The Validity of "1 = 0.999...", the Concept of Infinity, and the Unconscious Extension of the Concept of Equal Sign

This paper examines the validity of "1 = 0.999..." and the concept of infinity, and points out the unconscious extension of the concept of equal sign. To view "1 = 0.999..." as valid, it is necessary to adopt the concept of infinity. However, infinity must be handled by replacing it with the concept of the finite. The meaning of the equal sign is also extended from "identical", which always held true for finite numbers, to "problem-free to treat as identical". The introduction of the concept of limits by adopting the concept of infinity has greatly developed mathematics. However, performing unconscious manipulation of concepts for that purpose is not logical. One should be conscious of which concepts are being written as preconditions.