how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional condition: $$ x \gt 1 $$ we know that a solution to this equation is specified by $$ x=\sqrt{2} $$ expressed in this way, it might be argued that the solution contains a quantity of information, $I$, which has the same order of magnitude as the information content of the question.
on the other hand it is evident that, measured by the activities of a suitably co-operative Turing machine, the quantity of information in the solution is in fact (something like): $$ ln \; \aleph_1 $$ how should i attempt to reconcile these two points of view?
In the sense of Kolmorogrov complexity the information content of $\sqrt 2$ is the length of the shortest program to calculate it. The exact value depends on your language, but it is not huge. It is certainly not infinite.