[ajkjk]

I guess we are talking about different things. It seems to me that it's trivial to imagine then conceptually. They go on forever and most of them never repeat? Sounds good to me. Sqrt(2) never repeats? sure, whatever. I never found the proofs of this stuff very interesting.

Now, I am a skeptic of their use in physics / science. But that's a different question, and more about pedagogy than the raw content of the theories.

[nilkn]

With that approach, all anyone has to say is that you'd have to provide infinite information to specify an example and that the way these objects interact is completely undefined; therefore you haven't defined or done anything at all. You are indeed simply imagining something -- and nothing more. You can imagine whatever you want, but nobody else is inclined to believe that what you imagine exists or behaves in the intended manner.

Beyond that, if a skeptic were inclined to accept the existence of objects with "infinite information content" by definition, they could then ask you to simply add two of them together. That would most likely be the end of it -- trying to add infinite non-repeating decimal expansions does not act intuitively. To answer this type of question in general, you would have to prove that the set of all infinite decimal expansions, if we grant its existence, has a property called completeness, as you would eventually discover that you would have to define addition x+y of these numbers as a limit: x+y = lim_{k -> infinity} (x_k+y_k) where {x,y}_k = the rational number obtained by truncating {x,y} after k digits. You must prove this limit always exists and is unique and well-defined. And even having done all that work, you still couldn't give a single example of one of these numbers without additional nontrivial work, so a skeptic could still easily reject all of this.

This is far beyond what you could reasonably expect the typical middle school student or even general member of the adult population to follow and far more difficult than simply defining complex numbers as having the form x+iy.

[ajkjk]

yes, I am describing imagining something. Imagine taking decimals and letting them go on without ending. That is conceptualizing them intuitively. It is easy.

I don't really know what you're arguing about. You are describing the sorts of things that have to be solved to construct them rigorously. But I don't know why. No one is talking about that.