My Math Forum Why is the set of rational numbers dense, and set of integers numbers not?

 Applied Math Applied Math Forum

 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: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, 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 $\displaystyle (x,y)$. Thanks from jenniferruurs 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

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, 04:55 AM   #4
Math Team

Joined: May 2013
From: The Astral plane

Posts: 2,162
Thanks: 879

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
 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, 06:12 AM   #5
Newbie

Joined: Dec 2018
From: Amsterdam

Posts: 26
Thanks: 2

Quote:
 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, 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
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, 08:26 AM   #7
Senior Member

Joined: May 2016
From: USA

Posts: 1,310
Thanks: 551

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, 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. Thanks from topsquark and romsek

 Tags dense, integers, numbers, rational, set

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post shunya Algebra 2 March 11th, 2014 02:26 AM Mighty Mouse Jr Algebra 6 May 11th, 2010 09:08 AM Mighty Mouse Jr Algebra 7 May 11th, 2010 04:29 AM EddSjd Algebra 10 September 16th, 2007 04:52 AM Spaghett Number Theory 0 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top