A natural construction for the real numbers

We propose a new construction of the real number system, that is built directly upon the additive group of integers and has its roots in the definition due to Henri Poincare [P, pages 230–233,] of the rotation number of an orientation preserving homeomorphism of the circle. The definitions of addition, multiplication and comparison of real numbers are very natural in our setting. The proposed definition of real numbers is illustrated with examples that are irrational, roots of an integral polynomial equation, but not expressible by radicals, or not root of an integral polynomial equation. I thank Sebastian Baader, Etienne Ghys and Domingo Toledo for stimulating discussions.