice method to share.
Footnotes:
1. hush, sensei