A Logical Calculus To Intuitively And Logically Denote Number Systems
  Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, we construct a logical calculus and deduce numbers to intuitively and logically denote number systems. 
Pith Xie