Talk:Irrational number

This article needs stuff on the theory of irrational numbers - that irrational numbers may be algebraic or transcendental, that they are infinite, etc.. Anthony Argyriou 17:14, 2 August 2007 (CDT)

What in the world do you mean by saying they are infinite? They are nothing of the sort. Michael Hardy 20:08, 3 August 2007 (CDT)

They are infinite in number, and in number density. It is believed that the infinity of the number of irrational numbers is a greater infinity than the infinity of the integers, though it's not known if the number of irrationals is properly aleph-1 or not.

Two or three proofs that particular numbers are irrational is not nearly as useful as more in-depth discussion of what exactly irrational numbers are, in general. Anthony Argyriou 01:13, 4 August 2007 (CDT)

That there are infinitely many of them I suppose I thought was obvious, but certainly that could be added. It is known that their cardinality is the same as that of the reals and greater than that of the rationals. If you accept the axiom of choice, then that's greater than aleph₁. As to what they are, that's stated at the beginning. One thing that is useful about the proofs of irrationality is that they show that it's very easy to see why irrational numbers must exist. Michael Hardy 11:25, 4 August 2007 (CDT)

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You wrote:

It is believed that the infinity of the number of irrational numbers is a greater infinity than the infinity of the integers, though it's not known if the number of irrationals is properly aleph-1 or not.

I find that a disturbing way of expressing things since it's likely to confuse anyone not already familiar with the answers to these questions. Let us note that:

• It is known, and provable by elementary (but not obvious) means that the set of irrational numbers has a greater cardinality than does the set of integers or the set of rational numbers.
• It is known (and somewhat more work to prove) that the set of irrational numbers has the same cardinality as the set of real numbers.
• Whether the set of real numbers, or the set of irrational numbers, has cardinality $\scriptstyle \aleph _{1}$ is independent of standard axiom systems for set theory. That fact is not at all elementary to prove. It is readily shown (if you allow yourself the axiom of choice) that the cardinality of the set of irrational nunmbers is at least $\scriptstyle \aleph _{1}$. I consider these questions off-topic for the present article.
• "Infinite in number density" is not a locution I've ever heard used for what I suspect you mean. I suspect you mean simply that between any two real numbers there is an irrational number. That is sometimes expressed by saying the set of irrational numbers is dense in the set of real numbers. Once you've got the fact that one particular number, e.g. √2, is irrational, then it's obvious: between two rationals a and b, the number

$a+{\frac {\sqrt {2}}{2}}(b-a)$

must be irrational, since if it were rational that would obviously imply that √2 is rational. Thus the fact that one irrational number exists seems more substantial than the quick corollary that says between any two reals there's an irrational number. (The fact that betwen any two reals there is a rational number is a much more substantial result.)

• Irrationality of some particular numbers is definitely of great interest. In particular, the fact that it was geometry that made it necessary to think about the concept in the first place is of considerable interest.

Certainly a lot of the information above should eventually get added to this article, but the things that are already there are in some ways more substantial. Michael Hardy 14:27, 4 August 2007 (CDT)

OK, I guess I'd better clarify something. That the cardinality of the set of all irrational numbers is the same as that of the reals ($\scriptstyle 2^{\aleph _{0}}$) is certainly on-topic for this article. But that stuff about $\scriptstyle \aleph _{1}$ seems quite off-topic. Michael Hardy 16:12, 4 August 2007 (CDT)