What is the square root of 42?

1 Answer
Sep 17, 2016

Answer:

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

Explanation:

#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.