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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: 7,660 Thanks: 2635 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
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 Originally Posted by v8archie Why "decimal" numbers? Do you realise that decimals are just a representation of numbers?
I have to compare the denominator line argument with the diagonal argument and powerset of N. As binary system is used by powerset of N, it lefts decimal system for explaining the diagonal argument.

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. Thanks from 123qwerty
March 12th, 2016, 11:56 AM   #5
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 Originally Posted by Country Boy 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.
When studying numbers, we have to write them using a system. Decimal system is used by Cantor in diagonal argument, Binary system is used by him in power set of N proof.

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. Thanks from topsquark
 March 12th, 2016, 05:45 PM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,660 Thanks: 2635 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 3-dimensional space anyway.
March 13th, 2016, 04:36 AM   #8
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Quote:
 Originally Posted by Country Boy 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.
The "decimal numbers" in the title is in fact decimal numerals.

March 13th, 2016, 04:38 AM   #9
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Quote:
 Originally Posted by v8archie 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 3-dimensional space anyway.
Cantor used numeral system to prove uncountability in his diagonal argument and power set of N.

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