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I have tried learning about ratios and proportions from a couple of books, but I have problems with the approaches they take.

First of all, a definition is never actually given. Examples are given concerning only natural numbers. For example one definition I encountered was the ratio of object $x$ to object $y$ is $m:n$ if for every $m$ $x's$ there are $n$ $ y's$. Good enough but what about rational and real numbers ?

Secondly, if a ratio is supposed to be a relation between two quantities (which it is defined vaguely in many books) then why is it treated as a number, for example saying things like the ratio of a to b is greater than that of c to d and so on.

So the question is:

What is a definition of the ratio between two real numbers $a$ and $b$. Is it fair to say that $a$ and $b$ have ratio $m:n$ if $\frac{a}{m} = \frac{b}{n}$, for example?

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  • $\begingroup$ The ratio of the real numbers $a$ and $b$ is the real number $\frac ab$. You are freely allowed to expand or simplify that fraction if you so desire, and what makes two ratios equal are exactly the same as makes two normal fractions equal. $\endgroup$ – Arthur Dec 23 '15 at 14:17
  • $\begingroup$ I think you switched $m$ and $n$: if for every $m x$ there are $n y$ then the ration $x:y$ is $n\colon m$ ( think $m$ large compared to $n$). $\endgroup$ – Orest Bucicovschi Dec 23 '15 at 14:45
  • $\begingroup$ if for every 2 apples there are 3 oranges then the ratio of apples to oranges is 2:3,no ? $\endgroup$ – alexgiorev Dec 23 '15 at 14:47
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I don't know if this explanation will be sufficient for your purposes, but at a high level, a ratio $a/b$ is simply the product of $a$ and the multiplicative inverse of $b$.

Is that too abstract? I think that part of the problem here is that a "ratio" is rarely a well-formalized mathematical object: we use ratios in everyday discourse to describe relationships between two things using a rational number, but I rarely see the idea of a ratio mentioned formally in mathematics. I wouldn't overthink ratios -- you're fundamentally looking at arithmetic.

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  • $\begingroup$ Too late for not overthinking it. Thanks for bringing calm to my mind. $\endgroup$ – alexgiorev Dec 23 '15 at 14:19
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Your concerns are very much the same as the concerns of the ancient Greeks in the 4th century BC. Their solution is summarized in books 5 to 10 of Euclid's Elements, written about 300 BC in Alexandria, Egypt. Euclid's Elements.

It's really a deeply philosophical thing to ask whether a ratio is in fact a number. The solution in textbooks is to call an equivalence class of ratios a rational number. But that's really a cop-out. It doesn't say what the rational number really is. It's really one of those regularly perplexing things in mathematics. You start with some system which seems really "real", and then you construct something else from it, like the real numbers, the complex numbers, negative numbers, and so forth. But the concepts of mathematics keep getting more and more abstract. Another example is tensors, which have many definitions, all of them unsatisfactory, including equivalence classes (i.e. quotient spaces). But if you go back to the integers, if you look closely enough, you'll see that they're not really well defined either. They're just less difficult to accept than the "higher" number concepts.

And concerning the integers, around about 100 to 120 years ago, people were defining integers to be equivalence classes too. E.g. the number 3 is the equivalence class of all sets which contain 3 objects. That's pretty crazy, but a lot of books still use this definition of cardinality. My point is that even the integers are not above suspicion.

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