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I like to know in depth what really differs Pure Maths from Applied Maths. What are their respective applications? Also when this distinction was made in the history of Mathematics and why?

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closed as not constructive by Listing, Pete L. Clark, Asaf Karagila, Sivaram Ambikasaran, Byron Schmuland Dec 19 '11 at 20:11

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See here for one view. –  Dilip Sarwate Dec 19 '11 at 18:11
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"Pure" mathematicians care about the spectral theorem that says every real symmetric matrix can be diagonalized by an orthogonal matrix. "Applied" mathematicians care about the singular-value decomposition. –  Michael Hardy Dec 19 '11 at 18:26
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Pure mathematics is mathematics for which no one has yet found an application. Yes, this comment is a bit flippant but essentially accurate. Even mathematical topics which seem very abstract and "pure" (such as abstract algebra and category theory) have wide application in physics and computer science. –  ItsNotObvious Dec 19 '11 at 18:48
    
What I have heard that people often categorize trigonometry, and mechanics in applied maths whereas number theory and set theory are classified as pure maths. Why? –  Maxood Dec 19 '11 at 20:09
    
AM and PM are as different as day and night. –  Eric Lippert Dec 19 '11 at 23:57
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As has been hinted at above, the main difference between pure and applied mathematics is not really in what one studies, but in how one studies it. For example, the theory of distributions, as developed by Schwartz can thought of as a branch of functional analysis, and as such one could say that Distribution theory lives in the realm of pure mathematics. However, it is fairly well-known that distributions were used by physicists at least since Heaviside, and so one could think of distributions as mathematical objects used by applied mathematicians. Who's right?

Examples of this sort are abundant. Is knot theory part of algebraic/combinatorial topology? But what happens when Edward Witten takes a shot at it? Even modular forms are being used in string theory! My point is that the difference does not lie in the subject, or in the applications, but mainly in the people.

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I am tempted to write "nothing". And I did...

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