# What is the square root of 42?

##### 1 Answer

#### Explanation:

Note that

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

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