I have recently learned the proof for why you cannot "double" the cube, trisect the angle, and "square" the circle. I understand the whole analysis, assuming that a point is constructible if it is from $\mathbb{Q}\times\mathbb{Q}$. However, I am wondering, why do we assume that a point is constructable only if it is from $\mathbb{Q}\times\mathbb{Q}$.
My book give the definition that "We call a point in the plane constructable if it is constructable from $\mathbb{Q}\times\mathbb{Q}$, that is, from the set of all points in the plane with rational coefficients".
This definition seems to make sense. However, what guarantees that this is the case? I understand that the proof would not work if we considered a point being constructible from $\mathbb{R}\times\mathbb{R}$, but why not $\mathbb{R}\times\mathbb{R}$? What about a straight edge and compass?