Going back in history, mathematics originated as a practical science, as a tool to facilitate administration of harvest, computation of the calendar, collection of taxes, and so on. But even i early Greek society, the study of mathematics had one main goal: the understanding of humankind's purpose in the universe according to a rational scheme. Thus developed a mathematics investigated more in the spirit of understanding rather than only of utility, and this has been a central and successful focus of mathematics since then.

The constant development of new and more sophisticated technologies, in particular, the very fast progress of software and hardware technology, has contributed to a clear change in how mathematics should be studied and taught nowadays, and in how old and new mathematical theories can now be effectively and efficiently applied to the solution of current real-world problems.

Not everybody agrees on what Applied Mathematics means and which subjects or topics it includes. Differential equations may be applied mathematics for some (as it applies notions e.g. from analysis and linear algebra), while for others it is just another subject of pure mathematics, of course with several potential applications. However, we can make a first attempt to list topics:

(Pure) Mathematics: Topology, Abstract Algebra, Analysis, Linear Algebra.

Applied Mathematics: Dynamical Systems, Matrix Computations, Optimization, Financial Mathematics, Numerical Methods.

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