Rational numbers have many interpretations, but one of the simplest is as a ratio of one number to another. The fraction $1/2$ can be interpreted as the ratio 1:2 (i.e. one apple for every two oranges). Rational numbers are also considered an extension of the number system of integers $\mathbb{Z}$ to (nearly) close it under division.

I would seem like a natural extension of the number system to include ratios comparing three or more quantities. Is there any mathematical description, in terms of a "number system" or otherwise, of ratios which consist of three or more parts, such as $1:2:5$ (i.e. one apple for every 2 oranges for every 5 papayas)?

  • 2
    $\begingroup$ It seems to me that you must distinguish $(a:b):c$ from $a:(b:c)$ for this to be meaningful. When this is done, then note that, for example, $(a:b):c = a:bc$. so we can reduce back to the case of a simple ratio. This can be iterated to describe $n$-part ratios, for any integer $n$ $\endgroup$
    – M_B
    Jun 26 '17 at 3:30
  • $\begingroup$ @M_B I don't understand your comment, What do the parentheses mean? How do you suddenly go from (loosely speaking) two rational degrees of freedom to one? $\endgroup$
    – JiK
    Jun 26 '17 at 19:19

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).

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    $\begingroup$ So in other words you're saying that (unlike the 2-part case) it doesn't have a ring to it? $\endgroup$
    – user541686
    Jun 26 '17 at 6:59
  • 9
    $\begingroup$ I mean, even in the two-part case you can't treat all ratios as forming a ring if 1:0 and 0:1 are both to be allowed. Really, such ratios belong to the projective rational line (which includes a point "at infinity") rather than the rationals themselves. $\endgroup$ Jun 26 '17 at 10:45

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