August 20th, 2017, 10:58 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Dedekind's Theory of Irrationals
I have gone through one video lecture about Dedekind's theory of Irrational numbers. I just want to make sure that what I have understood about it is right! Please let me know if I need to know something about it. We divide the whole set of numbers into two classes which is known as a cut/section. Let us say L and R the lower and upper classes respectively. Properties: 1. Each element in the upper class is greater than the elements in the lower class. 2. Each element should belong to either of the classes. 3. If the cut is a rational number, then L had the greatest number and R will not have the least number or 4. If the cut is a rational number, then L doesn't have a greatest number and R has the least number. 5. If the cut is an irrational number then both the classes will not have the greatest & least members respectively. Since the aggregate of the classes are larger than the rational numbers themselves, the additional members are called as irrational numbers. Please go through it and let me know any more points ðŸ˜Š 
August 21st, 2017, 12:14 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Hi For 2 you should note any irrational number cut does not belong to L or R 3 and 4 can be combined by stating L contains all real numbers less than the Dedekind Cut , R contains all real numbers greater than or equal to the Dedekind Cut. If a cut is a rational number then it belongs ro R. Last edited by agentredlum; August 21st, 2017 at 12:25 AM. 
August 21st, 2017, 12:16 AM  #3  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
Thanks for that ðŸ˜Š  
August 21st, 2017, 12:21 AM  #4  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
 
August 21st, 2017, 12:21 AM  #5 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Note my edited post , I forgot to include the word 'cut' between the words 'number' and 'does' 
August 21st, 2017, 12:23 AM  #6  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Quote:
@#$%!!! I fixed it , thank you Last edited by agentredlum; August 21st, 2017 at 12:28 AM.  
August 21st, 2017, 10:40 AM  #7 
Senior Member Joined: Aug 2012 Posts: 2,045 Thanks: 584  Which set of numbers? If you already have the irrational numbers, then what is the point of the exercise? The point is to construct the reals out of the rationals. At the start of the construction you don't have any irrational numbers. 
August 21st, 2017, 07:09 PM  #8  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
At first, there are only Rational numbers and we divide it into two classes, later when we found that there are gaps in between Rational numbers, we have named the additional numbers as Irrationals and together the Reals.  
August 21st, 2017, 07:36 PM  #9  
Senior Member Joined: Aug 2012 Posts: 2,045 Thanks: 584  Quote:
 
August 21st, 2017, 07:41 PM  #10 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 
Thank you. It's nice to get challenges from you That way I can learn more things 

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dedekind, irrationals, theory 
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