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real number

Synonym: 
real, $\mathbb{R}$
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

54C30 no label found26-00 no label found12D99 no label found

Comments

I've always understood the real numbers as having a relatively simple axiomatic definition (e.g. [0]). I'll admit I didn't properly understand this entry, but it didn't even seem to acknowledge the existance of this kind of definition. Is it worth me writing such a definition, either for inclusion on this article, or as a seperate article?

[0] http://www.maths.ox.ac.uk/current-students/undergraduates/lecture-materi...

When the entry discusses the real numbers being 'the complete ordered field', this is kind of ultra-shorthand for the axioms in your link.

There's already a set of axioms like this on pm though:
http://planetmath.org/encyclopedia/AxiomaticDefinitionOfTheRealNumbers.html

I suppose one of the main reasons to do constructions like the one in the entry you commented on is that otherwise you don't know if there's anything out there that actually satisfies these axioms...

Ahh, I hadn't spotted that, thanks!

Further, 'sum' and 'product' are closed operations in $\mathbb{R}$. The reference given by Adam does not make mention to that fact.

> Further, 'sum' and 'product' are closed operations in
> $\mathbb{R}$. The reference given by Adam does not make
> mention to that fact.

actually it does - the first couple of lines say:

For every pair of real numbers a, b \in R there is a unique real number a + b, called their ‘sum’.
For every pair of real numbers a, b \in R there is a unique real number a · b, called their ‘product’.

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