> Oh, and some of them (either nearly or exactly half) are negative.

Well, it doesn't quite work that way. Yes, you can put the negative integers in 1:1 correspondence with the positive integers, but you can also put them in 1:1 correspondence with the positive integers that are divisible by 2, or 3, or 3,000... or, for that matter, the negative integers that are divisible by three thousand. All of those sets have the same cardinality (often called "countably infinite").

There are, however, provably more real numbers than there are integers.

It is unknown whether there are any sets whose cardinality lies between the cardinality of the set of integers and the cardinality of the set of real numbers.

Bait taken I see :)

Hands up anyone who hasn't considered a stay at the infinite hotel.

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...

There is a sense in which the even integers can be considered "half" of all the integers: https://en.m.wikipedia.org/wiki/Natural_density

"Counting stuff" can be generalised to infinite sets in different ways: cardinality is one (arguably the most common) of them, but natural density or measure are others.