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Is there a (more or less complete) theory of real numbers, where every number except for $1$ is defined by a combination of arithmetic operations acting on $1$.

I know we can build any whole number by using addition and subtraction of 1s, and any rational number by using division of whole numbers.

We can also build any irrational number by using infinite series with rational terms or infinite products or infinite continued fractions.

So it is not a question about whether it is possible to build all real numbers that way, but rather is where some treatise on the subject?

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  • $\begingroup$ Have you heard of Dedekind cuts? The idea is to identify a real number $r$ with the set $(-\infty,r)\cap\Bbb Q$. A set of that form is called a cut. Formally: A cut is a subset $A$ of $\Bbb Q$ that has no biggest element and is downward closed ($x\in A$ and $y<x$ implies $y\in A$); the reals is then defined to be the set of cuts. $\endgroup$ – Akiva Weinberger Sep 10 '15 at 21:22
  • $\begingroup$ Also look up "[equivalence classes of] Cauchy sequences". The idea of this construction is to identify a real number with the set of all sequences of rational numbers that converge to it. $\endgroup$ – Akiva Weinberger Sep 10 '15 at 21:23
  • $\begingroup$ Thank you, but I think Dedekind cuts are not really operational. How exactly do we 'cut' a rational number into a sum of two irrationals by using only four arithmetical operations? We need to know one of them already to find another. At least, it is useful in that way - by knowing at least one irrational number we can potentially obtain an infinite amount of them $\endgroup$ – Yuriy S Sep 10 '15 at 21:27
  • $\begingroup$ Well, the Cauchy sequence version might work better for you; it doesn't use the four basic operations, but it uses the limiting operation, which you've been implicitly using (in your infinite sums, for instance). $\endgroup$ – Akiva Weinberger Sep 10 '15 at 22:35
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There are numerous constructions of the real numbers, some of which may be along the lines of what you are looking for (do note your question in not entirely clear). You may be interested in the survey article here (or its arXiv version) which surveys many constructions.

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  • $\begingroup$ Thank you for the link. Now that I think about it, it was more of a philosophical or even psychological question. I am not satisfied with the usual definitions of algebraic and transcendental numbers, so I was looking for something more concrete. $\endgroup$ – Yuriy S Sep 10 '15 at 21:07
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    $\begingroup$ Your paper is very nice (and you are very kind to mention my little contribution to the subject). There is a paper by F.A. Behrend "A contribution to the theory of magnitudes and the foundations of analysis" Math. Z. 63, 345-362 (1956) which I rate very highly in this connection. In particular, Behrend shows that is quite simple to define the reals using decimal expansions and simple logic to define the additive structure and prove that the additive group is complete and totally ordered. He then gets the multiplicative structure by some neat reasoning about order-preserving endomorphisms. $\endgroup$ – Rob Arthan Sep 10 '15 at 21:14
  • $\begingroup$ thanks @RobArthan I will certainly look at that article. $\endgroup$ – Ittay Weiss Sep 10 '15 at 21:17

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