Question

What's the square root of 146?

Answer 1

`sqrt(146) = [12;bar(12,24)] = 12+1/(12+1/(24+1/(12+1/(24+...))))`

Explanation:

Note that `146 = 144+2 = 12^2+2`

Being of the form `n^2+2` it has a continued fraction of the form:

`[n;bar(n,2n)] = n+1/(n+1/(2n+1/(n+1/(2n+1/(n+1/(2n+...))))))`

which since the continued fraction does not terminate, is irrational.

So:

`sqrt(146) = [12,bar(12,24)]`

`color(white)(sqrt(146)) = 12+1/(12+1/(24+1/(12+1/(24+1/(12+1/(24+...))))))`

`color(white)(sqrt(146)) ~~ 12+1/(12+1/(24+1/12)) = 42049/3480`