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I refer here to the question “Can every real number be represented by a (possibly infinite) decimal?” asked by WakeUpDonnie Jun 2 '13 (at 21:43).

I have few a follow up questions which are related so I will present them both here:

Question No 1 is a about the sum of all the terms $a_n/10^n$ to represent the number
$a= 0.a_1a_2a_3…a_n…$ (to use an example of a real number between 0 and 1).

I have previously asked a question about the sum of $(−1)^n/n^2$ for all N, i.e. not the limit of the series but the actual sum of an infinite of number of terms , $(−1)^n/n^2$.

I got the comment: Mathematics does not have "an actual sum of an infinity of numbers" in store. – from Christian Blatter last November. This distinction - between the limit of a series and a sum of an infinite number of terms - actually seems reasonable to me when I think of it.

Nevertheless, when we talk about representing a real number as an “a (possibly infinite) decimal” expansion, it seems that we do equate these things, i.e. a sum of an infinite number of terms $a_n/10^n$ for the general case of a real number (between 0 and 1 in my example).

So my question No 1 is: Do you believe we should accept Blatter's view in the general case and accept an exception for an infinite sum of positive terms, to the effect that this sum equals the corresponding limit?

Question No 2. Do I need the axiom of choice to describe (or refer to) my general real number (between 0 and 1): $a= 0.a_1a_2a_3…a_n…$

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Mikael Jensen
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"Sum of infinitely many elements" always has to mean something other than "repeatedly apply the binary sum", since there is no way to make literal sense of repeatedly applying the binary sum infintiely many times.

You can give alternate definitions in the various cases. e.g. we can define the infinite sum of positive numbers can be defined as the least upper bound of all possible finite sums taken from your set of numbers.

Of course, this turns out to give the same answer as the limit of partial sums definition.

There are even summation methods that are motivated for completely different reasons than convergence, and yield different summation operators. e.g. in the summation method given by zeta regularization, we get identities like

$$1 + 2 + 3 + 4 + \ldots = -\frac{1}{12}$$

whereas this would sum to $+\infty$ if we used the limit-of-partial-sums version of infinite summation.

For question number 2, the $a$'s can be defined directly by a formula

$$ a_n = \lfloor 10^n a\rfloor - 10 \lfloor 10^{n-1} a \rfloor $$

so you don't need the axiom of choice at all. (note this also gives the digits to the left of the decimal place as well)

  • Regarding question No 2: Thank you. I was thinking about the initial “acquisition” of the number which may not be relevant here. I realize that you may use the number to calculate $a_n$ once you “know” the number, and that the transfer process (necessary for you to "know" the number) would require an infinite number of steps (in the general case) just like your decoding process. – Mikael Jensen Apr 13 '14 at 12:02