I am using the decimal approach because it is familiar to the students; the entry price for Cauchy sequences or Dedekind cuts is much higher. But ultimately I do not think it is satisfactory. Perhaps we just have to agree to differ.

Every decimal based approach to constructing the real number system that I have seen relies on defining addition/multiplication of two infinite decimals as limits of addition/multiplication of truncations of those decimals. So for example (0.111111…) x (0.222222…) will be defined as the limit of the sequence of multiplications:

0.1 × 0.2

0.11 × 0.22

0.111 × 0.222

etc

An approach that is quite natural when one considers that infinite decimals are themselves limits of sequences of finite decimals (since an infinite series is just the limit of a sequence of partial sums).

Kind Regards.

Here is an example which I will put in the notes. Suppose you have to calculate the digit after the decimal point in 0.414213562…+0.585786437…. Adding the digits so far we get 0.999999999…. But we don’t know whether this pattern continues. If each pair of digits sums to 9, then the sum of the numbers is 1, and the digit is zero. Otherwise, consider the first position where the digit sum is not 9. If it is less than 9, then the digit we are looking for is 9; if it is greater than 9, the required digit is 0. We have no idea how long we will have to wait to resolve this. Yet, of course, the longer we have to wait, the closer the sum is to 1.

Something you wrote above caught my eye as well sir:

“Please bear in mind, as you read what I have written, that the slogan of the course (taken from T. S. Eliot) is “precise but not pedantic”. So, for example, real numbers will not be defined by Dedekind cuts or by Cauchy sequences. This doesn’t mean that I think there is anything wrong with these constructions.”

I would agree that there is nothing wrong with these constructions if one is interested in more advanced stuff ,say topology, but for beginners is there anything wrong with just defining them as decimals?

Just speaking from personal experience, I don’t recall (when I was an undergrad at least) feeling enlightened by learning the Dedekind cut approach and certainly not the Cauchy sequence approach (spit). On the other hand I feel that going through the construction of R via decimals gave me a (slightly) better appreciation of elementary real analysis. Perhaps its just my applied bias talking.

Kind Regards

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Out of curiosity……

Analysis books seem to have an irrational (no pun intended) pedagogical bias against the decimal approach to R despite the alleged point of the first course in analysis being to understand Calculus better. Can anyone out there explain why?

PlanetMath • Relation Theory

When I was an undergraduate, the mathematics department was on the side of a hill; you came in at the ground floor and could go either up or down. This is my model. The students come in with at least some idea of what natural and real numbers, finite and infinite are; with the aim of being precise but not pedantic, I build on this to some extent.

I have no idea whether it will work, but it will be interesting to find out!

In the first year undergraduate course I teach countability is introduced at the end of the sections of the course discussing N, Q Z and R as one of the properties that distinguishes R from Q but otherwise not a huge amount is said about infinity. It will be interesting to see what you have to say about it!