Abstract
There arc two methods to extend the set of rational numbers to the one of real numbers.
One of them is due to Dedekind and another due to Cantor. Their ideas are respectively based on
the concept of "cut" and on the one of "fundamental sequence". Both are complete theories on the
real numbers which give theoretical definitions of fundamental properties of real numbers, such as
the continuity (or completeness). But, unfortunately it seems not easy for students of high schools
to understand their theories without studing the general set theory (and the mathematical logic).
Nevertheless it would be expected at least for all students belonging to any courses of sciences to
study this subject as a part of liberal arts. For this purpose we need to compose a continuous
program on the study of "numbers" without theoretical gaps throughout junior high school, high
school and universitiy. In this paper, we try to evolve the theory on the extension of numbers,
especially the extension from the set of rational numbers to the one of real numbers based on
interminate deeimals and report a lesson practiced in a high school.
