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I would expect almost NO numbers are rational (integer or fraction) with an infinite number of Reals between each.

> I would expect almost NO numbers are rational (integer or fraction) with an infinite number of Reals between each.

You're right (technically correct, which is the best etc.)! That is why "almost all" can mean everything except rational numbers.

in between any two real numbers, there is a rational number, and vice versa.

And yet somehow there are infinity times more reals than rationals...

Very hard to get your head around!

An fun, immediate, horrifying, consequence.

If I remove n elements from R, the remainder has n+1 connected components.

The complement of Z in R has |Z| connected components.

The complement of Q in R has |R| connected components.