They are not in the same class of countability. The devil lies in that detail of “at least one”. There are, in fact, an infinite number of rational numbers between any two irrationals, and an infinite number of irrationals between any two rationals… but that just brings us back to asking which infinity is larger. And the answer to that (which you can’t get just by looking at density) is that the infinity of the irrational numbers is larger than the infinity of the rational numbers.
To prove this, we first prove that the rationals have the same countability as the integers. That’s not so difficult: Every rational number can be represented by a pair of integers (the numerator and denominator in simplest form), and so the cardinality of the rationals is no larger than the cardinality of the set of pairs of integers. But the cardinality of the set of pairs of integers is the same as that of the set of integers, because you could put them on a 2d grid, start at the origin, and spiral out from there.
Then, we have to prove that the cardinality of the real numbers is greater than the cardinality of the integers. The usual way to do this is Cantor’s diagonalization argument: For any list of real numbers you might make (i.e., a correspondence between integers and real numbers), you can always find a real number that is nowhere on that list, by constructing one that differs from the first number in the first digit, differs from the second number in the second digit, and so on. So there can’t be a list of all of the real numbers, and so the count of real numbers must be strictly greater than the count of the integers.
I need to clarify. We were talking about numbers with every possible finite sequence of digits, not about normal ones. I see that neither yours nor the one with single zeros added are normal. However, I was asking whether the number of added zeros made a difference to the weaker condition of every possible finite sequence, and why. Does that weaker condition have a name, BTW?
The one with single zeroes added is normal. To be abnormal, the added digits have to create an uneven distribution in the limit. Adding a single zero between all strings has no effect, because eventually the strings get arbitrarily long–i.e., millions, trillions, any number you can think of. An extra zero compared to an arbitrarily long string is irrelevant. The same would be true if I added a million zeroes between each number, because again even a million is tiny once you get a little ways out.
My example was abnormal, though, because it doubled the number of digits, with half of them being zeroes. So no matter how far out you go, the fraction of zeroes is always >50%.
There are weirder examples. Say instead that I insert one zero after the first number, then two after the second, then four, and so on: 0.10200300004000000005…
Here, the number of zeroes grows so rapidly that it utterly dominates the distribution. The number of non-zero digits is negligible, and approaches 0% in the limit. And yet that number contains all possible strings as well.
Not sure. Though these constructed numbers could be considered Champernownesque numbers.
One thing this thread is confusing is the difference between the numbers and the representation of them. Decimal representation (nor any other base) is not the most natural or obvious. Think about how you would add two infinite decimals. You add finite decimals by starting on the right, but there is no right. You could try starting on the left, but then you continually have to carry and correct the previous digit. Which might have been a 9 and then you have to do it again. And maybe again. If you can prove that this works, then try multiplying. Actually this was solved by an Italian-American mathematician Gian Carlo Rota back in the 70s, I believe.
There are other, better, definitions of the real numbers. Dedekind cuts, mentioned (not by name) above. Cauchy sequences.
One interesting example is the subfield of the algebraic numbers consisting of all numbers that satisfy a quadratic equation with whole number coefficients. These are the geometrically constructible numbers–the ones you can get using straightedge and compass. Showing the impossibility of trisecting an angle or duplicating a cube amounts to showing a cube root of an integer that is not the cube of an integer does not lie in that field. In case you were wondering \sqrt2+\sqrt3 is in the field generated by \sqrt6 and therefore satisfies a quadratic equation.
Also the approach of surreal numbers. In On Numbers and Games, Conway argues that this approach is simpler than any of the standard constructions:
Now if we define R in terms of Dedekind sections in Q, then there are at least four cases in the definition of the product xy according to the signs of x and y… This entails eight cases in the associative law and strictly more in the distributive law. Of course an elegant treatment will manage to discuss several cases at once, but one has to work very hard to find such a treatment… The main advantage of an approach like the present work [ie. using surreals] is that there is just one type of number… I think this makes it the simplest so far, from a purely logical point of view.
The point being that using the surreal approach, all numbers – integers, rationals, reals, as well as objects traditionally viewed as completely different things like infinite numbers, infinitismals, ordinals, etc., are all instances of the same thing, all using the same very simple definition. You don’t have, for example, the duplication of the definition of rationals as you get when using Dedekind cuts.
Surreal numbers are fine, but if you just want real numbers, they go too far.
Here is another definition, due to Emil Artin. Call a function f:N\to Z (N = natural numbers, Z = all integers) nearly constant if the two variable function |f(n+m)-f(n) -f(m)| is bounded over all n and m. And say that such a function is nearly 0 if it is bounded. Then the set of nearly constant functions modulo the equivalence relation of their difference being nearly constant is the reals. This is relatively obviously equivalent to Cauchy sequences