# What is the square root of 42?

Sep 17, 2016

$\sqrt{42} \approx \frac{8479}{1350} = 6.48 \overline{074} \approx 6.4807407$

#### Explanation:

$42 = 2 \cdot 3 \cdot 7$ has no square factors, so $\sqrt{42}$ cannot be simplified. it is an irrational number between $6$ and $7$

Note that $42 = 6 \cdot 7 = 6 \left(6 + 1\right)$ is in the form $n \left(n + 1\right)$

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.