Definition talk:Absolute Value

I don't think the stuff on ordered integral domains is consistent with the $p$-adic numbers (or most non-Archimedean values) -- this kind of stuff operates as valuations in valuation rings [1] --Linus44 16:31, 20 March 2011 (CDT)

Fair comment, I moved it about.

Ultimately what I'm trying to get to is where everything about orderings which are taken as "axiomatic" in standard works on analysis (at the start of the book there's usually something which says: "We take as axiomatic these facts about the real numbers ...") are derived from more basic properties derived from the most skeletal structures of all - and as far back as I've managed to get so far is the "ordered integral domain" where the "ordering" is derived from that most basic of concepts the "positivity property". So there's going to be a certain amount of restructuring needed on the analysis pages (at least) where the concepts need to be redefined in that context.

Now the "abstract absolute value" is interesting in that it fulfils the conditions derived from the "absolute value" as introduced based on the "ordered integral domain" which is the bit you commented about. Which comes first? The abstract concept, from which we deduce that the concrete version as defined on the OID and demonstrated to fit these conditions? Or the absolute value as defined on the OID from which we introduce the concept for the general field (and I can see where this goes: clearly the complex modulus is an abstract absolute value and so $(\C, \cdot)$ is a valued field.

Or we do both at once and prove that absolute value as defined via the positivity property on the OID is the only such function meeting these conditions on the OID? (Is that even true?

I need to take a step back and come back to it when I've thought about something else for a while. My brain works glacially slowly nowadays. --prime mover 17:15, 20 March 2011 (CDT)

I guess to some extent it's a matter of taste.

My feeling is that analytically the absolute value is just a measurement of distance -- the existence of such a function classifies $\R$ as a valued field, but it is just a convenience of the archimedean property and completeness that this so closely relates to ordering, and $\R$ is more naturally an ordered field.

But I've only really studied the $p$-adics and discrete valuations in this context, I've only touched upon ordered fields, formally real fields etc., so I don't understand exactly what the relationship is, and I skip the page whenever I see ordered abelian groups mentioned...

BTW, I'll probably be quiet for a a couple of months, I'm doing the Part III exams soon and I'm every bit as scared as they want me to be. --Linus44 13:13, 21 March 2011 (CDT)

Various Classes of Numbers

Since the natural numbers, rationals, etc. are all subsets of the reals, why are they even addressed in the article, since if the definition holds for the reals, it will hold for all subsets of the reals.

cos it is all right?

Also, it might be helpful to clarify the fact that the complex numbers are not totally ordered by the less-than or equal to sign, but that does not mean that they cannot be totally ordered by any relation. In fact, (correct me if I'm wrong) the Well-Ordering Theorem proves this is not the case (the theorem is equivalent to the Axiom of Choice, however) - that in fact, there must exist a relation that totally orders the Complex Numbers. -Andrew Salmon 15:02, 10 September 2011 (CDT)

Yes, this could be done on another page, I suppose. The philosophy of this site is not to put everything on the same page. One page, one result. Don't want to clutter this page up with abstruse set-theory results based on an axiom not universally accepted.

Anyway, the fact is mentioned. "Complex numbers $\C$: As $\C$ is not an ordered set, the concept as defined here can not be applied. The notation $\size z$, where $z \in \C$, is defined as the modulus of $z$ and has a different meaning."--prime mover 15:14, 10 September 2011 (CDT)