Which is bigger: $\sqrt{1001} - \sqrt{1000}$, or $\frac{1}{10}$? 
Which is bigger: $\sqrt{1001} - \sqrt{1000}$, or $\frac{1}{10}$?

I can calculate the answer using a calculator, however I suspect to do so may be missing the point of the question.
The problem appears in a book immediately after a section called 'Rules for square roots'with $\sqrt{ab} = \sqrt{a}.\sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ as the given rules.

We want to find the sign of $\sqrt{1001} - \sqrt{1000} - \dfrac{1}{10}$.  Multiplying by the obviously positive $\sqrt{1001} + \sqrt{1000} + \dfrac{1}{10}$ we get $(\sqrt{1001})^2 - (\sqrt{1000} + \dfrac{1}{10})^2 = 1001 - 1000 - \dfrac{2\sqrt{1000}}{10} - \dfrac{1}{100} = \dfrac{99}{100} - \dfrac{2\sqrt{1000}}{10}$
Since $\sqrt{1000} > 30$ this quantity is obviously negative.  So $\dfrac{1}{10}$ is larger.
