



New Member


Oct 19, 2009, 03:12 PM


Prove between any two rational numbers, there is a rational number:
Hey,
I'm having some trouble solving this problem:
Prove or disprove the following: between any two rational numbers, there is a rational number.
Cheers,



Expert


Oct 19, 2009, 04:06 PM


Hmm. How about this:
Consider two rational numbers A and B, where B>A. The difference between these number is D = BA. Hence A+D = B. Note that D is a rational number.
Now divide D by a known irrational number that is >1, such as sqrt(2). Call this E, where: E = D/sqrt(2). E is definitely irrational, and positive, and smaller than D. So A < A+E < A+D. Remember A+D = B. So: A < A+E < B, and A+E is irrational. Hence no matter the values of rational numbers A and B, there is always an irrational number that lies between.



Junior Member


Oct 19, 2009, 04:10 PM


Originally Posted by ebaines
Hmm. how about this:
Consider two rational numbers A and B, where B>A. The difference between these number is D = BA. Hence A+D = B. Note that D is a rational number.
Now divide D by a known irrational number that is >1, such as sqrt(2). Call this E, where: E = D/sqrt(2). E is definitely irrational, and positive, and smaller than D. So A < A+E < A+D. Remember A+D = B. So: A < A+E < B, and A+E is irrational. Hence no matter the values of rational numbers A and B, there is always an irrational number that lies between.
Sorry ebaines, I think you misread the question. It asks for the proof of a rational (not irrational) number between two rational numbers.



New Member


Oct 19, 2009, 08:03 PM


Any solutions?



Junior Member


Oct 19, 2009, 09:57 PM


Originally Posted by klbcooldude
Any solutions?
I'm going to work from ebaines's example.
Consider two rational numbers A and B, where B>A. The difference between these number is D = BA. Hence A+D = B. Note that D is a rational number.
Now divide D by a known rational number that is >1, such as 2. Call this E, where: E = D/2. E is definitely rational, and positive, and smaller than D. So A < A+E < A+D. Remember A+D = B. So: A < A+E < B, and A+E is rational. Hence no matter the values of rational numbers A and B, there is always an rational number that lies between.
Basically I took ebaines explanations, and changed irrational to rational values, hope you didn't mind me doing that ebaines :o



Expert


Oct 20, 2009, 05:35 AM


Originally Posted by Nhatkiem
Sorry ebaines, I think you misread the question. it asks for the proof of a rational (not irrational) number between two rational numbers.
Oops  I guess I did misread it! Proving the existence of irrational numbers between any two rational numbers just seemed more... interesting. But as you suggest, changing the sqrt(2) to any rational number > 1 does the trick to prove that that there's always another rational number between any two rational numbers. It's just a short additional step to prove that there are an infinite number of rational numbers  and an infinite number of irrational numbers  between any two rational numbers, no matter how close together they are.


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