Jtdylktuy

In grade school and high school, I was taught a real number is a number with a decimal expansion--that is, a finite sequence of digits followed by a decimal point followed by an infinite sequence of digits.

When I moved on to studying analysis, I was introduced to the Dedekind cut construction of the real numbers, and then proved every real number could be expressed as a decimal expansion.

Now presumably, the real numbers could also be rigorously constructed as decimal expansions, very similarly to the Cauchy sequence construction.

Is there a particular reason why, when Dedekind performed his original construction of the reals, he chose to use cuts rather than decimal expansions (or base 2 expansions for that matter)? Is there some unforeseen difficulty in rigorously constructing the reals by means of decimal expansions?

One problem is that multiple decimal representations correspond to the same real number. Of course, this is easily solved, though.

I think the real point was foundational: defining reals as decimal expansions is defining them as sequences of rationals (the Cauchy definition is via equivalence classes of sequences of rationals - the decimals approach picks out a "canonical" Cauchy sequence for a given real). By contrast, Dedekind cuts define a real as a set of rationals. On the philosophical side, sets are slightly simpler than sequences, and Dedekind was very interested in developing the foundations of mathematics.

Dedekind's definition is also more natural in that it doesn't fix a base: so it really defines the real numbers without making any arbitrary choices.

The OP asked

Is there some unforeseen difficulty in rigorously constructing the
reals by means of decimal expansions?

They might be interested to know that Simon Stevin (1548–1620) actually constructed the real numbers using decimal expansions (1585), and that was the first 'formal' construction.

Now it is fairly certain that his arguments would not pass muster in our modern world.

As a mental representation of numbers, the expansion is concrete and easier to visualize than Dedekind cuts. But I couldn't find a rigorous development of constructing the real numbers via expansions (can someone comment with links?).

You might think that developing a representation of numbers using base $2$ is less arbitrary than base $10$. In point of fact, I've been investigating these exact questions in the past month or so. Some might find it amusing to observe some of this work/torment so I present it in the next section.

The following is a 'logic exercise' and not what you would find in notes/papers constructing the real numbers as base b expansions.

Here we define the non-negative real numbers as binary expansions, also describing and sketching out the theory allowing us to add two numbers (we don't define multiplication).

Young people with mathematical talent are fascinated when told that

$tag 1 frac{1}{2} + frac{1}{4} +frac{1}{8} +frac{1}{16} + dots = 1$

With some further 'geometric hand-waving', they can be convinced that every non-negative number 'point' on the line has the form

$tag 2 m + sum_{n=1}^infty frac{b_n}{2^n} ; text{ with } m in mathbb N text{ and } b_n in {0,1}$

Using this high school representation theory, we can define addition on $mathbb R^{ge 0}$ with a 'symbolic/logic algorithm' that can handle adding $a$ and $b$, where

$tag 3 a = sum_{n=1}^infty frac{a_n}{2^n} text{ and } b = sum_{n=1}^infty frac{b_n}{2^n}$

This is all about putting our numbers in standard/simple form where we use the 'add em up' relation $text{(1)}$ and its variants, namely

$tag 4 sum_{n=k}^infty frac{1}{2^n} = frac{1}{2^{k-1}}$

Algorithm

The number $a$ and $b$ are in standard form, so we know that $a + b lt 2$. So, the first question is to determine if $a + b ge 1$, in which case we simplify by 'peeling off' a $1$.

The following is crucial here:

Theorem 1: For any $zeta ge 1$, if $a_zeta = 0$ and $b_zeta = 0$, then the coefficients of the $2^{-n}$ terms for $0 le n le zeta - 1$ are 'finalized' by adding together

$tag 5 sum_{n=1}^{zeta-1} frac{a_n}{2^n} + sum_{n=1}^{zeta -1} frac{b_n}{2^n}$

So, by the theorem, if $a_1 = 0$ and $b_1 = 0$, the sum $a + b$ will be less than $1$.

The algorithm will use the theorem, finalizing all the coefficients to any degree of precision required.

If $zeta$'s can be found of arbitrarily large size, then 'cranking out' the coefficients is no problem. But if not, we can put the algorithm to work on the largest $zeta$ found (if any), and then the algorithm must simplify

$tag 6 sum_{n={zeta+1}}^{infty} frac{a_n}{2^n} + sum_{n={zeta+1}}^{infty} frac{b_n}{2^n}$

For each term $n$, the algorithm can check if $a_n$ is $1$. If it is it can 'take it away' and put it in a '$c$-ledger'. If not, it can take $b_n$ away and put it in '$c$-ledger'. When this is completed, using $text{(4)}$ the '$c$-ledger' will simplify to one term, resulting in

$tag 7 frac{1}{2^{zeta}} + sum_{n={zeta+1}}^{infty} frac{d_n}{2^n}$

The $zeta$ slot is ok since there is nothing else to add there, and the algorithm terminates.

We know that this well-defined algorithm is an associative and commutative binary operation by removing our 'forgetful functor' and jumping back into any modern developed theory of the real numbers. But trying to prove it from this platform might not be worth the effort (we will let an AI machine come up with the arguments).