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The replacement of mathematical objects by others which resemble them in certain respects. It is possible to study in this way numerical characteristics and qualitative properties of an object by reducing the problem to the study of simpler or more convenient objects (e.g. of objects with readily calculable characteristics or with properties which are known). Number theory deals with Diophantine approximations, in particular with approximations of irrational numbers by rational numbers. Approximations of curves, surfaces, spaces and mappings are studied in geometry and in topology. Certain branches of mathematics deal almost exclusively with approximations — e.g. the theory of approximation of functions and the theory of numerical methods in analysis.