What's the square root of 146?

1 Answer
Aug 10, 2018

#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#