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February 26th, 2019, 05:04 AM   #1
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Why is the set of rational numbers dense, and set of integers numbers not?

If we have two sets:
  1. Set one is the set of rational numbers with the usual less-than ordering
  2. 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?
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February 26th, 2019, 05:10 AM   #2
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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 $\displaystyle (x,y)$.
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Last edited by idontknow; February 26th, 2019 at 05:13 AM.
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February 26th, 2019, 05:24 AM   #3
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Quote:
Originally Posted by idontknow View Post
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?
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February 26th, 2019, 05:55 AM   #4
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Originally Posted by jenniferruurs View Post
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
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February 26th, 2019, 07:12 AM   #5
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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?
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February 26th, 2019, 08:18 AM   #6
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Originally Posted by jenniferruurs View Post
So for the integers an example would be that there is no number between 2 and 3. Is this correct?
Well... yeah!
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February 26th, 2019, 09:26 AM   #7
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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.
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February 26th, 2019, 10:02 AM   #8
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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.
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