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Today the term ''applied mathematics'' is often used in a much broader sense. Mathematicians generally consider it incorrect to conflate applied mathematics (a subset of mathematics) with applications of mathematics (the actual act of applying mathematics to real-world problems). But scientists and social scientists who utilize mathematics in their work do not usually make this distinction.
An example may serve to sharpen the somewhat fuzzy lines separating traditional ''applied mathematics'' from [[pure mathematics|''pure mathematics'']] and distinguishing both of these from ''applicable mathematics'', or the use of mathematics as a tool. A "smooth" function, such as cos(''x''), can be represented by a [[Taylor series]] containing a countably infinite number of terms. ''Pure mathematics'' is concerned with the problem of proving that the Taylor series exists, and with the closely associated problems of determining its coefficients and its circle of convergence, or the ___domain in which it is valid. ''Applied mathematics'' addresses the more practical problems of how the series may best be calculated; how many terms must be included to achieve a desired level of precision; and how best to tabulate the resulting values, or perhaps encapsulate the Taylor series within a computer algorithm. Finally, a surveyor who consults the tabulated or computerized values of cos(''x'') while making trigonometric calculations is not really ''doing'' mathematics – he is simply using results the mathematicians have derived to complete a surveying project.
Some branches of mathematics – [[differential equations]] ([[Ordinary differential equation|ODE]]s and [[Partial differential equation|PDE]]s), [[matrix theory]], [[continuous modelling]], [[probability]], and [[statistics]] – are widely applicable to many fields of science and technology. Others – such as [[numerical analysis]], [[scientific computing]], [[information theory]], [[cryptography]], [[graph theory]] as applied to [[network theory|network analysis]], and theoretical [[computer science]] – have fueled the rapid proliferation of digital computers. Problems associated with computer technology have, in their turn, provided the motivation for mathematical advances in all these fields. And the increasing power and speed of the computers themselves have opened new possibilities in [[computational topology]] and [[computational geometry]].
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