  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.
