My Math Forum Question about Dedekind's paper "Continuity and Irrational Numbers"
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 July 11th, 2017, 02:42 PM #1 Newbie   Joined: Jul 2017 From: Wahkiacus, WA, USA Posts: 2 Thanks: 0 Question about Dedekind's paper "Continuity and Irrational Numbers" Hi all, I've been working through this paper (available at https://www.gutenberg.org/files/21016/21016-pdf.pdf) and am confused about a couple points. I was hoping someone here could clarify things for me. I use his terminology, which might be different from the current terminology. In section 6 (at the bottom of page 10 in the pdf), Dedekind is reducing addition with real numbers to addition with rationals (if I understand him correctly). At one point in the proof, he states that α + β = c2 + p, where α and β are real numbers and c2 and p are rational numbers. But since α and β can be irrational numbers and as far as I understand an irrational number cannot be the sum of two rationals, why isn't this equation invalid? What am I missing? Similarly, a couple lines later he states that α - 1/2p is a number in A1 and β - 1/2p is a number in B2. But if I understand his notation correctly, A1 and B1 are classes in the domain of rational numbers and α - 1/2p and β - 1/2p can be irrational numbers. Again, I can't figure out what I'm missing. Thanks!
July 11th, 2017, 03:25 PM   #2
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 Originally Posted by Yosef At one point in the proof, he states that α + β = c2 + p, where α and β are real numbers and c2 and p are rational numbers. But since α and β can be irrational numbers and as far as I understand an irrational number cannot be the sum of two rationals, why isn't this equation invalid? What am I missing?
Didn't look at the paper but the sum of two irrationals can be rational, which seems to be the case you're describing. What's the context of Dedekind's argument?

 July 11th, 2017, 11:21 PM #3 Senior Member   Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT Dedekind defines a real number to be what he calls a cut (now called a Dedekind cut). A cut is a subset of the rational numbers. In the section you cite, Dedekind is defining the sum of two cuts. This sum also will be a cut and hence will be a subset of the rationals. Leaving out some details, that's why the sum of two reals can be expressed in terms of rationals. Thanks from topsquark and Yosef
 July 12th, 2017, 03:40 AM #4 Banned Camp   Joined: Dec 2012 Posts: 1,028 Thanks: 24 $(\sqrt(2)-1)+(2-\sqrt(2))=?$ Thanks from Yosef
 July 12th, 2017, 10:46 AM #5 Newbie   Joined: Jul 2017 From: Wahkiacus, WA, USA Posts: 2 Thanks: 0 Thanks for the replies! Rereading the section with your points in mind, I think what he's doing is defining the operation of adding two reals as producing a certain cut from two other cuts. Then the equations are only dealing with rational numbers, to show that this newly defined operation doesn't contradict any of the definitions regarding rational numbers.

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