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October 25th, 2012, 12:31 PM  #1 
Member Joined: Aug 2012 Posts: 72 Thanks: 0  Rational number
How I prove that beteen every some real numbers there is a rational number?

October 25th, 2012, 12:37 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,641 Thanks: 625  Re: Rational number
Look at the decimal expansions of the two real numbers and find the first decimal place where they differ. Insert a finite decimal expansion in between. 
October 25th, 2012, 02:11 PM  #3 
Member Joined: Aug 2012 Posts: 72 Thanks: 0  Re: Rational number
What if they are irrational?

October 25th, 2012, 08:39 PM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Rational number
We can approach any real number as close as we like by a rational approximation. Suppose a, b are distinct, irrational, real numbers. There is a fixed, finite distance between them but we can make the distance between a and our rational smaller than any fixed distance thereby putting our rational in between a and b. This is probably wrong but heres another idea... suppose 2 distinct, real, irrational numbers a and b give interval [a... b] divide both by a now you have the interval [1... b/a] or [b/a...1] the ordering doesn't matter and b/a is most likely irrational but definitely fixed in value. You can approach 1 by a rational approximation as close as you like and since b/a is a fixed distance from 1 you can always insert a rational between them. This should imply that there is a rational between any 2 distinct irrational real numbers, the size of the interval changes when dividing by 'a' but it is still a finite interval with fixed endpoints and 1 as an endpoint is very convenient to approach. 
October 25th, 2012, 09:04 PM  #5  
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Rational number Quote:
Another approach would be to take two distinct real numbers and multiply both of them by an integer sufficiently large such that we may insert the integer between them:  
October 26th, 2012, 01:16 AM  #6  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Rational number Quote:
 
October 26th, 2012, 03:10 AM  #7 
Member Joined: Aug 2012 Posts: 72 Thanks: 0  Re: Rational number
Another question: How I prove between two every irrational number there is irrational number?

October 26th, 2012, 03:37 AM  #8 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361  Re: Rational number
Difficulty with [color=#40FF40]mathman[/color]'s approach comes when the difference of the two numbers is an integer. In the questions, There must be (at least) two different numbers. For the latest question, multiply both numbers by a factor s.t. the denominators are equal. Between the numerators, there is an irrational number hence the quotient is irrational. 
October 26th, 2012, 04:50 AM  #9 
Member Joined: Aug 2012 Posts: 72 Thanks: 0  Re: Rational number
what is s.t.?

October 26th, 2012, 06:29 AM  #10 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361  Re: Rational number
such that


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