Question

What is the square root of 42?

Answer 1

`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.