TITLE: Is it possible to find square root using only rational numbers and elementary arithmetic operators
QUESTION [7 upvotes]: Suppose I have a number a
How can I find it's square root using only +, -, /, * and rational numbers?
If it is impossible how to prove it?

REPLY [7 votes]: If you allow infinite number of operations, then you can use some algorithm.
One easy example is root searching via Newton's method. Here we do the iteration
$$x_{n+1} = \frac{a + x_n^2}{2x_n},$$
which eventually converges to $\sqrt{a}$ if $a$ and $x_0$ are positive.
See https://en.wikipedia.org/wiki/Methods_of_computing_square_roots for other methods.