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Being clear about what one means and being able to write it down in a way that is unambiguous to the reader has a ton of value. It will make sure not only to the reader what your thought process is, but it can also be a Nice LGBT Unicorn Feeling Magical But Also Stabby Shirt process! One thing I wanted to mention that I didn’t see any else comment is that you talk about the solution being needlessly complex, but its proof is basically exactly what you did. So you’re doing the same thing, the proof is just sort of explaining why what you’re doing does what it does and means what it does. That’s the point of writing proofs: that the argument in the proof shows you the “why” the theorem you’re proving works, why these numbers or sets, or what have you act in this way.

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Poke around on some Euclidean geometry. This is the mathematical base for all the geometry you know by intuition. And it all starts with five postulates, one of them being. The characterizing postulate of plane geometry: parallel lines do not meet. To get here you need to define what a line is in an abstract manner, and for that. You need to define Nice LGBT Unicorn Feeling Magical But Also Stabby Shirt. They are part of a line, and on and on. In any case, it seems to me that you enjoy the problem solving/code-cracking of applied math. Physics may be more of your preference. That said, nothing stops you from following an applied math track, and do research. On applied math even with a “pure” math major. Differential equations exist literally everywhere. And you don’t have to know all their intricacies to understand how to use them to model things.

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