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The opposite is true. Take the interval [0, 1], "sample a random real number x" from it. P(x is rational) = 0, P(x is irrational) = 1. Informally, the overwhel
by aportnoy 8y ago
The opposite is true. Take the interval [0, 1], "sample a random real number x" from it. P(x is rational) = 0, P(x is irrational) = 1.
Informally, the overwhelming majority of real numbers are irrational.
- mavhc 8y agoWhat's weird is there's always a rational number between any 2 irrational numbers, yet there are way more irrational numbers
- lisper 8y agoAll 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.
- thaumasiotes 8y agoCorollary: 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.
- throwawaymath 8y agoCan 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.
- zeeboo 8y agoDo you agree both sets are countable? If so, any two countable sets can be put into bijection by composing their bijections to the naturals.
- throwawaymath 8y agoOh, right. Yeah I guess that makes sense by definition.
- aportnoy 8y agoAny infinite set is either countable or uncountable.
- throwawaymath 8y agoI'm aware, but that's not what I meant.