Differentiating between points and their coordinates In Croatian mathematical textbooks it is customary to use the notation $A(x,y)$ to mean that $(x,y)$ are the coordinates of point $A$. The coordinates of the point are clearly differentiated from the point itself. However, I have never encountered this notation in foreign (English) literature. Rather, I’ve seen this written as $A=(x,y)$. This confuses me because if interpreted literally the notation says that point $A$ is the ordered pair of coordinates $(x,y)$ with no differentiation made between the two whatsoever.
Is this notation shorthand or are points and ordered pairs of real numbers indistinguishable in this paradigm? I get that it makes no difference in practice since every n-dimensional Euclidean space is isomorphic to $\mathbb{R}^n$ but I’m still curious as to the precise meaning of this notation. If the notation should be taken literally, i.e. points are tuples, what notation should be used to express coordinates for points in other Euclidean spaces? Any insight is appreciated :)
 A: I think there are three distinct concepts here.

*

*Point is a geometric term. Euclid's axioms can speak about points all day without mentioning coordinates even once.

*Vectors are a concept from linear algebra. You can have vector spaces without caring too much about a basis, and even do algebraic operations on these.

*Coordinates are just pairs of numbers (in 2d). So at that level you have essentially arithmetic objects and operations.

Now all of these are related. And the typical relationship that of a model, which is a structure satisfying a set of axioms. The pair of numbers with the operations you learned satisfy the axioms of a vector space, hence they can be used as a model for that vector space. Likewise a vector space satisfies the axioms of Euclidean geometry, so it can serve as a model for that. So this "model for" is a more nuanced concept instead of "the same as". I've expressed similar views before in this answer to the question of whether the Euclidean plane is the same as $\mathbb R^2$.
Note that none of the concepts speak about names. You can say "I have a point called $A$, and I describe that using a vector which I'll represent (with respect to the standard basis) as these numbers". But you can also say "I have a point, and I describe that using a vector $A$ which has these coordinates". The former is closer to the $A(x,y)$ notation, the latter closer to $A=(x,y)$ but as always, notation is to a large part convention with fuzzy meanings unless you define them more strictly for your specific use case.
I recall that my teachers at some point tried hard to distinguish between the point in the plane, and the vector pointing from the origin to that point. Not sure if that distinction served any specific purpose, but I do recall that my mental distinctions blurred over time. Perhaps that's the key thing: as long as you know that some concepts are distinct at a certain level, you can very often ignore the distinction during most steps, and still focus on the distinction when it matters.
One great benefit of $A=(x,y)$ is that it allows you to have $(x,y)$ itself as an anonymous point. As constructions grow more complicated, naming every intermediate point can become tedious at some point. And the second notation allows you to name only things you care about, while still staying consistent.
