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Here is a possibility, taken from Tristan Needham, Visual Complex Analysis (Oxford Univ. Press).            The advantages of this theorem are: it is certainly not obvious, "it would require a great deal of ingenuity" to prove this without complex numbers, it is elementary planar geometry, and it is ...


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(comment) Why the need and is there really a pedagogical benefit to a non-standard presentation of complex numbers? This feels more like something that appeals to you, that thus you want to push on students. But without considering if it really benefits them or why it wasn't done before. Or even if the non-standard approach is detrimental. Other than ...


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I understand why you think a "pure" mathematics course should avoid "applications". But applications don't have to be in the "applied mathematics" sense of the term; they can be examples of how studying one area of mathematics helps you understand another, or even just be interested in another. For example: To fully understand things like Cauchy-Schwarz, ...


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I will dodge the "should" part of the question, and try to address the "objective" issues. If you want to teach a class with fewer proofs and more applications,"Serge Lang's book is not the way to go. The math books that are most concerned with "applications" are the ones written by and for engineers. Serge Lang's Linear Algebra book, like all his other ...


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