Real and Imaginary Numbers: Set of All?

[Deveno]

I don't really see why this must be true.

really? I'm surprised at you.

how does one define "an infinitely repeating decimal"? that is, how can a person be sure that the infinite sum:

\sum_{k=1}^\infty \frac{3}{10^k}

actually "converges"?

ok, the usual proof that 0.\overline{333} = 1/3 goes like this:

x = 0.33333...
10x = 3.33333... <--here. this step. how does one justify what "multiplying an infinite decimal MEANS?"

9x = 3
x = 3/9 = 1/3.

it all looks very convincing, but the step i am calling into question is a serious issue. looks like "hand-waving" to me. why? because even a little thought tells us "infinite things" DON'T behave like finite things. so if you have an infinite sum, you need to back up any claims about working with that sum.

and that notion of "convergence" is what we use to justify that "infinite decimals" mean something. so my point is, UNTIL you have a solid notion of what a "convergent series" IS, talking about "infinite decimals" doesn't make any SENSE.

so if you are going to use infinite decimals, you are already pre-supposing some facts about a completion of "some number system" (i don't care if the numerical (digital) representation is base 10, or base 2, or base p).

so yes, one of the drawbacks of "decimals" (or binary representations, doesn't matter), is that many rational numbers don't have a terminating decimal form. and the "proofs" that high-school kids are taught (how to find/covert fractions to decimals) are true, but not VALID. I'm not saying this is bad pedagogical practice, we're taught what the areas of several regions are, without having a good definition of "area", either.

******

let me back up a bit, and address the main topic. the real question that we are dancing around, is: what is a number? this is a very good question, and there are several good candidates:

a) real numbers
b) complex numbers
c) rational numbers
d) computable numbers
e) constructible numbers
f) algebraic numbers
g) matrices
h) elements of a division algebra (perhaps commutative, preferably)
i) vectors
j) p-adic numbers

for various reasons, people believe that one or more of these are "unsatisfactory", because they do not capture some "intuitive" idea of what a number should BE.

MY feeling about the answer to this question, is another question: what is a number supposed to DO?