0.999...
0.999... is a repeating decimal that represents the number 1. The three dots represent an infinite list of "9" digits. Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is 1; that is,
Despite common misconceptions, 0.999... is not "almost exactly 1" 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 intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only elementary arithmetic and the Archimedean property: for each real number, there is a natural number that is greater (for example, by rounding up). Other proofs generally involve basic properties of real numbers and methods of calculus, such as series and limits. Why some people reject this equality is a question studied in mathematics education.
In other number systems, 0.999... can have the same meaning, a different definition, or be undefined. Every non-zero terminating decimal has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all positional numeral systems that represent the real numbers.