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From Wikipedia, the free encyclopedia
In mathematics and more precisely in algebraic number theory, modular arithmetic is a set of methods allowing one to solve problems concerning the integers. These methods derive from the study of the remainder obtained after applying a Euclidean division.
Even though its origins may be retraced to antiquity, historians generally associate its birth with the year 1801, date of publication of Carl Friedrich Gauss's book Disquisitiones Arithmeticae (Latin, "Arithmetical Investigations"). His new approach, based on a greater abstraction, helped verify famous conjectures and simplify the proof of important results. Despite number theory being the natural domain of these methods, the consequences of Gauss's ideas spread in other fields such as algebra and geometry.
The 20th century changed the status of modular arithmetic. On the one hand, other methods were necessary to progress in number theory. On the other, the development of many industrial applications forced the creation of algorithms using modular arithmetic techniques. These essentially resolved questions brought up by information theory, a branch now mainly considered as applied mathematics.