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Comment by mavhc | Hacker News Reader

What's weird is there's always a rational number between any 2 irrational numbers, yet there are way more irrational numbers

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All this becomes pretty intuitive if you think about numbers in terms of their decimal expansions. A "random number" is one with a random digit in each of its (infinitely long) list of decimal digits. A rational number is one where, at some finite point, all of the digits start to repeat in some finite pattern. The odds of that happening by chance are zero.

Likewise, if you take any two randomly generated list of decimal digits, at some point there will be a digit in the same place that is different between the two. At that point, you can construct a rational between the two by choosing the smaller digit and then adding "1000....".

But yeah, it's kind of weird.

Corollary: rational numbers must serve double duty separating many different pairs of irrational numbers.

You can keep going with things that sound weird:

There are infinitely many rationals between any two irrational numbers.

There are as many rational numbers between any two finite irrational numbers as there are between positive and negative infinity.

Can you clarify that last point? I don't think it's strictly true. Aside from countable and uncountable infinities, you can have larger and smaller infinites as well. Unless every set S of all rational numbers between any irrational x and irrational y is isomorphic to the set P of all rational numbers, I don't see that this is correct. And I don't immediately see that you can put them into 1-1 correspondence.

Do you agree both sets are countable? If so, any two countable sets can be put into bijection by composing their bijections to the naturals.

Oh, right. Yeah I guess that makes sense by definition.

Any infinite set is either countable or uncountable.

I'm aware, but that's not what I meant.

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