terminology - Once and for all - "Rational numbers" - because of ratio, or because they make sense?

This is a question I'm sure was asked before but I can't find it. There are many sources claiming that the term "rational number" for the elements of $\mathbb{Q}$ comes from the word "ratio", since a rational number is the ratio of two integral numbers. However, there are also other sources that claim that it is because they "make sense" as opposed to irrational numbers.

None of the sources I've encountered were very much convincing as to why their claim is correct. Rational and irrational numbers were known already to the Greeks but I don't know if that's the terminology they used and what it meant then.

So my question is simple: What is the meaning of "rational" in rational number? What's more important is that any claim will be backed up with a reliable and convincing source.

Answer

This is not a complete answer, but too long for a comment. I think that to definitively answer your question would require access to (or knowledge of) both Renaissance era math texts and books/papers from the 1700s (when math in Latin started to be translated to English).

Once upon a time the Greeks used the word "logos" to mean what we think of today as a ratio (a scaling factor; one quantity divided by another). In the 1600's, Greek mathematical text was translated into Latin and the word "ratio" was used for "logos". In Latin, "ratio" meant something that was reasoned out, calculated, or thought through. You can perform all of these actions using logic. But if you are reasoning out, calculating, or thinking through a numerical computation (like evaluating $\frac{a}{b}$), you might have what we today call a "ratio".

So I would say the answer is both. Most recently, a "rational number" is what we today call a "ratio" - it's one number divided by another (specified to two whole numbers). But if you look a little further back in the etymology, the reason that "one number divided by another" is today called a "ratio" is because that happens to be something that you would reason out. And so with that underlying etymology, a rational number is a number that "makes sense" as the end result of some logical thought.

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Just because, here are my two other favorite math etymology items.

• "radical" comes from Latin for "root": "radix". (Pronounced properly, this sounds a lot like "radish".) So why is $\sqrt{}$ called a radical sign? Probably because $\sqrt{2}$ is a root of $x^2-2$. But why are zeros of polynomials called "roots"? Does this have anything to do with other modern uses of "radical": applying to politics, ideas, chemistry, Chinese character sets? Yes, it does. Think about squares and sides. In all these instances, something "radical" is "off to the side". None of this this has anything to do with "radius", despite the apparent similarity.


• "polygon" is often translated as a many-sided figure. Certainly, "poly" means many. But the "gon" actually means corner or angle. In modern Greek, "goneis" means elbow. So I like to think of "polygon" as a many-elbowed figure. "Ortho" means straight/direct (think orthodontia: straightening teeth, and orthodoxy:direct interpretation). So "orthogonal" means something like "having straight corners", which we would translate to "having right angles".