Mr Andreas Ryve of Mälardalen University, Västerås and Eskilstuna, Sweden, is the youngest professor in mathematics didactics in the world.
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Pure Mathematics is the branch of mathematics that deals only with mathematics and how it works - it is the HOW of mathematics. It is abstracted from the real world and provides the "tool box" of mathematics; it includes things like calculus. Applied mathematics is the branch of mathematics which applies the techniques of Pure Mathematics to the real world - it is the WHERE of mathematics; it includes things like mechanics. Pure Mathematics teaches you HOW to integrate, Applied mathematics teaches you WHERE to use integration.
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That is one of the unsolved problems of mathematics.