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In mathematics, 0.999… is a real number denoted by a recurring decimal, which is also written as , or 0.(9). Notably, this number is exactly equal to 1. In other words, "0.999…" represents the same number as the symbol "1". Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.
The equality has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of mathematics education have studied the reception of this equation among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number, an expectation that infinitesimal quantities should exist, that arithmetic may be broken, an inability to understand limits or simply the belief 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 is even a system in which an object that can reasonably be called "0.999…" is strictly less than 1.
