0.999.
In mathematics, 0.999... (also written as 0., 0. or 0.(9)) is a notation for the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal is a numeral system, numeral that represents the smallest number no less than every number in the sequence (0.9, 0.99, 0.999, \ldots) ; that is, the supremum of this sequence. This number is equal to1. In other words, "0.999..." is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematical rigor, mathematically rigorous mathematical proof, proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In #Alternative number systems, other systems, 0.999... can have the same meaning, a different definition, or be ...
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Repeating Decimal
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is , whose decimal becomes periodic at the ''second'' digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals. The infinitely repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this decimal representation is called a terminating decimal rather than a repeating decimal, since the zero ...
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