Real Probability: Rational vs Irrational Numbers

[BillhB]

So from what I've been reading rational numbers are a countable infinity, while the irrationals are an uncountable infinity. So the number of irrational numbers > the number of rational numbers. Irrational numbers can "normal irrationals" or transcendental numbers, or at least that is what I've read. This seems pretty intuitive, a random number would more likely be irrational than rational.

So I was thinking, given a infinite random number generator would a given real be more likely to be represented as either a rational number or as a root of polynomial than transcendental? Or is this comparison impossible since I'd guess that transcendental numbers being a subset of an uncountable infinity are also an uncountable infinity? Or is this not true?

Any information would be great, or where to start reading about set theory.

 

[Deedlit]

The roots of rational polynomials are known as the algebraic numbers; this is a countable set. So, if a random number generator selected a real number uniformly between 0 and 1, say, it would select a transcendental number with probability 100%.

 

[BillhB]

The roots of rational polynomials are known as the algebraic numbers; this is a countable set. So, if a random number generator selected a real number uniformly between 0 and 1, say, it would select a transcendental number with probability 100%.

Just read that on the wikipage that roots of rational polynomials are countable, I missed it during the first reading.

Thanks.

So question answered.

So where's the proper place to start reading about set theory? My background only includes one proof based course on linear algebra, and I'm currently taking a course on ordinary differential equations.

 

[micromass]

Just read that on the wikipage that roots of rational polynomials are countable, I missed it during the first reading.

Thanks.

So question answered.

So where's the proper place to start reading about set theory? My background only includes one proof based course on linear algebra, and I'm currently taking a course on ordinary differential equations.

Get the book by Hrbacek and Jech. PM me if you want more information or help!

 

[BillhB]

Get the book by Hrbacek and Jech. PM me if you want more information or help!

I'll check it out! Much appreciated.

 

[Zafa Pi]

So from what I've been reading rational numbers are a countable infinity, while the irrationals are an uncountable infinity. So the number of irrational numbers > the number of rational numbers. Irrational numbers can "normal irrationals" or transcendental numbers, or at least that is what I've read. This seems pretty intuitive, a random number would more likely be irrational than rational.

So I was thinking, given a infinite random number generator would a given real be more likely to be represented as either a rational number or as a root of polynomial than transcendental? Or is this comparison impossible since I'd guess that transcendental numbers being a subset of an uncountable infinity are also an uncountable infinity? Or is this not true?

Any information would be great, or where to start reading about set theory.

A subset of an uncountable set is not necessarily uncountable, it could have just one element.

 

[FactChecker]

A subset of an uncountable set is not necessarily uncountable, it could have just one element.

But the compliment in the Real numbers of a countable set must be uncountable.

 

[Zafa Pi]

But the compliment in the Real numbers of a countable set must be uncountable.

True enough, but I was referring to BillhB's statement, "I'd guess that transcendental numbers being a subset of an uncountable infinity are also an uncountable infinity? Or is this not true?"

 

[FactChecker]

True enough, but I was referring to BillhB's statement, "I'd guess that transcendental numbers being a subset of an uncountable infinity are also an uncountable infinity? Or is this not true?"

Oh. Sorry. I missed that part.