Image:999 Perspective.png|300px|right In mathematics, the repeating decimal 0.999… which may also be written as $0.\backslash bar\{9\}$, $0.\backslash dot\{9\}$ or $0.(9)\backslash ,\backslash !$, denotes a real number equal to the number one. In other words, the notations 0.999… and 1 represent the same number within the real number system. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all integer bases, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the only representation. Even more generally, any positional numeral system contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of fractions and the structure of a simple fractal, the Cantor set. They also occur in a classic investigation of the infinitude of the entire set of real numbers. 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 equality among students, many of whom initially question or reject this equality. Many are persuaded by an appeal to authority from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common errors triggered by counterintuitive behavior of the real numbers; for example that each real number has a unique decimal expansion, that nonzero infinitesimal real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems can be constructed bearing out some of these intuitions, and in some of which the equality is false. Though these number systems are different to the standard real number system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in mathematical analysis.

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