RE: Antwort: [math-fun] Floor function question
Hartmut writes: << The real numbers are characterised as being the unique totally ordered field, s.t. for every number x there exists a natural number n (= 1+...+1) with n>x. The set of all n with n>x has a unique minimum m =: ceiling(x). floor(x) is then defined in the usual way. A total ordering is necessary (complex numbers), that every number is eventually surpassed by the naturals is necessary as well (surreal numbers).
Wouldn't *any* subfield F of the reals R also be a totally-ordered field satisfying the same condition? --Dan
I forgot: Additionally, the field has to be order complete as well! Then the reals are unique. My math gets rusty. Remark: ceiling can be extended to the surreal numbers in the obvious way and then is an ordinal. I can't see an easy definition of floor in this case, though! Cheers, Hartmut
floor(x) = -ceiling(-x) ? hartmut.holzwart@allianz.de wrote:
I forgot: Additionally, the field has to be order complete as well! Then the reals are unique. My math gets rusty.
Remark: ceiling can be extended to the surreal numbers in the obvious way and then is an ordinal. I can't see an easy definition of floor in this case, though!
Cheers, Hartmut _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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