Preferred Coordinate Notation

Question

Let's say I have a point A in $\mathbb{R}^2$. Should I write its coordinates as $(x_a, y_a)$, or $(a_x, a_y)$? Sometimes I see something like $(a,b)$, but if I had a triangle with vertices A, B and C, it seems like it would get confusing. I'm also intrigued by this because of programming, since point_x and point_y seem much better than x_point and y_point in this context.

I'm sure the rule of thumb is to be consistent to whichever style I pick, but I was curious if there was a convention among the mathematical community.

Answer 1

Two dimensions are not that common in mathematics, you tend to have $d$ dimensions and pick $x\in \mathbb{R}^d$, its entries are then usually denoted by $x_i$ with $i\in \{1,\dots, d\}$. In particular you don't use the $x$ and $y$ coordinates, but the $x_1$ and $x_2$ coordinates. This generalizes much better since you would run out of letters quickly otherwise. Since two dimensions are somewhat non-standard, both of your notations would be fine (but non-standard).

In programming there are tuples or lists which you typically access with square brackets. E.g. in python

a = (1,0,0)
a[0] # first coordinate

again you would only use x_point and y_point in the case where you truly only want two coordinates. For this you could also use structs, e.g. in julia

struct Point2d
    x
    y
end

p = Point2d(1,0)
p.x # x coordinate
p.y # y coordinate

there is also the concept of named tuples which lets you avoid defining a struct but have the same p.x access option. They might be defined as p=(x=1, y=2).