find a close rational to the square root

If you have some number X and you know the perfect square less than that number A^2 and the one greater B^2, you can get a rational number close to the square root by:

For example X = 13, 3^2 is less than 13 and 4^2 is more, the difference between them is 7 and 13-9 is 4:

Or X=15:

or X=89

I think it might actually get better the larger X is...