In this paper are introduced the ordinal integers ,the ordinal rational numbers ,the ordinal real numbers ,the ordinal p-adic numbers ,the ordinal complex numbers and the ordinal quaternion numbers .It is also introduced the ordinal characteristic of linearly ordered fields. The final result of this series of papers shall be that the four different techniques of transfinite real numbers ,of non-standard enlargment fields of the real numbers ,of surreal numbers ,of ordinal real numbers give by inductive limit or union the same class of numbers known already as the class No and that would deserve the name the "infinitary totally ordered Newton-Leibniz realm of numbers ".