What is the square root of 42?

1 Answer
George C.
Sep 17, 2016

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.

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