sky15 blog 1 ths is the 3rd post (4)

The equality 0.999… = 1 has long been taught in textbooks, and in the last few decades, researchers of mathematics education have studied the reception of this equation among students, who often vocally reject the equality. The students' reasoning is often based on an expectation that infinitesimal quantities should exist, that arithmetic may be broken, or simply that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object that can reasonably be called "0.999…" is strictly less than 1.