Mathematical fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules.

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Chess notation

Chess notation is the term for several systems that have developed to record either the moves made in a game of chess or the position of pieces on a chessboard. The earliest systems of notation used lengthy narratives to describe each move; these gradually evolved into terser notation systems. Currently algebraic chess notation is the accepted standard and is widely used. Algebraic notation has several variations. Descriptive chess notation was used in English- and Spanish-language literature until the late 20th century, but is now obsolete. There are some special systems for international correspondence chess. Portable Game Notation is used when working with computer chess programs. Systems also exist for transmission using Morse code over telegraph or radio.

Recording moves

Some move-recording notations are designed mainly for use by human players; others are designed for use by computers.

Notation systems for humans

In recognized competitions, all players are required to record all the moves of both players in order to resolve disputes about whether a player has made an illegal move and what the position should now be. In addition, if there is a time limit rule that requires each player to complete a specified number of moves in a specified time, as there is in most serious competition, an accurate count of the moves must be kept. All chess coaches strongly recommend the recording of one's games so that one can look for improvements in one's play. The algebraic and descriptive notations are also used in books about chess.