Math intution

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Can anybody tell me tips on how I can develop intution in mathematics. I am Senior. I get great score, but I want to push up limits, i.e. I want to be fast and accurate and also do math questions without any paper.

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  • #2
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In addition to this I would like it if you recommend me on math books(precalculus) that have difficult problems and questions.
 
  • #3
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Unfortunately, there's no 'trick' to it. It's a matter of practice. And that means hard work. Practice what you're doing a lot, and you'll notice it'll come more intuitively in time. Likewise, if you don't do math problems for a significant amount of time, you will likely find that you will still be able to solve many problems, but you'll lose the 'intuition' to do such problems quickly. This merely means that all the tricks that would previously come to mind immediately upon seeing a very specific part of a problem are no longer there, and the key to get them there again is by, you guessed it, practice.

The best book on precalculus I can recommend is Precalculus by James Stewart. Especially because of all the exercises - precalculus theory can very easily be found online. It's the many, many exercises that makes this such a good book (in my humble opinion).
 
  • #4
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I would hit a concept from multiple angles. Draw pictures, make simple cases then gradually increase the complexity, do lots of problems. After you feel comfortable with the concepts try coming up with counterexamples: cases where the theorem doesn't apply.

But the secret, which was taught to me by a great professor, is to experiment with math. Try jumping straight to the problems without reading the section at all and see if you can guess at the solutions. It will develop your intuition and build confidence because if you came close to guessing right then you know by your own ways you're capable of figuring it out. If you guessed wrong then at least you got all the bad guesses out of the way. Don't let the heavy abstract pure math people tell you that math isn't about experimenting and guessing. The professor that taught this method is the pure math professor at my undergrad school and it has helped me immensely over the years.
 
  • #5
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Can anybody tell me tips on how I can develop intution in mathematics. I am Senior. I get great score, but I want to push up limits, i.e. I want to be fast and accurate and also do math questions without any paper.
Speed and computational accuracy shouldn't be your primary goal. There's more to math than that. The most important thing is to always ask why, and try to find a satisfying, intuitive answer if possible, through whatever means necessary (you may consult any number of books, ask people, think hard about it yourself).


In addition to this I would like it if you recommend me on math books(precalculus) that have difficult problems and questions.
I don't know if I would recommend putting more time into precalculus. You have to ask yourself what your goal is. Do you want to be a math major in college, an engineer, or a physicist? You should focus your effort towards whatever will help you the most in the long term, not just precalculus, assuming you are already doing well in it. Not just that but what you find interesting. That may help dictate your choices.
 
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Can anybody tell me tips on how I can develop intution in mathematics.
Practice and experimentation. The more you use a particular sort of mathematics to solve problems, the better you become at using that sort of mathematics to solve problems.

I want to be fast and accurate and also do math questions without any paper.
This is quite different than having an intuitive understanding of mathematics. (or, at most, it's intuition about a small aspect of mathematics)

But anyways, this also involves practice, but possibly of a different sort. It may help to develop your own techniques.
 
  • #7
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So wait, you just want to get better at calculations in your head? Because in my opinion, math intuition concerns more conceptual type things. I was going to recommend some competition style problem solving books for better intuition, but that doesn't sound like what you're actually looking for.
 
  • #8
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I don't know if I would recommend putting more time into precalculus. You have to ask yourself what your goal is. Do you want to be a math major in college, an engineer, or a physicist? You should focus your effort towards whatever will help you the most in the long term, not just precalculus, assuming you are already doing well in it. Not just that but what you find interesting. That may help dictate your choices.
I wish to be a particle physicist (theoretical physics). I believe that I must have a very strong basis in maths so I need to complete some maths in high school. Please recommend me the topics to complete (other than the syllabus) and books with difficult questions.
 
  • #9
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I wish to be a particle physicist (theoretical physics). I believe that I must have a very strong basis in maths so I need to complete some maths in high school. Please recommend me the topics to complete (other than the syllabus) and books with difficult questions.
Well, since you mention building your intuition, I would recommend Lines and Curves: A Practical Geometry Handbook. It will give you practice at understanding things visually. One of the tricks to math is to look at things just the right way so that they become obvious. You have to keep thinking about the proofs and trying to visualize them until they make sense and everything is obvious. Also, there are many exercises in the book. Another one you might try is Geometry and the Imagination.

For physics, you could read Feynman's lectures in physics.

That ought to be enough to keep you busy.
 
  • #10
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homeomorphic, how about books on algebra( linear, abstract...) and I have feynman's lectures but should I study them right now or in my undergrad days?

homeomorphic thanks in advance
 
  • #11
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homeomorphic, how about books on algebra( linear, abstract...)
I didn't really learn basic linear algebra from a book (I took a course that used Strang, but I really learned it later, pretty much in later classes, plus, just deriving everything on my own), so it's hard to recommend anything for that.

I also wouldn't really recommend the books that I learned abstract algebra from, so no recommendations there from me. It's not that they are bad, but I don't think they would be the place to start. You might try reading Symmetry by Hermann Weyl, which I still haven't read, but have been meaning to read. I think it should be a good prelude to studying abstract algebra.

Quote from John Baez:

"Hermann Weyl, Symmetry, Princeton University Press, Princeton, New Jersey, 1983. (Before diving into group theory, find out why it's fun.) "

You could also try Galois theory, by Ian Stewart.

and I have feynman's lectures but should I study them right now or in my undergrad days?
It's good to get an early start if you have the time and inclination. Even if you have to take a class and repeat the material, it will reinforce it and you will learn it better. You could look into whether you can skip introductory physics in college. There are AP tests and various other options, but I am not an expert on those. Don't think of it as a race because haste makes waste, but if you are serious about physics, it pays to start early. I am finishing a PhD in math, now, and I know if I had gotten serious about it in high school, I would be in a better position now. The thing is, in high school, your competition is pretty weak, so you don't really know what you are up against. So, you spend a lot of time thinking maybe everything will just be easy and relaxing. Meanwhile, your competitors may be learning many things and getting ahead. Because I got a late start, I'm busy playing catch-up now.
 
  • #12
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I agree with you homeomorphic, a head start helps in the future. So what parts of math should I cover now? I mean I will take Calculus and nothing else in highschool but what can I add by myself like abstract algebra, which will in the future give me a head start like you pointed?
 
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  • #13
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I agree with you homeomorphic, a head start helps in the future. So what parts of math should I cover now? I mean I will take Calculus and nothing else in highschool but what can I add by myself like abstract algebra, which will in the future give me a head start like you pointed?
I would say don't bite off more than you can chew. Even the stuff I have mentioned so far will probably keep you busy. You have to walk before you can run.

Right now, I would say, rather than trying to learn a lot of material, you are probably better off just trying to understand what math is really about, how to think about it, build intuition.

Some suggestions are How to Solve It by Polya and the Art and Craft of Problem Solving by Paul Zeitz. How to Read and Do Proofs.

Here are two books for algebra that I haven't read, but they sound really good:
A Book of Abstract Algebra by Charles C Pinter and Visual Group Theory by Nathan Carter.

Linear algebra is a good subject that doesn't require calculus prerequisites, but like I said, I don't know what books to recommend for that.

Maybe just focus on one or two books at a time.
 
  • #14
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....Try jumping straight to the problems without reading the section at all and see if you can guess at the solutions. It will develop your intuition and build confidence ....
I often do exactly this. I will even tackle an assignment before even going over the material. I do, however, always have to actually read the material being questioned.
 
  • #15
You might also consider learning one- or two-dimensional cases, then try to generalize them to higher dimensions.
 
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