Wikipedia:Reference desk/Archives/Mathematics/2019 April 5

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 5

Mathematical Symbol for Irrational numbers?

Hello! I am wondering if there is a symbol or word that represents all irrational numbers, like the Aleph-naught, which represents all real numbers. I am also wondering if there is a symbol like it for imaginary numbers. FalloutCraftr! (talk) 01:15, 5 April 2019 (UTC)

So first of all, this isn't your question, but ℵ₀ does not represent "all real numbers", but rather the cardinality of the set of all natural numbers.

As far as I know there is no standard symbol for the set of all irrationals. Irrationals are defined by exclusion; that is, they're just real numbers that aren't rational.

So you can write R\Q for the set of all irrationals, where R is the real numbers, Q is the rational numbers, and the "\" symbol represents set difference.

Some people would prefer to use blackboard bold for R and Q; this is a matter of taste (my taste is that blackboard bold is mostly for blackboards). Also if you go to the number article you'll very likely find some symbol that somebody uses for the irrationals — but as I say, I don't believe any such symbol is really standard. --Trovatore (talk) 02:14, 5 April 2019 (UTC)

As for the OP's second question: I guess they meant "non-real" numbers rather than "imaginary" numbers, and if I'm right than the answer is analogous to Trovatore's answer: Z\R C\R. HOTmag (talk) 07:43, 5 April 2019 (UTC)

The purely imaginary numbers are usually written as iR. The non-real complex numbers are C\R. —Kusma (t·c) 07:56, 5 April 2019 (UTC)

• (e/c) For imaginary numbers (or non-real complex numbers), there's a stackexchange thread here. The complex numbers with nonzero imaginary part would be C\R. In numerical analysis it is actually useful to have a computer datatype for imaginary numbers that's separate from the type for complex numbers,[1][2] but most programming languages don't support this even if they support a complex number type. 173.228.123.166 (talk) 08:03, 5 April 2019 (UTC)

Thanks for those references. I was wondering why Imaginary was included but had never got around to looking it up. Dmcq (talk) 10:46, 5 April 2019 (UTC)

Following Bourbaki's General Topology, as in our article on blackboard bold, $\mathbb {J}$ or J is most often used for the irrationals. The irrationals can be considered as the Baire space (set theory) and have a unique natural representation as continued fractions, that is as sequences of natural numbers N^N, which give them a universal complete metric: every Polish space, complete separable metric- is a continuous image of the irrationals. See also [3].John Z (talk) 13:38, 5 April 2019 (UTC)

I thought about mentioning Baire space but it didn't really seem like what the OP was asking about. The identification between the irrationals and Baire space works only for topology/descriptive set theory (nothing convenient happens to the algebraic properties of the irrationals, for example). For those purposes, the main reason for bringing in the irrationals seems to be that it gives you a nice concrete visualization of what Baire space "looks like" topologically, which otherwise might be harder to grasp. But even there, starting with the reals and removing any countable dense subset would do as well.

As for this J, at least in the descriptive set theory community, it doesn't seem to have caught on. Moschovakis used ${\mathcal {N}}$, which doesn't really seem to have caught on either, in spite of his huge influence on the field. Mostly I've seen ω^ω or occasionally ^ωω. Those of course are for Baire space, not for the irrationals specifically. --Trovatore (talk) 18:06, 5 April 2019 (UTC)

Aside to John Z (talk · contribs): I'm curious what leads you to claim that the continued-fraction homeomorphism between Baire space and the irrationals is the "unique natural" one. It's certainly a cute one. It's explicit, intuitive, easy to understand. But unique natural? Why, exactly?

As a small piece of contrary evidence, I'd point out that the ordering on Baire space given by pulling back the usual ordering on the irrationals is not a very obvious one at all. You look for the first location where two sequences differ, but then you decide which one is greater in opposite ways, depending on whether the location of first disagreement is even or odd. I doubt that strikes anyone as a "natural" order on Baire space. --Trovatore (talk) 22:38, 6 April 2019 (UTC)

On second look, maybe you were ascribing naturalness to the representation of the irrationals as continued fractions, not the homeomorphism with Baire space. That does seem a bit more plausible. --Trovatore (talk) 22:44, 6 April 2019 (UTC)

Yes, that last is what I meant, my sentence should have been ordered more naturally. I'd say the induced Baire space ordering is like the usage of J as a symbol; not completely standard, the best and most universal and natural, but as good as it gets or better than one might expect. :-)John Z (talk) 17:26, 8 April 2019 (UTC)