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February 26th, 2019, 04:04 AM  #1 
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  Why is the set of rational numbers dense, and set of integers numbers not?
If we have two sets:
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, 04:10 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 514 Thanks: 80 
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 $\displaystyle (x,y)$. Last edited by idontknow; February 26th, 2019 at 04:13 AM. 
February 26th, 2019, 04:24 AM  #3 
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  
February 26th, 2019, 04:55 AM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,162 Thanks: 879 Math Focus: Wibbly wobbly timeywimey stuff.  
February 26th, 2019, 06:12 AM  #5 
Newbie Joined: Dec 2018 From: Amsterdam Posts: 26 Thanks: 2  
February 26th, 2019, 07:18 AM  #6 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions  
February 26th, 2019, 08:26 AM  #7 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  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, 09:02 AM  #8 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706 
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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dense, integers, numbers, rational, set 
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