# In what course are the number systems typically constructed?

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?

• I constructed a number system in Math 102, which was only used by the CS major. – Joshua Sep 14 '16 at 2:38
• Math 101 (focused for math majors/minoring in math) in Univ of Eastern Finland :) – mike3996 Sep 14 '16 at 9:40

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 $\mathbb{N}$).

• Is $\mathbb N$ so hard to construct? To be honest, thought, I can't recall whether any course I had any interaction with constructed both $\mathbb N$ and $\mathbb R$ in a formal way. – David K Sep 13 '16 at 23:16
• So a lot of it is just deciding what your set theory is, so that it even makes sense to "construct $\mathbb{N}$" (construct it from what?) Once you've chosen ZF or ZFC, $\mathbb{N}$ as a set is the ordinal number $\omega$, which is easy enough to define (it is identifiable as "the set the axiom of infinity tells you exists"). But now you have to set up the operations. That part is a bit more involved (just as it is when you construct $\mathbb{R}$). – Ian Sep 13 '16 at 23:18
• I see little merit in constructing the natural numbers from the ZF(C) axioms. I think the axioms for $+,\cdot$ and $\leq$ are way easier to accept than many of the ZF(C) axioms. (I also don't see why we have to start with the Peano axioms...) – Stefan Perko Sep 14 '16 at 8:54
• @StefanPerko Except again, now you're describing what $\mathbb{N}$ is, not constructing it. Just as people have objected to synthetic approaches to $\mathbb{R}$ in this thread, so one might object to synthetic approaches to $\mathbb{N}$. Of course ZF/ZFC are synthetic, but one thing we learned in the 20th century is that eventually a formal system is inevitably synthetic. – Ian Sep 14 '16 at 12:38
• @Ian Yes, that's what I'm saying. How are the ZF axioms anymore "obvious" than the axioms for $\mathbb{N}$? The axioms for $\mathbb{R}$ are imo waaay harder to accept, so I think there is a quite difference to accepting the axioms for $\mathbb{N}$. I can see that constructing $\mathbb{N}$ from the set theory axioms is the way to go if you are studying ZFC set theory specifically, but other than that I don't see it. – Stefan Perko Sep 14 '16 at 13:21

The book Numbers by Ebbinghaus et al. is a fascinating, readable and rigorous survey of the number systems of modern pure mathematics.

• I'd never even heard of this book, so +1 from me. As an added bonus, it goes above and beyond the usual number systems, talking about $p$-adics, octonions, nonstandard analysis, and numbers constructed from games à la Conway. – Will R Sep 13 '16 at 23:44

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).

• See also Foundations of Analysis by Edmund Landau, or The Anatomy of Mathematics by Kershner & Wilcox. – bof Sep 13 '16 at 23:10

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:

Finally, the classic reference on this whole topic (the progression from $\mathbb{N}$ to $\mathbb{R}$) is: