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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?

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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
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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
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Me fail English? That's unpossible! youtube.com/watch?v=8iSD9lPVY6Q –  Gerry Myerson Aug 26 '12 at 23:40
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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
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3 Answers

up vote 3 down vote accepted

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

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

There are some discussion about this sort of things in English.SE e.g. 1, 2, 3.

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“Regula” is certainly the Latin word for “rule”. The Latin prefix “in”, meaning “not”, takes various forms depending on the first sound of the word it’s attached to. –  Lubin Aug 27 '12 at 3:39
    
@Lubin good to know. My point really was that ir- comes often with words starting with r. –  user2468 Aug 27 '12 at 3:41
    
See also 4. The prefix in- usually becomes ir- before r, becomes im- before b, m, or p, and il- before l, a process known as sandhi/assimilation. –  ShreevatsaR Aug 27 '12 at 6:42
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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.

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"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
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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
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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
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