Thursday, 8 January 2015

elementary number theory - Definition of real by infinite series instead of their Cauchy limits

Looking at Wikipedia´s definition of real numbers I choose a variant one of the alternative definitions, using Cauchy limits. However, Instead of taking a limit I choose the number to be represented by an infinite expansion in a base (here taken as 10), that is an infinite set of numbers 0-9, and I simplicity I take numbers between, and including, 0 and 1. I haven´t used the term "set" but I realize my numbers may be taken to be that.



I now state that ab if 0.a1a2a3a4a50.b1b2b3b4b5 if an=bn for all n.



Furthermore, I have a>b if the first an>bn in a´s and b´s expansion respectively including the first number 0 or 1.
For addition and multiplication I establish the infinite series by starting from the highest decimal just as one might do for final numbers. That is, the generating process is similar to addition and multiplication of infinite series such as Euler first did, solving the “Basel problem” of the sum of (1/n)2. (I understand it was perhaps not considered rigorous but I believe it is when the corresponding series converge).



The rules imply that there is a zero corresponding to the expansion 0.00000 and that




1.00000>0.99999



and



1.000000.99999=0.00000...



This is agreeable since it brings to mind the intuitive (or perhaps naive) idea from the number line, that numbers are actually “touching” each other. Another issue is that I cannot see a process to “get away” from 1.00000 to any “next” or other near number since 1(0.99999)n0 for all n. It therefore brings on the problem that for a not identical to b, I still have ab0.



In Wikipedia´s article on Peano axioms I couldn´t find the sign >, only (except in the section “Equivalent axiomatizations”) but I take it that a-b cannot be 0 if a>b (and that is perhaps how > is defined).




I can introduce “=” or “almost equal” so that a and b are “almost equal” if ab0 but I believe that would take me back to the normal definition of real numbers using Cauchy limits.



If I go down the road where 1.00000>0.99999 all my “new real numbers” would be strictly ordered and I could add and multiply.
Is there any difference (other than the unusual use of <) between this and the real numbers defined in Wikipedia?

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