stoichiometry - Law of Multiple Proportions: What is the significance of small whole numbers?

I'm new to Chemistry and in my textbook, it describes the Law of Definite Proportions (aka Proust's Law) and then goes on to describe the Law of Multiple Proportions.

The example they give is carbon monoxide and carbon dioxide, where the mass ratio of oxygen to carbon in carbon dioxide is $2.67:1$ and the mass ratio of oxygen to carbon in carbon monoxide is $1.33:1$.

I understand the above example, and I understand that when you compare them in the following way:

$2.67~\mathrm{g}$ (proportion of oxygen to $1~\mathrm{g}$ carbon in carbon dioxide)$/ 1.33~\mathrm{g}$ (proportion of oxygen to $1~\mathrm{g}$ carbon in carbon monoxide) $= 2.00$, a small whole number.

$$\frac{\text{Mass oxygen to 1 g carbon in carbon dioxide}}{\text{Mass oxygen to 1 g carbon in carbon monoxide}} = \frac{2.67~\mathrm{g}}{1.33~\mathrm{g}} = 2.00$$

My textbook doesn't describe what a small whole number is, or what it means … just that, in this example, $2.00$ is a small whole number.

What is the significance of the $2.00$?

I understand that carbon dioxide has double the oxygen. Is this where the $2.00$ plays a role?

Answer

There's no universally accepted hard cut-off, as far as I'm aware, which determines the range of numbers that would constitute "small" mass ratios. Generally, all textbook examples select compounds that yield single-digit mass ratios.

The significance of the ratios being small (as well as their constancy and the limited number of different mass ratios for binary compounds actually found in nature) is that they allow for reliable conclusions to be made about the stoichiometric proportions in which elements combine based on crude empirical data alone, as well as leading to conjectures about the number of possible configurations, bond arrangements, and oxidation states yielding said combinations.

The law is valid for a large variety of binary compounds simply by virtue of the fact that there is typically a limited number of permutations in which two given elements can be combined to yield a stable compound. For example, C and O have only two, N and O have only four, metals have a limited number of oxidation states, therefore giving rise to relatively few possibly formulas for binary ionic compounds, etc. The law breaks down for cases in which two elements can combine to form long chains or complex molecules (e.g., large hydrocarbons and [mostly hypothetical] silicon analogues), when the mass ratios can sometimes become quite large.