# What is the square root of 337?

May 13, 2016

$\sqrt{337} \approx 18.35755975$ is not simplifiable since $337$ is prime.

#### Explanation:

$337$ is prime - it has no positive factors apart from $1$ and itself.

As a result, $\sqrt{337}$ is not simplifiable.

It is an irrational number which when squared (multiplied by itself) gives you $337$. Its value is approximately $18.35755975$.

Since it is irrational, its decimal representation neither terminates nor recurs.

It has a continued fraction expansion which does repeat, namely:

sqrt(337) = [18;bar(2,1,3,1,11,2,4,1,3,3,1,4,2,11,1,3,1,2,36)]

$= 18 + \frac{1}{2 + \frac{1}{1 + \frac{1}{3 + \frac{1}{1 + \frac{1}{11 + \frac{1}{2 + \frac{1}{4 + \frac{1}{1 + \ldots}}}}}}}}$

To construct rational approximations for $\sqrt{337}$ you can truncate this continued fraction.

For example:

sqrt(337) ~~ [18;2,1,3,1] = 18+1/(2+1/(1+1/(3+1/1))) = 257/14 ~~ 18.357