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Edit:

To those voting to close as 'opinion based', the core question (which should satisfy the 'opinion based'-criteria) is this: Is there any specific reason why an imaginary number can't be classified irrational; additionally, are the properties of traditionally irrational numbers, when on the imaginary-plane, different from those numbers when on the real-plane?


Original Post Follows:

I was discussing irrational numbers and mentioned $πi$ when, to my surprise, I was confronted by a colleague who declared that $πi$ was not irrational since irrational numbers have to be real. I didn't agree with him, but after some googling, to my surprise he was right.

By current definitions of irrational, a number must be real to be irrational. Ironically, transcendental numbers are defined as being able to be real or complex.


Online Definitions:

In mathematics, the irrational numbers are all the real numbers which are not rational numbers

 

In mathematics, a transcendental number is a real or complex number that is not algebraic


Is there any specific reason why an imaginary number can't be irrational (other than the definition of irrational). As far as I know, $πi$ contains all of the properties of an irrational number (other than not being on the real-axis).

Albert Renshaw
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  • A guess (I don't really know the answer): probably to avoid making $i$ itself irrational, because it is definitely not rational. –  Feb 03 '18 at 22:11
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    Depends on the definition. If you define the rational numbers as ratios of integers, then all rationals are real, so, by that definition **all** non-real complex numbers would be irrational. We can sort-of talk about rational complex numbers, as $\mathbb Q[i]$, but there's no particular reason to prefer those over other rings, like $\mathbb Q[\sqrt{-3}].$ So we tend to use "rational" and "irrational" only in the case of the reals. But $\pi i$ is trascendental. – Thomas Andrews Feb 03 '18 at 22:12
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    I think this is probably pure semantics. An artifact of the fact that irrational numbers were "discovered" at a time when "numbers" was synonymous with $\mathbb{R}$. – Tim kinsella Feb 03 '18 at 22:14
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    To build upon @ThomasAndrews' comment, $\mathbb Q$ ("rational numbers") is the simplest field of characteristic 0, so it does have a distinguished status among the fields. $\mathbb Q[i]=\{a+bi\mid a,b\in\mathbb Q\}$ is merely one extension of this field using roots of an irreducible polynomial $x^2+1$ - but *that* polynomial has no special status among other polynomials whose roots you can take to extend $\mathbb Q$. –  Feb 03 '18 at 22:18
  • @AlbertRenshaw If you called $\,\pi i\,$ an irrational, then you would probably also want to call $\,i\,$ an integer. But integers and rationals are *defined* to be real numbers. They have counterparts in the complex field, see for example [Gaussian integers](https://en.wikipedia.org/wiki/Gaussian_integer) and [Gaussian rationals](https://en.wikipedia.org/wiki/Gaussian_rational), though they don't share all properties. – dxiv Feb 03 '18 at 23:01
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    P.S. That this question received $\,3\,$ downvotes and a vote to close ("*opinion based*" ?) shows that MSE can sometimes be too pedantic with its own rules. +1 from me. – dxiv Feb 03 '18 at 23:04
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    @dxiv sent this to colleague the second I posted the question: https://i.imgur.com/VWlF5XG.png – Albert Renshaw Feb 03 '18 at 23:11
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    Just because you saw a definition of “irrational number” somewhere, that doesn’t mean that it’s the generally accepted definition of irrational number. My position is that *of course* $\pi i$ and $i$ are irrational, and I think that most mathematicians would agree. (But who am I to say?) – Lubin Feb 03 '18 at 23:24
  • @Lubin I agree fully. However, I saw lots of posts online claiming this wasn't the case (to my surprise). Here https://www.quora.com/Are-imaginary-numbers-always-irrational-Why-or-why-not is just an example of one with two credible sources (I see you yourself, given your history, are a credible source as well) That's why I wonder if there is a reason for such definitions in irrationality. I think Timkinsella is right in saying that irrational was simply defined before imaginary numbers existed If this is the case I'd like to at least edit the wikipedia to reflect such but need a source – Albert Renshaw Feb 03 '18 at 23:32
  • I strongly disagree with most of these comments. Irrationals are real numbers that are not rationals. There are lots of ways to identify $\Bbb{R}$ as a subset of a larger space, and it seems weird to me to declare that everything in such a set that isn't in $\Bbb{R}$ is therefore irrational. – Sort of Damocles Feb 03 '18 at 23:48
  • @dbx I think most mean to say that in the form a+bi; the number is irrational if a and/or b is irrational, 3+2i would still be 'rational'. 0+πi would not however. (I can't speak for everyone though) – Albert Renshaw Feb 04 '18 at 00:01
  • @dxiv 4 votes to close as opinion based, not sure what that is all about. Hopefully my edit will help clarify for future readers/voters – Albert Renshaw Feb 04 '18 at 00:49
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    There's no "irony" in complex transcendentals. The criterion for deciding algebraic-vs-transcendental is whether a number is-or-isn't a root of a non-constant polynomial with (real) integer coefficients. Such polynomials (eg, $x^2+1$) can have complex roots, so the algebraic-vs-transcendental discussion already ventures beyond the real line. (BTW: I'm with you among the inclusionists that say non-real complex numbers are irrational, but it's not a big concern to me, as context should make one's usage clear. Note that there's lack of consensus about whether the "natural numbers" includes zero.) – Blue Feb 04 '18 at 01:12
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    @AlbertRenshaw `3+2i would still be 'rational'` That was the point of my first comment. By the same logic, you would have to say that $a+bi$ is an integer if $a$ and $b$ are integers, but that's already circular and, besides, most everyone agrees that $\,i = 0 + 1 \cdot i\,$ is not an integer proper. "*Irrational*" is a narrower definition than just "*not rational*". Many things are not "rational", equilateral triangles for example, but they are not "irrational" either. All this is, of course, just my own personal / biased / whatever one wants to call it opinion ;-) – dxiv Feb 04 '18 at 03:46
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    @AlbertRenshaw There is also a good filiation/construction reason why irrationals are *reals* which are not rational. Every definition by negation, such as ***ir***rational implicitly assumes a "universal set" against which the complement is taken. If you were to rebuild the entire algebra from axioms, the notion of "rational" would come up soon after the integers, and that of "irrational" once you built the real numbers. At that point the "universal set" is the real numbers, since nothing larger exists, yet, so it's natural that "irrational" would refer to a "*real* which is not rational". – dxiv Feb 04 '18 at 03:46
  • This isn't a question of semantics, nor is it one of opinion. The question as asked is about definitions, and there is a clear right answer. The scope has expanded weirdly in the time since this question had been active, and I'm going to vote to close now too. – Sort of Damocles Feb 04 '18 at 06:04

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If you defined irrational numbers as $\mathbb C \setminus \mathbb Q$ rather than $\mathbb R \setminus \mathbb Q$, then you would be in the uncomfortable position of calling both $i+1$ and $\sqrt 2+\pi i$ irrational, even though the first looks almost like a rational, even an integer, whereas the second looks more like what we expect from an irrational.

Instead, it's cleaner to define Gaussian rationals as those complex numbers $a+bi$ where both $a$ and $b$ are rational. So the first example above is a Gaussian rational (in fact a Gaussian integer), whereas the second is not.

Théophile
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    Indeed, there is no compelling reason why one cannot *extend* the definition of irrational number from the reals to the complex (or other number systems). However the conventional definition is that of *real irrationals*, so there is some need for an author to explain their extension when the notion of *complex irrationals* is to be entertained. – hardmath Feb 04 '18 at 01:37