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I know that we can go from real numbers to complex numbers and then to quaternions and then to octonions and, if I remember correctly (I have read something about this sometime in the past but do not remember the details), we can build "numbers" of "dimension" $2^n$ (reals have $n=0$), complex ones have $n=1$, and so on and so on...)...

I guess that, generally, these structures with growing $n$ can and do generally lose some of its properties so to call them "numbers" is somewhat imprecise and counterintuitive but I would like, despite that fact, to know what are the main ideas behind this construction, and are there any other construction(s) that generalize the concept of number in some other ways?

Although the question is not rigorous enough I know that some of you know, more or less precisely, what exactly I want to know, so if you can help with clarification of these issues go for it, and, please, make your exposition as elementary as possible.

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    $\begingroup$ This is the Cayley-Dickson Construction: en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction $\endgroup$ – Travis Dec 14 '17 at 17:38
  • $\begingroup$ ...but NB there are many other ways to generalize the notion of 'number'. $\endgroup$ – Travis Dec 14 '17 at 17:38
  • $\begingroup$ @Travis If you know enough about all of this, I would like to see your answer, if you would like to answer this. $\endgroup$ – user480281 Dec 14 '17 at 17:39
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    $\begingroup$ In one direction, Clifford algebras are a natural path after quaternions and octonions... In another direction, p-adic numbers, in a third one dual numbers, etc. $\endgroup$ – Jean Marie Dec 14 '17 at 17:42
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    $\begingroup$ . . . . and there are the hyperreal numbers, which form a non-Archimedean ordered field. $\endgroup$ – Michael Hardy Dec 14 '17 at 17:54
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If you want to describe numbers as "something like the integers", then probably the right generalization would be rings. This seems the most comfortable to me: you have addition, subtraction and multiplication, and things distribute as you would expect. So, you can say it is "like the integers" in those respects.

Classically, fields (which are special cases of rings), were called "number systems" since they generalized things like the real and rational numbers. The idea behind number systems here was generally to express magnitudes of lengths in a plane geometry. Actually, every ordered field (indeed every ordered division ring) can be interpreted that way. The interpretation still holds up mostly for unordered fields, although then using the word "magnitude" loses its meaning.

If you wished to include all the things you get with the Cayley-Dickson construction on the reals, then you'd have to go further to nonassociative rings.

On the other hand, if you are more interested in numbers as counting things, then you should take a look at cardinal numbers. These capture sizes of sets, essentially. There is cardinal arithmetic and multiplication, but since the class of cardinal numbers isn't a set, it does not really fall into the same category as rings.

If not quantity but actually order matters to you, then you should look at ordinal numbers. These capture the same properties that $\mathbb N$ has regarding sequencing and well-orderedness.

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