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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.)

billpg
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    Historically, I think the notion of "proportion" or "ratio" $a:b$ (colon...) made sense for any "numbers" $a$ and $b$, even if one was $0$. Maybe not for both $0$? And equality $a:b=c:d$ was the expected $ad=bc$. Unlike "fractions", there wasn't really any addition, but that seems not to have been what these things were used for. It may be that this was an algebraicization of classic Greek discussion of "proportion", without reference to "number". I have no references, so I'm not making this an answer... Good rationalization about etymology, though! :) – paul garrett Jul 22 '21 at 17:48

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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.

Troposphere
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