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Author Topic: Are irrational numbers separated by an infinite number of rationals?  (Read 2706 times)

Offline AndroidNeox

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Is this true?

"Between every two rational numbers are an infinite number of irrational numbers. Between every two irrational numbers are an infinite number of rational numbers."
« Last Edit: 10/05/2013 20:17:11 by chris »


 

Offline JP

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Re: Numbers Rational and Irrational
« Reply #1 on: 08/05/2013 01:56:58 »
Yes.
 

Offline graham.d

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Re: Numbers Rational and Irrational
« Reply #2 on: 08/05/2013 10:25:47 »
Furthermore, there are more irrational numbers than rational numbers.
 

Offline evan_au

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Re: Numbers Rational and Irrational
« Reply #3 on: 08/05/2013 12:36:00 »
Quote
1. Between every two rational numbers are an infinite number of irrational numbers.
I agree with this. I understand that Cantor showed that the set of rational numbers (=fractions) is "countable": you can define a 1:1 relationship between all fractions and the whole numbers.
However, the set of irrational numbers is uncountable, ie it is infinitely more than the set of rational numbers.

Quote
2.Between every two irrational numbers are an infinite number of rational numbers.
I don't see how this is consistent with (1). If you pick any two of the infinite set of irrational numbers "between" two "closest" rational numbers, then is this a counterexample?

For how to count by infinities, see: http://en.wikipedia.org/wiki/Aleph_number
 

Offline JP

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Re: Numbers Rational and Irrational
« Reply #4 on: 08/05/2013 15:40:56 »
Quote
2.Between every two irrational numbers are an infinite number of rational numbers.
I don't see how this is consistent with (1). If you pick any two of the infinite set of irrational numbers "between" two "closest" rational numbers, then is this a counterexample?

Pick two irrational numbers that are not equal.  To find a single rational number between them, compute their decimal expansions.  At some digit, you can truncate one of them and generate a number that lies in the interval between them.  Since it has a finite decimal expansion (due to truncation), it's rational.  Now, you can add any 1/N with N a big enough integer to this rational and remain in the interval--and the result will be rational.  Since there's an infinite number of integers that will ensure you remain in the interval, you have found infinitely many rational numbers between these two irrational numbers.
 

Offline evan_au

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Re: Numbers Rational and Irrational
« Reply #5 on: 09/05/2013 13:06:12 »
OK, I guess dealing with infinities still doesn't seem entirely intuitive and rational...
 

Offline graham.d

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I think it can be understood by non-pure-maths-people (most of us) by the idea that there are different sized infinities! The infinity of irrationals is bigger that the infinity of rationals. I'm sure a mathematician can express this more rigorously.
 

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