The origin labelled on a graph: $0$ or O?

Question

When one draws a graph, say in the x,y plane, we label the origin with a circular/elliptical symbol.

Now is this a $0$ (zero), or is it O (for Origin), or simply just a circle/ellipse?

Can it be either?

Answer 1

I think it would depend on context. When we're treating the $x,y$ plane as a description of a Euclidean plane, and we're doing things such as naming points $P = (x_1,y_1)$ and $Q = (x_2,y_2)$, then I would write $O = (0,0)$ (using the letter O as the name of the point).

On the other hand, if $x$ and $y$ are coordinates of a vector space, then there is indeed a zero vector whose name is $0$ (numeral zero) and whose coordinates are $(0,0)$. So in that case it makes more sense to use the numeral zero to label the "origin".