Talk:Irrational number

Article Checklist for "Irrational number"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developing article: beyond a stub, but incomplete
Underlinked article?	Yes
Basic cleanup done?	Yes
Checklist last edited by	Anthony Argyriou 17:14, 2 August 2007 (CDT)

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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)