Are ratios like $4:0$ or $0:4:0$ defined? I saw such ratios being used to describe the phenotype ratio in a mono-hybrid cross – tall plants:short plants $=0:4$.
[Math] Are ratios with zero defined
ratio
Related Solutions
The notation $a:b$ emphasizes a relative relationship between $a$ and $b$, and the notation $\frac{a}{b}$ emphasizes an operation on two elements $a$ and $b$.
But ultimately the two symbols represent the same thing (at least, when using integers): a relative size between $a$ and $b$. If you think carefully about what it means for two ratios to be equivalent, you'll find that the definition of equality of $\frac{a}{b}$ and $\frac{c}{d}$ is just that the ratios $a:b$ and $c:d$ are equal.
Actually, I do believe that I've seen posters from certain countries actually use "$:$" for division, to the confusion of the rest of us.
One difference in these two notations is that you can link a lot of ratios together at once like this: $1:2:4:7$. This expresses a bunch of ratios at once: $1:2$, $2:4$, $4:7$, $1:4$, $2:7$ etc. If these were ratios of ingredients in some mixture recipe, then you could rather handily increase and decrease the size of your recipe as you desired using this notation.
But this does not translate over to the slash notation, which becomes problematic if you're thinking of the slash as an operation.
This is a bit of a reach, but one way to think of it is that $a:b$ is kind of like "a division operation you are postponing." This is why you can stack them together because no operation is intended. (If you used slashes, the urge would be to carry out the operations until you have a single fraction, but this would require parentheses to make the expression unambiguous.)
To represent a multi-part "ratio" $a_1:\cdots:a_n$, where each $a_i$ is an integer, I would suggest an element of the projective space $\mathrm{P}_\mathbb{Q}(\mathbb{Q}^n)$ (see Wikipedia) which is the set of equivalence classes of $$\mathbb{Q}^n\setminus\{(0,\ldots,0)\}$$ under the equivalence relation $\sim$, where $$(a_1,\ldots,a_n)\sim(b_1,\ldots,b_n)\iff \text{there is some $\lambda\in\mathbb{Q}$ such that }a_i=\lambda b_i \text{ for all }i$$ Denoting the equivalence class of $(a_1,\ldots,a_n)$ as $(a_1:\cdots:a_n)$, you can rigorous statements like $$(1:2:5)=(3:6:15)\qquad (1:1)=(7:7)=(\tfrac{1}{3}:\tfrac{1}{3})$$ However, this is not really a "number system" in the same way $\mathbb{Q}$ is (it has no natural ring structure).
Best Answer
They are indeed well-defined, so long as at least one number is non-zero. For the given example, it just means no tall plants were observed in the cross.
The ratio colons are sometimes used to denote homogenous coordinates, where another ratio representing some point can be obtained by multiplying all numbers in that ratio by the same number. The origin (all numbers zero) is excluded, and does not represent any point.