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March 11th, 2016, 02:34 PM  #1 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Cardinality of the set of decimal numbers
Cardinalities of the set of decimal numbers and ℝ are discussed using denominator lines and rational plane. On the rational plane, a vertical line is referred by its abscissa M. Because the points of a vertical line represent the quotients i/M which have the same denominator M, the vertical line at abscissa M is called denominator line of M. Please read the article at PDF Cardinality of the set of decimal numbers PengKuan on Maths: Cardinality of the set of decimal numbers or Word https://www.academia.edu/23155464/Ca...ecimal_numbers 
March 11th, 2016, 03:42 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,622 Thanks: 2611 Math Focus: Mainly analysis and algebra 
Why "decimal" numbers? Do you realise that decimals are just a representation of numbers?

March 11th, 2016, 04:43 PM  #3  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
People are used to decimal numbers, it is simpler to understand.  
March 12th, 2016, 04:52 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
There is NO such thing as "decimal numbers". There are "decimal numerals" which is a specific way of representing numbers. Decimal or other representation has nothing to do with "cardinality" of the real numbers.

March 12th, 2016, 11:56 AM  #5  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
On cannot get rid of numeral system.  
March 12th, 2016, 04:50 PM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
You still don't get it! Being "rational" or "real" or a "countable" set is completely independent of the system you use to write it. And you don't have to "use a system". You can study numbers in ways that have nothing to do with the way numerals are written.

March 12th, 2016, 05:45 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,622 Thanks: 2611 Math Focus: Mainly analysis and algebra 
I usually write numbers as $a$ or $b$ or $n$ or $x$ or $\phi$ or $\pi$ or $\theta$ or ... In the whole of my university degree I only remember one occurrence of a real number (that was not zero or one) being written in a numerical representation. It came in tensor calculus and the equation was $$\delta_{ii} = 3$$ which is only true in 3dimensional space anyway. 
March 13th, 2016, 04:36 AM  #8 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  The "decimal numbers" in the title is in fact decimal numerals.

March 13th, 2016, 04:38 AM  #9  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
 

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