Real Probability: Rational vs Irrational Numbers

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In summary, irrational numbers can be either "normal irrationals" or transcendental numbers. Irrational numbers can "normal irrationals" or transcendental numbers, but I think it's more likely for a given real number to be represented as a rational number than as a root of a polynomial.
  • #1
BillhB
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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.
 
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  • #2
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%.
 
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  • #3
Deedlit said:
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.
 
  • #4
BillhB said:
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!
 
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micromass said:
Get the book by Hrbacek and Jech. PM me if you want more information or help!

I'll check it out! Much appreciated.
 
  • #6
BillhB said:
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.
 
  • #7
Zafa Pi said:
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.
 
  • #8
FactChecker said:
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?"
 
  • #9
Zafa Pi said:
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.
 

Related to Real Probability: Rational vs Irrational Numbers

1. What is the difference between rational and irrational numbers?

Rational numbers are numbers that can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as a ratio and have non-terminating decimal expansions.

2. How are rational and irrational numbers used in probability?

Rational numbers are used in probability to represent the likelihood of an event occurring, while irrational numbers are used to represent the uncertainty or randomness of an event.

3. Can irrational numbers be used to calculate probabilities?

No, irrational numbers cannot be used to calculate probabilities as they cannot be expressed as a ratio. However, they can be used to represent the uncertainty in a probability calculation.

4. Are all irrational numbers real numbers?

Yes, all irrational numbers are real numbers. Real numbers include both rational and irrational numbers.

5. How are rational and irrational numbers related to each other in probability?

Rational and irrational numbers are both used in probability to represent different aspects of chance and uncertainty. They are related in that they both play a role in determining the likelihood of an event occurring.

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