What is the square root of 42?

##sqrt(42) ~~ 8479/1350 = 6.48bar(074) ~~ 6.4807407##

##42=2*3*7## has no square factors, so ##sqrt(42)## cannot be simplified. it is an irrational number between ##6## and ##7##

Note that ##42 = 6*7 = 6(6+1)## is in the form ##n(n+1)##

Numbers of this form have square roots with a simple continued fraction expansion:

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

So in our example we have:

##sqrt(42) = [6;bar(2, 12)] = 6+1/(2+1/(12+1/(2+1/(12+1/(2+...)))))##

We can truncate the continued fraction early (preferably just before one of the ##12##'s) to get good rational approximations for ##sqrt(42)##.

For example:

##sqrt(42) ~~ [6;2,12,2] = 6+1/(2+1/(12+1/2)) = 337/52 = 6.48bar(076923)##

##sqrt(42) ~~ [6;2,12,2,12,2] = 6+1/(2+1/(12+1/(2+1/(12+1/2)))) = 8479/1350 = 6.48bar(074) ~~ 6.4807407##

This approximation will have approximately as many significant digits as the sum of the significant digits of the numerator and denominator, hence stop after ##7## decimal places.