Yeah, this comes to mind quickly because I've encountered the question of approximation of rational numbers by irrationals, in the form of a lecture about continued fractions I attended 6 years ago.
It's the algebrist in me talking, but quite often, you'll get insight on these sort of things by knowing intuitively the overarching structure of whatever objects you are working with.
So the best thing you can do to get more confident about these, aside of course from practicing on problems, is to widen your understanding of the material by getting interrested in many little things here and there.
It's hard to fully explain what I mean, especially as english isn't my first language, but you have to somehow get "intimate" with the material of your courses, if you want proofs to come naturally to you.>>12130464>(I like exotic spheres, and that we have a one-parameter family of differentiable structures on R^4, but these are only interesting, and not that useful)
When I first heard of it, I got very excited at the idea,and I remember finding this book :https://www.worldscientific.com/worldscibooks/10.1142/4323
Now, a quick research let to that article :https://www.arxiv-vanity.com/papers/hep-th/9411151/
I don't know enough physics (though the sentence "The analysis is based on the A. Connes’ construction of the standard model." from the article did give me a slight hard-on) to figure out how promising it all is, but the idea that they might use exotic differentiable structures in physics is quite exciting, and apparently not entirely dead.