I've never seen a definition that addresses non-real numbers as either rational or irrational. I think the definition as written is just fine, and standard.

Are there any other entries in PM that rely on the proposed broader definition (or that are incorrect because of this narrower definition)?

Roger

> By the way, the Springer on line encyclopedia is curiously
> indeterminate about this. They say that an irrational is a
> "number that is not a rational number" but go on to discuss
> it in terms of a geometric length incommensurate with 1. All
> the examples are on the real line.

I think I might replace the word "curiously" with "horribly" there. In some sense, there's no such thing as a number. One would be hard-pressed to find a common definition which included all objects that we've come to know as numbers.

In the particular case, I think it's fairly well agreed upon that without any preceding text to the contrary, an irrational number is a real number that is not rational. If pressed, I would call e*i complex irrational.

Cam

> Mathworld simply does not say, but again all the examples are on the real line.

The second sentence of the Mathworld article:
"Irrational numbers have decimal expansions that neither terminate nor become periodic."

Such numbers must be real.

mathcam asks: I agree with the first (unquoted) part of your post, but I think this last definition is unsatisfactory -- What are the rational numbers in an arbitrary field of characteristic zero (i.e. what are the integers)?

Let K be a field of characteristic zero. Then there is a "one" element in K, call it 1, and n times 1 = 1+...+1 (n times) is not zero for any natural n (because of the assumption on the characteristic). The numbers n x 1 are your integers. Also, since K is a field, every n x 1 (not zero) has an inverse which we call 1/n. Thus, there is a copy of Q (or isomorphic to Q) inside K.

Alvaro

Leaving aside issues of what is correct or standard terminology,
I like to use the word "rational" to describe complex numbers whose
real and imaginary parts are rational numbers. Likewise, I would
call elements of algebras such as quaternions, polynomials, and
matrices rational when they have all rational coefficients. The
unifying theme here is that the rational elements form a subalgebra
which is obtained by starting with the generators of the algebra and
applying the basic operations of addition, subtraction, multiplication, and division a finite number of times (i.e. excluding
infinite sums and such operations).

A lot of the issue here has to do with the nature of this website, in
particular the fact tht it is being written as a collection of more
or less independant entries. For instance, if I was writing a book
or even an article in which polynomials whose coefficients were
rational numbers or complex numbers whose real and imaginary parts
were rational numbers played an important role, I would simply
define the term "rational polynomial" or "rational complex number"
suitably and not consider the matter further because this use of the
terminology would not conflict with standard usage but be an
extension of the term "rational number" which would be of use in the
context of my work. However, here, the usual situation is for a
trerm to be defined in one entry and used in a differrent entry
which makes for a tricky situation when dealing with non-stnadard
extensions of standard terminology.

I certainly concede that every field contains a subring naturally isomorphic to the integers, but for example, in the characteristic 0 field of the p-adic rationals, this definition does not coincide with the p-adic integers. In C_p, the complex p-adics, a realm in which we would actually like to make a distinction between rational and irrational, the distinction is even more exacerbated.

Cam

> I will definitely edit my entry according to the PM
> convention, as most everyone who has contributed to this
> thread seem to support it; however, I still find it very odd
> to use the convention that "irrational" and "not rational"
> do not mean the same thing.
>
> Warren

I guess I'd still argue they do mean the same thing. The subtlety is that defining the complement of a set is a relative, and not absolute, notion -- It requires the knowledge of a bigger set. I don't think either of us would argue that a banana is an irrational number simply because it's not a rational number.

So the distinction between the two approaches is only in answer the question "If it's not in the rationals, where is it?" The convention that we're espousing is that we take the complement of the rationals inside the real numbers, whereas you take the complement inside the (perfectly valid, but less common) complex numbers.

Cam

> I certainly concede that every field contains a subring naturally
> isomorphic to the integers, but for example, in the characteristic 0
> field of the p-adic rationals, this definition does not coincide with
> the p-adic integers. In C_p, the complex p-adics, a realm in which we
> would actually like to make a distinction between rational and
> irrational, the distinction is even more exacerbated.

Every field has a simple field as subfield, i.e. smallest (inclusion) subfield of F. If a characteristic of a field is p>0, then this subfield is isomorphic to Z_{p} (notice that p is prime). If a characteristic is 0, then this subfield is isomorphic to Q.

So if you consider any field F of characteristic 0, then you can assume that Q is a subfield of F. There's even more - there's no other subfield of F isomorphic to Q (since every subfield of F contains Q and Q is simple). So then definiton makes sense:

Irrational element (yeah... this is my definition :) ) in char 0 field F is any element in F\Q.

Irrational number is irrational element in R = real numbers.

Also C_{p} is isomorphic (as fields) with C = complex numbers. Of course Q_{p} can be embedded in C_{p}, so Q_{p} is nothing else then subfield of C = complex numbers. So I don't understand where's the problem?

Of course all of this is as a bit strange. I don't know whether there's use of irrational elements (instead of numbers).

joking

> I still find it very odd to use the
> convention that "irrational" and
> "not rational" do not mean the same thing.

Dear Warren, "Irrational" (from Latin "irrationalis" containing the negative prefix "in" and the adjective "rationalis" 'rational, reasonable') in general means 'not rational', but when speaking of numbers the established meaning is obviously 'non-rational real'. And I think it would give us no benefit, if we would call "irrational" the non-rational elements of any extension field of Q (e.g. of Q(i), R(X), C[[X]]).
Jussi

Hi Warren, I think this meaning of the terms "imaginary" and "purely imaginary" is very rational and practical but quite rare in America! Historically, "imaginary number" has of course meant 'non-real number', i.e. 'non-real complex number'.
Jussi

P. S. -- BTW, I and some other people can say that also 0 is purely imaginary (but not imaginary!). What could prevent it? Then all purely imaginary numbers form an additive group and (R-module).