I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there?
In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.
The decimal places are just a way of representing it.
The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
No, it really isn’t. The real numbers can be constructed in a number of ways, and it is more common to define them as either Dedekind cuts, or equivalence classes of Cauchy sequences of rational numbers.
Personally, I’d go with the sideline cut definition.
Dang autocorrect. “sideline” should be Dedekind
Or maybe we can know them equally well? The function f(x) = x(0^(sin(πx)^2)) for example "requires" infinities, but only returns integer values.