Talk:Real number

"Real number" or "real numbers"?

I'd prefer, overall, this article to be at real numbers, rather than at the singular name. After all, unlike ordinal numbers, a single real number does not really have any property that can be expressed without referring to other real numbers, unless the (non-standard) choice of which construction to use is made.

(Note I accidentally started the other article before I noticed the singular article had already been copied over. I'm now not sure which one to turn into a redirect.)

Thoughts?

Philipp Rumpf 13:56, 3 February 2007 (CST)

I vote for 'real number'. The language of the article is not straightforward. I wish my memory was good enough to quote my second grade classmate Michael Cohen (Mike, are you out there?) ((whose father was a professor at the Institute for Advanced Study)) when he explained the term to our disapproving second grade teacher. Can you simplify the language? NancyNancy Sculerati MD 16:18, 3 February 2007 (CST)

Any reasons for preferring "real number"? It's possible to write an article about a "real number" (something along the lines of "a real number is a rational number, or a mathematically consistent description of how a new number would be larger than some rationals, and smaller than some other rationals, but not necessarily rational itself") - but that would make our choice for one preferred definition (Dedekind cuts, in this case), which might be controversial, but the current article is more about "the field of real numbers".

Hmm, I was just asked by someone physically present to define real number, and went for "it's an orderable number that's not infinitesimal and not infinitely large, i.e. smaller than some natural, and larger than the reciprocal of some natural". Admittedly, it only covers the positive case, but (while not very mathematical), I must say I kind of like the definition. Starting with the totality of all imaginable "numbers", you must delete those that are infinitely small, in the "close to zero" sense, or infinite, or not orderable on a line, and all those which would result in one of the first three by arithmetic operations, but you're still left with quite a few, and those are the real numbers.

Back to your comment, what do you think could be improved? It's really hard for some mathematicians, including myself, to still realise which bits of an introductory article on a subject I'm familiar with are inaccessible to other readers.

Philipp Rumpf 16:52, 3 February 2007 (CST)

"Computers can only approximate most real numbers"

The statement "Computers can only approximate most real numbers" is a little obtuse. One problem is that there is no way that I know of to tell if the computer representation carries all the digits. If our computer carries the equivalent of a five decimal digits, then we don't know if 2.7183 represents e or just what it says. Also, computers can represent numbers in other ways than digits, for example as shown in the maths article, $\pi \!$ can be represented by that very symbol. And we are using computers to write all this. So we might have to talk about how numbers are represented (e.g. decimal, binary, number of bytes,...) Peter D. Noerdlinger 01:39, 26 March 2007 (CDT)

The rest of the paragraph seems to explain the statement fairly well, in my opinion. -- Jitse Niesen 21:26, 28 March 2007 (CDT)

Continuum Hypothesis

I'm not sure that the statement of the continuum hypothesis here is correct. The article Continnum Hypothesis at Mathworld matches my understanding, namely that it is the postulate that $c=\aleph _{1}$. Greg Woodhouse 10:10, 29 March 2007 (CDT)

Hi Greg. I'm glad to see you here, but I'm afraid I have to disagree with the new version of the article.

You and MathWorld are right that the Continuum Hypothesis states that $c=\aleph _{1}$, where c is the cardinality of the real number. However, I think the current version of the article does not explain well what $\aleph _{1}$ is. It is not the cardinality of the power set of the integers. In fact, the reals and the power set of the integers have the same cardinality. Instead, $\aleph _{1}$ is the cardinality of the next bigger set after $\aleph _{0}$ (see Aleph-1 at MathWorld or Aleph number at Wikipedia]). So, $c=\aleph _{1}$ means that the reals is the next bigger set after the integers.

Hence, I reverted that change. -- Jitse Niesen 05:32, 30 March 2007 (CDT)

I believe you're right here. My understanding is that $\aleph _{i+1}=2^{\aleph _{i}}$ is essentially(?) the generalized continuum hypothesis, and not a definition -- my mistake. Now that I think about it, what really made me do a double take is the notation $2^{\omega }$ because I think $2^{\alpha }$ really only makes sense if $\alpha$ is a cardinal. Am I missing something? Greg Woodhouse 08:41, 30 March 2007 (CDT)

I never understood ordinal numbers, but I think you're right that $2^{\omega }$ doesn't make much sense. Wikipedia say that it is defined and that $2^{\omega }=\omega$ which is definitely not what we want here. -- Jitse Niesen 10:02, 30 March 2007 (CDT)

Perhaps $2^{|S|}$ where S is a set of order type $\omega$? You obviously understand ordinals better than I do, because I assumed the GCH in my original text without even realizing it! Greg Woodhouse 10:40, 30 March 2007 (CDT)

I've been told that it's okay to use $2^{\omega }$ in this context [1]. However, I think it's potentially confusing (at least for us), so I reformulated the sentence. -- Jitse Niesen 08:08, 31 March 2007 (CDT)

Audience

I'd like to point out that this article should be maximally useful for the average university-educated person. This means that the first paragraph needs to be rewritten so that people who don't already know what "real numbers" are can understand it. After all, if I need an article about what the real numbers are, am I going to be able to understand "order completeness, Cauchy completeness, or the nested interval property"?

This doesn't mean that it has to be dumbed down. It can have all sorts of impressive technical apparatus. It just has to be as comprehensible as possible to the university student. So begin with an explanation comprehensible to people who don't know what real numbers are. Then you can impress us with your knowledge of Cauchy completeness, etc. Please see Article Mechanics for more such remarks. --Larry Sanger 12:32, 29 March 2007 (CDT)

I think you're right. I've attempted a rewrite. I don't think it eliminates any advanced concepts that are not discussed later in the article. Greg Woodhouse 13:39, 29 March 2007 (CDT)

Cauchy sequences

The section on Cauchy sequences and completeness includes an aside on applications of the Cauchy criterion to analysis/calculus. The specific example given is establishing convergence of the exponential function. This is an important point, and I'm reluctant to just delete it, but it is also not really relevant to the main article. Then again, it does serve the purpose of providing some motivation for the concept of a Cauchy sequence. I'm somewhat torn on this one. Greg Woodhouse 14:14, 29 March 2007 (CDT)

Intro

In mathematics, real numbers represent quantities that may or may not have exact expressions as fractions (${\frac {3}{4}}$) or decimals (0.75).

This seems like a really lousy opening sentence. It puts the emphasis in entirely the wrong place. What sorts of exact expressions they have should be WAY down below the beginning. I'll be back..... Michael Hardy 17:59, 14 April 2007 (CDT)

I agree that it's awkard. What do you suggest? Greg Woodhouse 19:05, 14 April 2007 (CDT)

Simpler Definition?

How about a simpler (maybe informal) definition that can also be understood be non-mathematicians? E.g., "Numbers with an infinite decimal representation". Alexander Wiebel 11:59, 26 June 2008 (CDT)