Given an irrational number, is it possible to represent it using only rational numbers and square roots(or any root, if that makes a difference)?
That is, can you define the irrational numbers in square roots, or is it something much deeper than that? Can pi be represented with square roots?
Answer
The smallest set of numbers closed under the ordinary arithmetic operations and square roots is the set of constructible numbers. The number $\sqrt[3]{2}$ is not constructible, and this was one of the famous greek problems: the duplication of the cube.
If you allow roots of all orders, then you're talking about equations that can be solved by radicals. Galois theory explains which equations can be solved in this way. In particular, $x^5-x+1=0$ cannot. But it clearly has a real root.
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