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I am a pure math major (and now econ as well!). My knowledge of applied math is limited to what friends have told me. As the name suggests, it s geared toward applications, such as probability, statistics, stochastic processes, game theory, optimization, graph theory, and the like. That is basically all I know.
Pure math, by definition, is theoretical. If you have little desire to write proofs, then don't major in pure math because that's pretty much all you'll be doing. Minoring in pure math doesn't require you to take algebra and analysis so you'll do few if any proofs in that case.
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