0.999... instance of repeating decimal
In mathematics, 0.999... (also written as 0.9 or 0..9) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence. This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1  –  rather, "0.999..." and "1" represent exactly the same number.
There are many ways of showing this equality, from intuitive arguments to mathematically rigorous 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 other systems, 0.999... can have the same meaning, a different definition, or be undefined.
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation.  For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it.  This has been the subject of several studies in mathematics education.Rigorous proof
The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number 0.999...9, with n nines after the decimal point, is denoted 0.(9)n. Thus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on. As 1/10n = 0.0...01, with n digits after the decimal point, the addition rule for decimal numbers impliesDedekind cuts
In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers less than x. In particular, the real number 1 is the set of all rational numbers that are less than 1. Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the formApplications
One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:1⁄7 = 0.142857 and 142 + 857 = 999.
1⁄73 = 0.01369863 and 0136 + 9863 = 9999.E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If it can be proved that if a decimal of the form 0.b1b2b3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.
type of example of manifestation of representation of