# Why are “irrational numbers” not named “nonrational numbers”?

Possible that I'm misunderstanding the concept of irrational numbers, but seems like the term nonrational would be much more clear. Why is "irrational" more clear than "nonrational"?

UPDATE: Just to be clear, it would be true to say the terms “irrational numbers” and “nonrational numbers” have the exact same meaning, and neither is something the other is not, correct?

• The prefix "ir" does mean "non". It's a variant of "in", used before an "r".; e.g., "irresponsible", "irrefutable", "irregular", etc. – David Mitra Aug 26 '12 at 23:23
• Seconding @David M's comment: it's just a (perhaps slightly archaic) style of being "more euphonious" in forming negated adjectives and such. We don't say "non-possible", nor even "in-possible", but "im-possible" for some similar reason. Linguistic, not mathematical. – paul garrett Aug 26 '12 at 23:34
• Me fail English? That's unpossible! youtube.com/watch?v=8iSD9lPVY6Q – Gerry Myerson Aug 26 '12 at 23:40
• Why do you think things are named based on what would be clearest? – Qiaochu Yuan Aug 26 '12 at 23:52
• ... aaaand not to mention that "clarity" itself is surely context-dependent. (Although when much younger I thought mathematics had the clearest self-description, I no longer believe this.) – paul garrett Aug 27 '12 at 0:09

It is from Latin "irrationalis" ... so you have to blame those old Romans for this form.

• Those bums didn't even speak English. :) – paul garrett Aug 27 '12 at 0:05

There is also the fact that the prefix ir- is often (in English language) used for words which start with r, e.g. irreducible (which I don't think come from Latin), irregularity. Same way you have il- for words which start with l, e.g. illogical.

The irrational numbers are the elements of $\mathbb{R}\backslash\mathbb{Q}$ , that is: the real numbers that are not rational.
This is not the same as non-rational since, for example, $i\in\mathbb{C}$ is not a rational number (since $i\not\in\mathbb{R}$ and in particular $i\not\in\mathbb{Q}$) so it is non-rational, but it is not an irrational number.
• "non" and "ir" are prefixes meaning "not". Chosing the "underlying" set to be $\mathbb C$ seems to be forcing things. – Pedro Tamaroff Aug 26 '12 at 23:24
• I'm not confident that $\sqrt{-1}$ is not irrational. True, the earlier tradition about "irrationals" only refered to real numbers, but that was under a tacit assumption that the real numbers were the "universe" of numbers, I think. I'd call anything in an algebraic extension of $\mathbb Q$ but not in $\mathbb Q$ "irrational". Presumably context would clarify. – paul garrett Aug 26 '12 at 23:32
• I'm with paul. Certainly in the context of Algebraic Number Theory and in Diophantine Analysis, $\sqrt{-1}$ is "an algebraic irrational". – Gerry Myerson Aug 26 '12 at 23:42