3

I'm looking for a book that develops the theory of real numbers in a rigorous way in terms of their decimal expansions. The exposition should be concrete and preferably aimed at mathematically talented high-schoolers. For example, it should be pitched at readers who haven't necessarily heard of the least upper bound property prior to reading this part of the text in question.

I'd be curious to hear about books in other languages as well.

Edit. To give you a better idea of what I'm talking about, I'm including an excerpt from the algebra textbook by Kiselev, where as an example he illustrates the meaning of $\sqrt{3} + \sqrt{2}$.

*Algebra* by Kiselev

My translation of the part at the bottom:

Adding numbers $\alpha$ and $\beta$ means finding a third number $\gamma$ which is greater than the sum of any approximations from below of these numbers, yet less than the sum of any approximations of them from above.

We accept without proof that for any two real numbers $\alpha$ and $\beta$, one and only one such number $\gamma$ exists.

Now this book doesn't actually answer my question, since it doesn't prove this statement or others like it. I am looking for something that actually carries these proofs out and makes an effort to be as accessible as possible while remaining rigorous.

Mike
  • 911
  • Try Tim Gowers https://www.dpmms.cam.ac.uk/~wtg10/decimals.html and there is a chapter in Hardy and Wright (both are essentially directed at bright first year undergraduates, but accessible to some bright high school students) – Mark Bennet Nov 14 '14 at 22:33
  • http://math.stackexchange.com/a/957183/1242 – Hans Lundmark Nov 14 '14 at 22:38
  • @MarkBennet The Gowers page isn't what I had in mind. It's just an overview. The section of Hardy and Wright takes real numbers as already existing, with known properties, and then defines and studies their decimal expansions. – Mike Nov 14 '14 at 22:47
  • @HansLundmark That text by Blatter is obviously directed at a mature reader interested in the question. I am thinking of something that could be read directly by a student. – Mike Nov 14 '14 at 22:56
  • I put it as a comment because it didn't answer the question - yet both have some insight for those who are interested. There are reasons why real numbers are developed without reference to base $10$ or other base - the three versions I know are via Cauchy sequences, or Dedekind cuts, or an axiomatic approach with every bounded set having a least upper bound. The last of these does not construct the real numbers - so one needs a model to prove that the axioms are consistent and the object exists. – Mark Bennet Nov 14 '14 at 22:57
  • @MarkBennet I understand that Cauchy sequences and Dedekind cuts are superior from a mathematical standpoint. However, I don't think they're as good for telling a high school student what a real number is, or what it might mean to add two real numbers. – Mike Nov 14 '14 at 23:02
  • Mike - one way of attacking this issue is to ask what it means to have a number line without gaps - intelligent high school students can engage with that. – Mark Bennet Nov 14 '14 at 23:06
  • I thought one of the reasons the Cauchy sequence formulation was useful was because it precisely captures the idea of approximating real numbers by decimal expansions. It is more general than that, but you can ignore that and observe that the sequence $3, 3.1, 3.14, 3.141, \ldots$ is exactly a Cauchy sequence. Then all the machinery developed for Cauchy sequences carries over directly to sequences of decimal expansions, except that it is simpler because it is more general. – MJD Nov 15 '14 at 00:37
  • You might be interested in Shafarevich's Discourses on Algebra, chapter 5. – Marius Kempe Nov 19 '14 at 20:16
  • @MariusKempe That source introduces the real numbers axiomatically and then studies their decimal expansion after the fact, rather than defining real numbers by their decimal expansions. – Mike Nov 21 '14 at 04:27
  • I know, but I still thought you might be interested in it. – Marius Kempe Nov 21 '14 at 05:05
  • @MariusKempe Thanks, it certainly seems like a good book. – Mike Nov 22 '14 at 04:07
  • You may read E. Landau's book Foundations of Analysis. –  Nov 23 '14 at 03:49

2 Answers2

1

Page 505 of Spivak's Calculus (the very final problem in the chapter on "Construction of the real numbers") contains an exercise that begins

"This problem outlines a construction of "the high-school student's real numbers." We define a real number to be a pair $(a, \{b_n\})$ where $a$ is an integer and $\{b_n\}$ is a sequence of natural numbers from $0$ to $9$, with the proviso that the sequence is not eventually $9$; intuitively, this pair represents $$ a + \sum_{n=1}^\infty b_n 10^{-n}. $$

With this definition, a real number is a very concrete object, but the difficulties involved in defining addition and multiplication are formidable $\ldots$"

He then outlines a program for defining addition and multiplication and proving their properties, one that can be carried out by a bright high-school student, albeit one that has read the rest of this book, and knows things about sequences and least upper bounds, etc. He notes, wisely, that the description of mutliplicative inverses is no fun at all.

That's hardly a "text", but it's reasonable, especially if you want to show it to very bright high schoolers. If you want to show it to others, I think you'll find that they mostly don't appreciate it, but perhaps I'm too cynical.

John Hughes
  • 93,729
  • A big part of my question is how authors may have solved the problem of presenting this in the most accessible way possible. Spivak doesn't need to do that because he's merely outlining the process - in half a page - for someone who has already read his calculus book, including such things as a proof that $e$ is transcendental, and of course a construction of the real numbers using Dedekind cuts. – Mike Nov 15 '14 at 01:24
  • I guess I'll side with Spivak here: I think that the most accessible way to present this is to do it right...which means you need to know things like least upper bounds, careful induction proofs, etc. You don't actually have to have read the whole book to appreciate it (I certainly hadn't read every word back when I encountered this in high school), but it's hard for me to imagine many students having the sophistication to understand the subtleties without a comparable experience with serious proofs and a bit of analysis. Anyhow, sorry this didn't satisfy your needs. – John Hughes Nov 15 '14 at 04:00
  • One more thought: a bright student might read Moise's "Elementary Geometry from an Advanced Standpoint" and find the section on Eudoxus's invention of (something equivalent to) Dedekind cuts at least moderately accessible; that might provide a decent foundation for understanding what's necessary to work through the Spivak problem. – John Hughes Nov 15 '14 at 04:02
0

If you can read German, here is a detailed development of the approach suggested by Gowers (see the link given by Mark Bennet), albeit in terms of binary fractions:

http://www.math.ethz.ch/~blatter/Dualbrueche_2.pdf

But note the following: Whichever approach you take, the amount of work to be done in order to verify all the details is about the same.

Christian Blatter
  • 226,825
  • Thanks for this link, which was also pointed out by Hans Lundmark above. I am looking for something addressed to an intelligent but relatively unsophisticated reader like a talented high school student. – Mike Nov 20 '14 at 05:13