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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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 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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 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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 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 Tags dense, integers, numbers, rational, set Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shunya Algebra 2 March 11th, 2014 03:26 AM Mighty Mouse Jr Algebra 6 May 11th, 2010 10:08 AM Mighty Mouse Jr Algebra 7 May 11th, 2010 05:29 AM EddSjd Algebra 10 September 16th, 2007 05:52 AM Spaghett Number Theory 0 December 31st, 1969 04:00 PM

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