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 February 26th, 2019, 05:04 AM #1 Member   Joined: Dec 2018 From: Amsterdam Posts: 38 Thanks: 2 Why is the set of rational numbers dense, and set of integers numbers not? If we have two sets:Set one is the set of rational numbers with the usual less-than ordering Set two is the set of integers numbers with the usual less-than ordering Why is the set of rational numbers dense, and set of integers numbers not? Density is that for all choices of x and y with x < y there is a ∈ A with x < a < y. Can somebody show me this via an example?
 February 26th, 2019, 05:10 AM #2 Senior Member   Joined: Dec 2015 From: Earth Posts: 820 Thanks: 113 Math Focus: Elementary Math Because two rational numbers are more close to each other than two integer numbers. Simply there are more values of $\displaystyle a$ that can be found in $\displaystyle (x,y)$. Thanks from jenniferruurs Last edited by idontknow; February 26th, 2019 at 05:13 AM.
February 26th, 2019, 05:24 AM   #3
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Quote:
 Originally Posted by idontknow Because two rational numbers are more close to each other than two integer numbers. Simply there are more values of $\displaystyle a$ that can be found in $\displaystyle (x,y)$.
What do you mean with this, can you give an example maybe?

February 26th, 2019, 05:55 AM   #4
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 Originally Posted by jenniferruurs What do you mean with this, can you give an example maybe?
You can always find a (countably) infinite number of rational numbers between any two rational numbers. You can't do that with the integers.

-Dan

February 26th, 2019, 07:12 AM   #5
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 Originally Posted by topsquark You can always find a (countably) infinite number of rational numbers between any two rational numbers. You can't do that with the integers. -Dan
So for the integers an example would be that there is no number between 2 and 3. Is this correct?

February 26th, 2019, 08:18 AM   #6
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Quote:
 Originally Posted by jenniferruurs So for the integers an example would be that there is no number between 2 and 3. Is this correct?
Well... yeah!

February 26th, 2019, 09:26 AM   #7
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Quote:
 Originally Posted by jenniferruurs So for the integers an example would be that there is no number between 2 and 3. Is this correct?
Yes, but it might be more illuminating to say that there are three integers between 3 and 7, but you cannot say that there is a finite number of rational numbers between 1/7 and 1/3.

 February 26th, 2019, 10:02 AM #8 Senior Member   Joined: Aug 2012 Posts: 2,424 Thanks: 759 You can always find a rational between any two rationals. You can't always find an integer between any two. For example there's no third integer between 2 and 3. Thanks from topsquark and romsek

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