On the one hand, $i(=\sqrt{-1})$ cannot be expressed as a ratio of integers, so, by definition, $i$ is not rational $\iff i$ is irrational.
However, the set of irrational numbers, $\mathbb{J}=\mathbb{R}\setminus\mathbb{Q}$ is defined to be the set of all real numbers that are not in $\mathbb{Q},$ but, clearly, $i \notin \mathbb{R}$, so it must be that $i \notin \mathbb{J}$, so $i$ is not irrational.
Clearly, the first two paragraphs seem to contradict each other, so I'm asking: is $i$ irrational or rational?
Best Answer
It depends on convention whether or not the scope of "irrational" includes non-real numbers. As you say, if the irrationals are defined as $\,\Bbb R\setminus \Bbb Q\,$ then $\,i\,$ is neither irrational nor rational. However, many authors use "irrational" to mean "not rational", i.e. $\,\not\in \Bbb Q,\,$ therefore $\,i\,$ is irrational. This is quite common usage in university-level algebra.
For example, in my 2006/3/8 sci.math post I remarked that if one searches books.google.com for "irrational algebraic" one finds such usage by many eminent mathematicians: e.g. John Conway, Gelfond, Manin, Ribenboim, Shafarevich, Waldschmidt (esp. in diophantine approximation, e.g. Thue-Siegel-Roth theorem, Gelfond-Schneider theorem, etc). See also other posts in that sci.math thread titled "Is $\,i\,$ irrational"?
Our of curiosity, I ran another Google Books search on "real irrational" numbers. Authors using such terminology presumably employ the more general definition of irrational numbers. Among such authors I found the following eminent mathematicians: Bombieri, Davenport, Dedekind, Euler, Hurwitz, Kronecker, Kirilov, Mahler, Lang, Ostrowski, Ribenboim, Weil.
However, it is not easy to find an explicit definition since higher-level textbooks assume the reader already knows basic terminology. I vaguely recall that someone (Gerry Myerson?) once posted to sci.math some links to definitions which make it unquestionably clear that the author employs the more general definition of "irrational". Perhaps someone can dig those up, or locate others.
In any case, it is a matter of convention. In most cases one can quickly infer the intended denotation from the context, so there is little chance for confusion.