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Approximation, in mathematics, is the use of solutions to a problem when they are not exactly correct.
Approximation is all that applied mathematics can give to the true physical picture, for two reasons. First, in experiments in the physical world, an exact answer can never be obtained: this is due to the limits of accuracy of measuring machines and to human error. Second, it is usually the case that simplifying assumptions are made when the mathematical analysis is undertaken, in the processes of abstraction and application (see applicability of mathematics). Sometimes, only approximate mathematical solutions can be found, as in chaos theory.
The difference this makes from the real world does not really have a harmful effect on science, provided that the scientist always realizes that it is there. If scientists make allowances for the errors which have resulted when they test their theories experimentally (and a large body of mathematics helps them to work out how large the error might be), they will be truly following the scientific method, whereby they will attempt to find closer and closer approximations to the truth. In this way, Newton\'s theory of gravitation was superseded by a better approximation when Einstein formulated the theory of relativity. SMcL |
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