0.999...
0.999... is a recurring decimal which is exactly equal to 1. In other words, the symbols 0.999… and 1 represent the same real number. Mathematicians have formulated a number of proofs of this identity, which vary with their level of rigor, preferred development of the real numbers, background assumptions, historical context and target audience. 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. Their 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 in the real numbers, as can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly.
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