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Study habits are personal. You have to study the best way that suits you. If you feel that you aren't getting anything out of just working problems in the textbooks, then there are other options available. For instance, this site. You can browse through questions that pique your interest and examine the questions being asked and the answers being given. ...


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Doing the examples helps if (i) they teach you something new, or (ii) they train you in applying some technique. You are complaining that you see too many type (ii) questions, and you might be right. What mix of (i) and (ii) is right naturally varies from one to the other, textbooks also want to give teachers a wide selection of problems (or at least ...


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The short answer is a strong NO. Mathematics is a product of sensory experience. Mathematical and philosophical learning has happened over a long period of human evolution that some of the sensory experience has been taken for granted. Hence, on the face of it, it might appear that mathematics is independent of sensory experience. If just the mind existed ...


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I think this is a very interesting question, which is hard to formulate precisely. It's also controversial. You might be interested in Misha Gromov's theory on the "Ergobrain." Here is an attempt at a summery of this idea. Gromov partly attempts to first pose and then answer questions similar to yours. Roughly speaking, Gromov thinks that much of ...


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I think that for math to be a possibility for me, I need the ability to conceive of an object. Symbols take the form of prototypical objects for most of math, in the sense that we can differentiate between two instances of a symbol and judge two instances of a symbol to be the same. Our use of symbols takes advantage of our latent ability to differentiate ...


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A single sample $t$ test is not "against a sample and a population". It is a test looking for evidence against some hypothesis regarding some parameter. For example, you might have the hypothesis that a population mean is $28$. That is, the hypothesis that $\mu=28$. Your sample value of $\bar{x}$ might provide evidence against this. Or your sample might be ...


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Just like in this answer, I would use blocks to visualize divisibility:


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By assumption we have $$a_i=c b_i,\quad i=1,\ldots,k$$ so in the given sum you can factor by $c$.


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I would recall/derive it simply by rewritting: Suppose $c | a_1, \dots, a_k$. Then $a_1 = ca_1', \dots, a_k = ca_k'$ where $a_i' = \frac{a_i}{c}$. Then, $$a_1u_1 + \dots + a_ku_k \\ = ca_1'u_1 + \dots + ca_k'u_k\\ = c(a_1'u_1 + \dots + a_k'u_k)$$ which is clearly divisible by $c$!


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I lighted upon this sterling answer by virtue of user Hepth at http://www.physicsforums.com/showthread.php?t=677222: I too get mental fatigue if I'm working too hard. Usually my problem is if I work on research (read papers/do math/program mathematica) for 8+ hours it takes another 4+ hours for my brain to slow down and I can relax. This leads to problems ...


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Then you just want to pick up a gcse/ A level core maths textbook, and keep going over examples, with things like factorisation, theres not much theory so it really is practice makes perfect.


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Do a mixture of both strategies. If you try to prove every statement in mathematics, then you will move at a snails pace. For instance, can't you just accept Fermat's Last Theorem without fully understanding 200 pages of the most advanced math in the world? On the other hand if you prove nothing, then you are missing out on the fun and not learning very ...


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I am no maths major but every learning field reduces to some basic values. In engineering we are taught dont answer first, find the question first. It sounds philosophical but it isn't. I would like to share what i do myself i first of all try to find the reason i am learning anything/subject.Then i try to make clear that what is my motivation for that ...



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