- published: 28 Apr 2011
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In mathematics, the repeating decimal 0.999… (sometimes written with more or fewer 9s before the final ellipsis, for example as 0.9…, or in a variety of other variants such as 0.9, 0.(9), or ) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999…" and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.
Every nonzero, terminating decimal (with infinitely many trailing 0s) has an equal twin representation with infinitely many trailing 9s (for example, 8.32 and 8.31999…). The terminating decimal representation is usually preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases (with a given base's largest digit) or in any similar representation of the real numbers.
The equality of 0.999… and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999… makes it equal to 1, but in some of these number systems, the symbol "0.999…" admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.
A symbol is a person or a concept that represents, stands for or suggests another idea, visual image, belief, action or material entity. Symbols take the form of words, sounds, gestures, ideas or visual images and are used to convey other ideas and beliefs. For example, a red octagon may be a symbol for "STOP". On a map, a blue line might represent a river. Numerals are symbols for numbers. Alphabetic letters may be symbols for sounds. Personal names are symbols representing individuals. A red rose may symbolize love and compassion. The variable x in a mathematical equation may symbolize the position of a particle in space.
In cartography, an organized collection of symbols forms a legend for a map.
The word derives from the Greek symbolon (σύμβολον) meaning token or watchword. It is an amalgam of syn- "together" + bole "a throwing, a casting, the stroke of a missile, bolt, beam." The sense evolution in Greek is from "throwing things together" to "contrasting" to "comparing" to "token used in comparisons to determine if something is genuine." Hence, "outward sign" of something. The meaning "something which stands for something else" was first recorded in 1590, in Edmund Spenser's Faerie Queene.
In computing, a directory is a file system cataloging structure which contains references to other computer files, and possibly other directories. On many computers, directories are known as folders, catalogs (catalog
was used on the Apple II, the Commodore 128 and some other early home computers as a command for displaying disk contents; the filesystems used by these did not support hierarchal directories), or drawers to provide some relevancy to a workbench or the traditional office file cabinet.
Files are organized by storing related files in the same directory. In a hierarchical filesystem (that is, one in which files and directories are organized in a manner that resembles a tree), a directory contained inside another directory is called a subdirectory. The terms parent and child are often used to describe the relationship between a subdirectory and the directory in which it is cataloged, the latter being the parent. The top-most directory in such a filesystem, which does not have a parent of its own, is called the root directory.