well it's hard to formally define them, but it's not hard to say "imagine that all these decimals go on forever" and not worry about the technicalities.
Imagine you have a ruler. You want to cut it exactly at 10 cm mark.
Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.
A reasonably defensible inference would be that adding a finite amount of precision adds a finite number of additional digits. That is a physically realizable operation. There's no obvious physical meaning to the idea of repeating that operation infinitely many times, so this is not clearly a meaningful way of defining or constructing real numbers. If you were trying to use this construction to convince a skeptic that irrational real numbers exist, you would fail -- they would simply retort that arbitrary finite precision exists and that you have failed to demonstrate infinite non-repeating, non-terminating precision.
The colloquial phrase 'infinite decimal' is perfectly intelligible without reference to whether it's an infinite amount of data or rigorously defined or whatever else.
There's a lot of trickery involved din dealing with the reals formally but they're still easy to conceptualize intuitively.
If I were a skeptic of real numbers, I’d tell you that talking about an infinite decimal expansion that never terminated and contains no repeating pattern is nonsense. I’d say such a thing doesn’t exist, because you can’t specify a single example by writing down its decimal expansion — by definition. So if that’s the only idea you have to convince a skeptic, you’ve already failed and are out of the game. To convince the skeptic, you’d have to develop a more sophisticated method to show indirectly an example of a real number that is not rational (for instance, perhaps by proving that, should sqrt(2) exist, it cannot be rational).
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.
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.
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.
Moreover, I disagree that you have imagined real numbers. I don’t think you’ve imagined a single real number at all in the manner you describe. Why should I believe you've even described anything that isn't rational to begin with? For instance, 0.999... is the same as 1. Why should I not think that whatever decimal expansion you're imagining is, similarly, equivalent to a rational number we already know about? Occam's razor would reasonably suggest you're just imagining different representations of objects already accounted for in the rationals. After all, an infinite amount of precision captured by an infinite nonrepreating string of digits could easily just converge back to a number we already know.
because when I picture 1.000 I am clearly imagining a real number. Likewise if I imagine pi, as defined any way you like.