# irrational number: etymology of the word

One usual meaning given to the expression "irrational number" is "number not logical or reasonable". In particular in Spanish, where the usual term is "número irracional" (without logic, madness).

This goes against what is the main characteristics of these numbers in mathematics: not possible to express them as a ratio (fraction, division) of numbers.

The etymology that google displays is:

where the latin term "rationalis" has four possible definitions:

• accounts (attributive)
• rational, reasonable, of or possessing reason
• that has a ratio
• syllogistic

Of these four options, it seems that the most near to the mathematical concept must be "that has a ratio", not the one about "reasonable".

For these reasons, it seems that the Spanish term "número irracional" (in Spanish) and similar descriptions in English or other languages are very inadequate, being better say "número no fracional" or similar.

Knows someone the history and etymology of the term?

• Let us explain the term "rational" ("irrational" is its contrary): it is issued from greek mathematics: all the numbers that they could "conceive" with their "reason", i.e. reasonnable numbers, where quotients $a/b$ because they were what they called "commensurable" (after all, 3/4 of a piece of cake is "commensurable" for us, and you can explain it to a "reasonable"10 years old, but $\sqrt{0.75}$ is not). – Jean Marie Sep 22 '17 at 10:03
• In French, the word ‘raison’ may have the meaning of ratio, and not only the meaning of reason. I don't see why the terminology should be changed. Why not change also asymptote, apothem, &c.? – Bernard Sep 22 '17 at 10:17
• This sounds like a question I would ask. Bravo! Words are important. (I remember being admonished in a coffee shop in Boston for using this term, with the idea that transcendental number had replaced it, only to subsequently find out that not all irrational numbers are transcendental.) – DukeZhou Feb 15 '18 at 21:13
• No one reads "número irracional" and interprets "número without logic"... That's absurd. Words have several meanings, and there is absolutely no reason for mathematical usage of words to be dictated by etymology. – Mariano Suárez-Álvarez Feb 15 '18 at 21:33
• Most simple groups are very complicated, there is nothing flat about flat modules, you cannot dissolve solvable groups, etc. – Mariano Suárez-Álvarez Feb 15 '18 at 21:37

## 1 Answer

Euclid used the word ἄλογον (alogon), the Latin word was a translation. When used in the Greek text of the Bible, "alogon" is translated as "absurd" or (King James Version) "unreasonable".

SEE: Euclid, Elements, x.1.3