Why are these curves' points called rational?

Question

While studying curves defined over finite field $\mathbb F_q$, it's said that the points of the curve are the rational points. Why is it said like this? For example, if $q$ is prime, aren't we just saying the curve only consists of the points with integer coefficients modulo $q$?

They're obviously rational, but only integers. Why are they called rational? In which case would the coordinates of the points be rational and not integer numbers (or polynomials with integer coefficients if $q$ is a power of a prime)?

Answer 1

It is a convention to call a point $P=(x_1,\ldots ,x_n)$ (on a algebraic variety over a field $K$) $K$-rational, or just rational, if all $x_i$ belong to the field $K$. See example $2$ for a "rational" point $P=(\sqrt{2},3)$ on the algebraic variety given by the equation $3x^2−2y=0$, where the coordinates are not rational numbers, but the point is $K$-rational with $K=\mathbb{Q}(\sqrt{2})$.

Answer 2

Yes, I was also baffled by this terminology initially. Some considerate authors use a less-confusing phrase, $K$-valued points. The amusing part of this convention is that despites Andrew Wiles' theorem the Fermat curve $x^4+y^4=z^4$ defined over the complex numbers, has a (actually uncountable number of) rational point(s)! For example $(i,i,\sqrt[4]2)$ is one such!