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). Hope this helps. Cheers, Hartmut