This is a follow-up question to Ratio when one entity is 0.
This question asks if ratios can have a zero, with answers that can be summarised as "It's complicated but no".
That's fine, but what are they called if they're not ratios?
X:Y means that same thing as a ratio when neither X nor Y are zero, but unlike ratios, you can say 1:0. This means is that for every one X, there's zero Ys. (0:1 would equivalently mean there's zero X for every one Y.)
(You could also say 58:0 but that's the same in practice as 1:0.)
Because X or Y might be zero, you shouldn't divide X by Y, or at least not without checking for zeros first.
If it turns out these don't have a name, can we please name it a "Billtio"?
(Pronounced bill-she-oh. I'm presuming ratios were invented by someone called Ray.)
I think what you're looking for might be the real projective line.
It consists of equivalence classes of points in $\mathbb R^2 \setminus\{(0,0)\}$, under the equivalence relation that relates $(a,b)$ to $(c,d)$ if there exists an $x$ such that $c=xa$ and $d=xb$.
If you restict to $a:b$ where $a$ and $b$ are integers (or rationals) you get a projective line over $\mathbb Q$ instead.