Does the range of the correlation coefficient or the covariance belongs to $\mathbb{Q}$

Question

I have a question that seems to me logic, but I haven't seen anywhere such a claim.

Could we claim that the range of the correlation coefficient or the covariance belongs to $\mathbb{Q}$, namely in the set of rational numbers? Is any theorem or something that gives a connection between them?

Answer 1

If your data points are themselves rational, then the covariance will be rational since the covariance formula involves only addition, subtraction, multiplication, and division. The formula for the correlation coefficient $r$ involves square roots, so might be irrational, but $r^2$ will be rational.