Abstract algebra seems to be more about general, abstract structures and patterns in mathematics, whereas number theory is more about properties of numbers and various connections that pop up between them.
So what course usually takes care of constructing the number systems? And are there any good introductory books out there that construct the number systems? Books that are especially good at explaining the construction of complex numbers?
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In my undergrad real analysis class, we constructed $\mathbb{R}$ given $\mathbb{Q}$ as a black box, using the Cauchy completion construction. One can also use the Dedekind cut construction.
Constructing $\mathbb{C}$ from $\mathbb{R}$ is an easy matter. Constructing $\mathbb{Q}$ from scratch is not quite so simple (mainly because you have to construct $\mathbb{N}$).
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The book Numbers by Ebbinghaus et al. is a fascinating, readable and rigorous survey of the number systems of modern pure mathematics.
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At least at my university, there is no course which actually teaches this material. We were presented with axioms for how the real numbers work. I would guess that this is the current trend, at least in my country (UK), but I can't speak for everyone.
First off, construction of the complex numbers is, for some reason, typically left to books which actually discuss some complex analysis. That is to say, excepting a few mentioned by others, I don't know of many books which talk about both the integers and the complex numbers. Some good books which do talk about complex numbers include:
All of the above contain a construction of the complex numbers in the first chapter. I should mention at this point that it might also be worth googling for resources, for example ProofWiki has a construction of the complex numbers. For non-complex number systems ($\mathbb{N},\ldots,\mathbb{R}$), a nice "do-it-yourself" version is included in Terence Tao's Analysis I notes, which can be found at this webpage, and these have also been published in a slightly more refined book format.
A great book which spends a good deal of time focusing on construction of the basic number systems (that is, excluding the complex numbers, as I recall) is:
I guess the main topic of this book could be described as "foundational issues"; hence why construction of numbers plays a prominent role.
On a similar note, introductory set theory textbooks often contain constructions of numbers (often up to and including the real numbers). For example, the following books:
Construction of $\mathbb{Q}$ from $\mathbb{Z}$ is I think covered in Alice in Numberland above. Nevertheless, I'd like to mention here that this construction is standard in abstract algebra: it's called the field of fractions of an integral domain. Constructing $\mathbb{Q}$ is the special case when the integral domain in question is $\mathbb{Z}.$ Two books which I know cover this are:
The references on the Wikipedia page linked to previously must also talk about this, but I haven't looked at those books.
The fact that construction of the reals appears in set theory texts is in part because of the role of set theory as a foundation for mathematics. However, classically, the construction of the reals belongs to real analysis. A construction of the reals is provided, for example, in the following books:
Finally, the classic reference on this whole topic (the progression from $\mathbb{N}$ to $\mathbb{R}$) is:
This last is really dry. There is nothing in this book except for theorems and their proofs. But that might be just what you're looking for.
These constructions often come up at the beginning of a first real analysis course. The natural numbers are defined from basic notions about sets, then the integers from the natural numbers, the rationals from the integers, and so on. There are many books that have these constructions. One that comes to mind is Royden's Real Analysis (now Royden and Fitzpatrick's).