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Given that the set of rationals is countable, and that the set of irrationals is uncountable, nearly "all" real numbers are irrational. In our experience, howe

by 3JPLW 12y ago

Given that the set of rationals is countable, and that the set of irrationals is uncountable, nearly "all" real numbers are irrational. In our experience, however, we mostly interact with rational numbers, and so irrational numbers seem to be the "strange" ones. But there are infinitely more of them! Were one able to devise a method to choose a real number at random, it'd be nearly impossible to ever get a rational number. As such, it would indeed be extraordinary to have some non-trivial function transform an irrational number (or two, like e^pi) in an unexpected way into a rational number. I imagine most mathematicians would take a very lopsided bet that they're irrational.

Dove 12y ago
This statement can actually be made stronger. Not only are the vast majority of real numbers irrational, they are transcendental[1] -- indescribable in algebraic terms. The numbers we can actually talk about and name are the countable algebraic numbers -- like the square root of two -- and a few transcendentals we know about. Not many. You could count them on your fingers and toes. The rest of the reals -- and it is most of that vast sea -- are completely inaccessible, unnameable, forever anonymous. [1] http://en.wikipedia.org/wiki/Transcendental_number http://en.wikipedia.org/wiki/Transcendental_number
- jsbgir 12y ago
  The set of transcendental but computable numbers is also countable, however: there is a countable number of algorithms. Therefore, the majority of the reals are uncomputable, like Chaintin's constant (the probability that a random algorithm halts). This can in turn be made stronger by noticing that the definition of Chaitin's constant is finite, and by the same argument, definable real numbers are countable. It follows that the majority of the reals are indefinable.
- bashinator 12y ago
  Seems nicely analogous to "regular matter" vs dark matter and dark energy in the Universe.
gizmo686 12y ago
The problem is that number like e and pi are not random irrational numbers, they are very specific irrational numbers with very specific properties.

by 3JPLW 12y ago

Dove 12y ago

This statement can actually be made stronger. Not only are the vast majority of real numbers irrational, they are transcendental[1] -- indescribable in algebraic terms. The numbers we can actually talk about and name are the countable algebraic numbers -- like the square root of two -- and a few transcendentals we know about. Not many. You could count them on your fingers and toes. The rest of the reals -- and it is most of that vast sea -- are completely inaccessible, unnameable, forever anonymous. [1] http://en.wikipedia.org/wiki/Transcendental_number http://en.wikipedia.org/wiki/Transcendental_number

jsbgir 12y ago
The set of transcendental but computable numbers is also countable, however: there is a countable number of algorithms. Therefore, the majority of the reals are uncomputable, like Chaintin's constant (the probability that a random algorithm halts). This can in turn be made stronger by noticing that the definition of Chaitin's constant is finite, and by the same argument, definable real numbers are countable. It follows that the majority of the reals are indefinable.
bashinator 12y ago
Seems nicely analogous to "regular matter" vs dark matter and dark energy in the Universe.

gizmo686 12y ago

The problem is that number like e and pi are not random irrational numbers, they are very specific irrational numbers with very specific properties.