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''.

  • $\begingroup$ The math is apparently like the food in Italy. Fwiw, I use an axiomatic approach for my Freshman calculus. I mention the construction of a set which follows the axioms is possible (but technical) and as such I leave it for a later course. I would not say this is typical, I think typically such issues are largely ignored as the focus is on computation (in US highschools even more than universities, but in both there exist exceptions like my course) . See page 26 of supermath.info/OldschoolCalculusII.pdf for my approach (which I think I got from Apostol..) $\endgroup$ Commented Jun 26, 2015 at 16:15
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    $\begingroup$ I'm interested to know how the real numbers are introduced at beginner level in different countries. What does "beginner" mean here? Here in the US, I think the typical approach is that these foundational issues are simply never discussed with the vast majority of the students we serve. The completeness property of the reals is something that one sees only if one is a university math major in an upper-division analysis course. $\endgroup$
    – user507
    Commented Jun 26, 2015 at 16:35
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    $\begingroup$ The algebra textbook by Kiselev, which was widely used in the USSR until the 1950s, defines real numbers by decimal expansions. It then gives a perfectly rigorous definition of addition via decimal approximations, and states without proof that there is a unique real number meeting the definition of the sum. It does similarly for multiplication. See this question for further details: math.stackexchange.com/questions/1022227/… $\endgroup$
    – Keith
    Commented Jun 26, 2015 at 20:23

2 Answers 2


Spivack's Calculus uses this approach:

At the beginning are stated some properties of "the number system" that amount to saying it is a complete ordered field (but not using that language). At the very end of the book (perhaps you reach it 9 months later, after doing rigorous calculus) there is a construction of a complete ordered field, and the proof that any two are isomorphic.


Here's my journey to understanding the real numberline:

Aged 15, Year 11:

I was first introduced to the idea of irrational numbers and shown that $\sqrt{2} $ and $\sqrt {3} $ are irrational. I new $\pi $ was special because it was transcendental but I knew little more about transcendence.

Aged 16, during lower sixth form:

My motivation for wanting to explore the question "what is a real number?" came from realising the difference in cardinality between the rationals and reals (through Cantor's diagonal argument). It really emphasised the difference between the set of rationals and set of reals.

Aged 17, during upper sixth form:

Learning of the density of the rationals and irrationals on the reals was also fascinating. I especially liked the idea that the real number line is full of "holes' if the rationals or irrationals are omitted.

Aged 18, during first year of undergraduate degree:

When it comes to the Completeness Axiom I found it fruitfull to consider and prove its equivalent statements. Personally I think "every decimal expansion converges" is the most intuitive statement of the axiom. However I didn't become convinced of its fundamental importance until I learnt of its equivalence to so many other propositions, especially: the principal of monotone convergence, Bolzano-Wierstrauss, Cauchy sequences, least upper bound property and nested intervals theorem.

I advocate the book below as a highly accessible introduction to real analysis:

Numbers and Functions: Steps into Analysis, R. P. Burns

  • $\begingroup$ And what about computable and non computable numbers? $\endgroup$ Commented Jul 9, 2015 at 16:12
  • $\begingroup$ Computable and non computable numbers are new to me. I don't see how they obstruct an understanding of the reals. $\endgroup$
    – Bysshed
    Commented Jul 9, 2015 at 18:47
  • $\begingroup$ Computable numbers are countable, so the fact that the set of real numbers is uncountable is essentially due to uncomputable numbers, that are numbers that we can never know with a given approximation. Not an obstruction, but an interesting properties. $\endgroup$ Commented Jul 9, 2015 at 20:20
  • $\begingroup$ Definitely interesting, do you have any recommendations for an introduction to the area? I'm struggling to understand that there are only countably many algorithms. $\endgroup$
    – Bysshed
    Commented Jul 9, 2015 at 22:14
  • $\begingroup$ there are many possible references. You can start from the Wiki page about computable numbers. More depth: projecteuclid.org/download/pdf_1/euclid.pl/1235422917. And you can find othere references in the answers to: math.stackexchange.com/questions/1078593/… $\endgroup$ Commented Jul 10, 2015 at 13:14

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