https://www.etymonline.com/word/irrational

> The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid.

https://www.etymonline.com/word/ratio

> The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos)

We can cross-check dictionary entries. The standard dictionary of Ancient Greek fully backs this up:

> λόγος

> II. 2 Math., ratio, proportion

The standard dictionary of Latin doesn't mention this particular sense. (A negative is harder to cite, but you can check it here: https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext... )

The senses that the Greek word and the Latin word have in common are those of reasoning in general and numeric computation in specific. You might guess that "irrational numbers" are named for their inability to be computed. (Or, if you're only looking at Lewis and Short, you might guess that they are named by reference to an inability to think methodically; this sense exists in Latin and indeed still persists in the English word "irrational". Insanity would usually be represented by another word, presumably something more like dementialis than irrationalis.)

However, ratio is the conventional translation of the Greek logos, and since we're told that "rational" (of numbers) comes from a translation of Greek, it seems fair to attribute an existing Greek sense to the translated word too. So the best analysis does appear to be that "irrational numbers" are named, as you might expect, for the fact that they cannot be represented as integer ratios.

[1] https://www.youtube.com/watch?v=1uLi1I3G2N4