Sums of roots to approximate numbers I was thinking about random sequences, and thought up of this problem. 
Can algebraic or rational be expressed as a sum or difference of integer square roots? 
For example, $5=\sqrt{25}=\sqrt{16}+\sqrt{1}$.
It is known that if the series is infinite any number can be approximated, but is it possible to get any algebraic number(rational or irrational) with a FINITE number of terms? 

Well certainly you cannot obtain a transcendental number as a sum of algebraics since algebraics are closed under sums. Nor can you obtain every algebraic number as a sum of radical integers (else every polynomial would be solvable!). But it's easy to see that one can obtain any rational integer.
Note: In fact Heine originally defined algebraic integers to be radical integers, i.e. the ring obtained by closing $\rm\:\mathbb Z\:$ under taking $\rm\:n$'th roots. Heine claimed that every solvable algebraic integer is a radical integer - a problem which is still open according to Franz Lemmermeyer. However, it is not too difficult to show that every quadratic integer is a radical integer, e.g.
$$ \frac{\sqrt{17}+1}{2}\ =\ \frac{\sqrt{17}+\sqrt{5}}{2}\ -\ \frac{\sqrt{5}-1}{2}\ =\:\ (7\ \sqrt{5} + 4\ \sqrt{17})^{1/3} - (\sqrt{5}-2)^{1/3}$$
