# How do you find a one-decimal place approximation for sqrt 50?

Oct 21, 2015

$\sqrt{50} \approx 7 + \frac{1}{14} \approx 7.1$

#### Explanation:

Since $50 = {7}^{2} + 1$ is of the form ${n}^{2} + 1$, its square root is a very simple continued fraction:

sqrt(50) = [7;bar(14)] = 7+1/(14+1/(14+1/(14+...)))

A rough approximation can be made by truncating early:

sqrt(50) ~~ [7;14] = 7+1/14 = 99/14 = 7.0dot(7)1428dot(5)

If we want a better one, just include more terms:

sqrt(50) ~~ [7;14;14] = 7+1/(14+1/14) = 7+14/197 = 1393/197 ~~ 7.071066

Actually $\sqrt{50} \approx 7.07106781186547524400$