I'm interested to know how the real numbers are introduced at beginner level in different countries. In my (old) experience of teacher in Italy there was some well defined steps:
1) Introduce the field of algebraic numbers, showing that there are numbers (radicals) that are not rationals. The classical proof is the irrationality of $\sqrt{2}$.
2) Introduce some transcendental number as $\pi$ , $e$ and other that we needs to calculate transcendental functions as exponential, logarithm, trigonometric, combined between them and with radicals. The set of this number is not defined in a really rigorous way, but it is used as a field, and we can thik that coincide with the field of computable numbers.
All this is given in the first years of high-school. The we begin with mathematical analysis and we need the Completeness Axiom for the reals.
In my experience this is introduced intuitively using the ''correspondence'' from the real numbers and the points of a straight line. Well, being an axiom, we have not to prove it, but it should be good to show that really we can do some set of ''numbers'' with such property. But the classical Dedekind or Cauchy constructions are difficult at this level so I'm interested to know how this question is solved in the real teaching experience.
The problem become more difficult when we note that a consequence of CA is that the set of real numbers is uncountable. I don't know if this fact is taught to the students a this level, but it seem to me very important. And if some student asks if this uncountable many numbers are the transcendental numbers that he knows what is the answer? We are forced to introduce the set of non computable numbers, that we can not compute in any way. Is this concept introduced at high-school level in some country ?
I've seen the similar question Good definition for introducing real numbers?
but I think that it is not a duplicate because I'm interested in the different experiences and possibly in different countries, to know if there are different approach or different ''schools of thinking''.