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.

• Somewhat off topic, but I couldn't help but to recall this wonderful strip. – Asaf Karagila Sep 15 '11 at 19:36
• IIRC people only began talking about rationals when stuff like $\sqrt 2$ showed up... anyway, this paper ought to be good reading. – J. M. is a poor mathematician Sep 15 '11 at 19:36
• $\sqrt{2}$ showed up in the time of the ancient Greeks. – Gadi A Sep 15 '11 at 19:37
• It appears that the confusion dates back at least to ancient Greece. Euclid used the word λόγος for proportions, but that is also the word for reasoned thought or argument (as well as for words and language in general, and also the Word of John 1:1 -- quite a multitasking lexeme). That seems to have been why they became to be called ratio in Latin in the first place, so it has been either-or-possibly-both for thousands of years. – Henning Makholm Sep 15 '11 at 19:37
• @HenningMakholm: So John 1:1 ("In the beginning was the λόγος ..." ) is a proof of some sort - akin to "Let $\epsilon > 0$ "?!!? – The Chaz 2.0 Sep 18 '11 at 20:24

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.

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".
• BTW, note that the similarity between "radix" and "radish" is not a coincidence. – Ilmari Karonen Sep 18 '11 at 21:02
• The brilliance of etymology. Great answer. – 000 Dec 8 '12 at 5:11

I was taught (decades ago in a mathematics course) that in Ptolemy's ancient Egypt, before Pythagoras discovered and proved the relationship between the sides of right triangles, the notion that there can exist numbers that cannot be expressed as a ratio of two integers was a crazy idea. Irrational, i.e., not able to be expressed as a ratio, eventually became a synonym for crazy.

It's easy to understand why people thought an irrational number was a crazy idea because any physical measurement of the distance between two points can be refined by magnification. For example, a measurement that is part way between say 5/32 and 3/16 of an inch can be perhaps be more precisely expressed as 11/64 inch. If that is not precise enough perhaps 21/128 or 23/128 will be the answer, and so forth. But Pythagoras proved that not to be the case.

I have been seeking a reference that supports that version of the etymology.

• You might also seek a reference for the claim that the Egyptians considered "the notion that there can exist numbers that cannot be expressed as a ratio of two integers" at all, even if only to denounce it as crazy. My gut feeling (also without references) is that the Egyptians were nowhere near such notions, and that even for the Greeks the question of rationality or irrationality was not about what they would call "numbers" (positive integers) but about ratios of lengths, areas, and such geometric quantities. – Andreas Blass Jul 1 '13 at 0:47