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I am a high school student considering the possibility of one day majoring in pure math. Today I happened to be looking over how to perform inverse trigonometric operations without the use of mathematical software (Precalculus). Once you understand these operations it can be easy to see why each equation comes to each conclusion.

As I look through several examples, each one seems to become more tedious when it could be quickly solved by taking a glance at the unit circle. This personal observation made me wonder if I will be expected to go through things like this as I move to higher levels in math. Will it be looked down on to use mathematical software or not "show my work" for obvious computations?

CaptainAmerica16
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    Higher math bears little semblance to K-12 "math" (which is really just acquiring a collection of tools one uses to do math, and usually in a tedious fashion). The calculus sequence, and to some extent linear algebra, have a decent bit of what you probably think of as "computation", but after that, not so much. In college proof-writing, it's generally assumed you know what you're doing with simpler concepts if you've gotten that far, so I could regularly get away with writing "after some computation, we arrive at..." or some such that leaves so much out it wouldn't have flown in high school. – Kaj Hansen Jul 22 '18 at 00:15
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    To drive the point home, if high school math is 100% computation, then undergrad math is more like 25-30% (will vary depending on your focus). Many problem set write-ups end up looking more like an English assignment from afar, with arguments organized into paragraphs and fewer specific numerals all around. – Kaj Hansen Jul 22 '18 at 00:22
  • Thank you for this. I understand the importance of the learning the basics, which is why I put forth so much effort into learning outside of what my teachers go over in class, but boy if it doesn't get difficult to "stay the course". I've had nearly hour-long classes going over 1 problem. Are university classes anything like that? – CaptainAmerica16 Jul 22 '18 at 00:29
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    In my college math classes, most of the professors required us to demonstrate in an at least semi-rigorous way that we understood a concept and to show enough detail to satisfy them. Take pretty much any example I guess: diagonalize a matrix, show that a function is analytic, calculate a probability, calculate the sum of an infinite series, etc. There were "computations" involved, but the focus wasn't on arithmetic manipulation. We were expected to know basic math and I usually received some mercy if I bungled the arithmetic but understood and demonstrated the process (steps in proof, etc.). – gcbound Jul 22 '18 at 01:25
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    But yet… but yet… In many cases, pencil-and-paper computation can give insights that pure thought can not. Often, I’ve spent an unreasonable amount of time and wasted too much paper on computation, looked at the whole mess, and realized that I should have seen why things were true by merely thinking clearly. Yet I felt sure that I would not have been able to do that clear thought without having spent the time on computation. – Lubin Jul 22 '18 at 02:32
  • @Lubin I can understand where you're coming from with this. I just wish it was more...I don't know... interesting. When I look at higher level math, I feel genuine excitement (like Christmas almost) at being able to understand even the smallest detail and then I come back to my current math classes and it doesn't feel the same. (For example, I studied basic fractals for a bit and it was literally one of the coolest things.) – CaptainAmerica16 Jul 22 '18 at 02:52
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    Indeed, computation can be booooring. There are times, though, when I find a certain mindless pleasure in it. – Lubin Jul 22 '18 at 02:56
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    Lubin makes a good point. I guess my point is more that college becomes less like high school in the sense that there's far less homework full of essentially the same problem with the numbers changed around. Computational ability remains essential, but the emphasis will shift to a theory perspective. That is, proving that formulae and algorithms actually work in the general case instead of just number-crunching through specific cases. Sometimes even discovering such things for yourself :) – Kaj Hansen Jul 22 '18 at 11:00

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