All principles and methods in mathematical analysis are based on the theory of real numbers and are taken as a starting point. Therefore, it is important to have a correct knowledge for the set of real numbers in order to have a profound knowledge of the principles and methods of analysis. There is a common point between two sets of numbers, called "ordered fields," in which both operations are defined and the size relation between any two numbers holds in the algebraic sense. However, there is an essential difference between the two sets. It cannot be described simply by algebraic...