Thank you CPhill and Problem,
First the queston in the heading is different from the question in the body of the post so CPHill and Problem have answered different questions.
ANONYMOUS - Please be much more careful next time you make a post.
CPhill's answer is much better because Problem's answer is an estimate. It is not an exact value for the question.
ALSO,
The second solution that Problem has given is wrong.
$$\sqrt{3}-\frac{4}{1+\sqrt{3}}\\\\
=\frac{\sqrt{3}(1+\sqrt{3})}{1+\sqrt{3}}-\frac{4}{1+\sqrt{3}}\\\\
=\frac{\sqrt{3}+3-4}{1+\sqrt{3}}\\\\
=\frac{\sqrt{3}-1}{1+\sqrt{3}}\\\\
=\frac{\sqrt{3}-1}{1+\sqrt{3}}\times\frac{1-\sqrt{3}}{1-\sqrt{3}}\\\\
=\frac{-(\sqrt{3}-1)^2}{1-3}\\\\
=\frac{-(3-2\sqrt{3}+1)}{-2}\\\\
=\frac{-(4-2\sqrt{3})}{-2}\\\\
=2-\sqrt{3}\\\\$$
.