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please be aware that this answers stems from personal experience and is meant more to apply when referring to books such as "calculus year one" or other more generic books that are intended to be large pools of practice problems in early college/high school caclulus/precalculus textbooks. Obviously there are books that are intended to be thouroughly read (...


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Overall answer: no. I struggled with the same problem as you for quite a long time and, in hindsight, I think I could have spent my time more wisely. Here are my current general guidelines at the time of this post. They may or may not work for you. The overall philosophy I employ is that exercises are usually there to get you comfortable with the material: ...


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Mathematics is kind of a subject that is fun as well as scary. I always know the concept and do one or two problems per concept. As solving each & every problem is time consuming and is of no use. My advice is practice all the concepts and theorems, and do two or 3 different types of problems which based on same concept. It helps us to know which ...


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i would suggest you read every problem , and in your head if you can see the direction pretty clearly then no need doing that , generally big texts do have repetition , but concise books meant for only problem solving without any theory do try to make sure each problem is unique . as far as second part of your research goes , i believe your approach gives ...


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I would pick up a copy of Baby Rudin and apply what you learn to real life. Does the set of objects in a room belong to the countable set $\Bbb{N}$? Are there examples of a cartesian product in this very room right now? All in all, focus on what's beyond the end goal: Applying this to your topological grad school courses, your research papers, and in ...


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You should find a good book or a good teacher if you want to appreciate the beauty of mathematics. If you personally experienced finding a subject beautiful and interesting then there will be no problem learning it even if you are a beginner. Note that by the terms good book and good teacher that is in accordance to your taste and therefore subjective.


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I'm confused as to how exactly [mathematics] helps me become more intelligent. Well the hope is not to just gain more knowledge (including techniques or tricks), but also to gain a deeper understanding of both logical and mathematical structures. Since you have only graduated from high school, you'll not know much about these. Originally, mathematics was ...


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Set up lines of latitude and longitude on the sphere so that the two points, call them $A$ and $B$, are on the same meridian. Then any motion from $A$ to $B$ can be "tracked" along the meridian by taking a moving point on the meridian that has the same latitude as the point moving from $A$ to $B$. The motion along the meridian (an arc of a great circle) ...


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Use a ball. Note that the less curvy a line is, the straighter it is (and therefore shorter). Then note that cutting a slice through the middle of the ball gets you the straightest line available. Which is a good definition of a great circle. (And as other answerers have pointed out, if the ball is edible then you will be nourishing bodies as well as ...


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There are two ways I tried with students. Case 1. Equator of ball On a plastic ball toy carefully tie a string around any great circle, ( use a smal cellulose tape/tab if needed, to prevent side slippage ,) for exactly one rotation. Make the string taut by pulling in opposite directions. The ball will be compressed, tension in taut string increases. ...


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You can always rotate the sphere so that points A and B are both on the equator. The idea then is you reduce your distance from point B the fastest if you head in the direction of point B, and that direction is along the equator.


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If you connect the two points by a rubber band in the shape of a meandering path on the sphere, it is intuitive the rubber band will snap into a great circle shape. Alternatively you can explain the geodesic as the path a magnetic marble would take if the sphere were a steel ball, and you let the marble roll along the surface of the sphere. It will roll in ...


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I have found it helpful to replace the sphere by an apple and introduce an "internal" observer by placing an ant on the apple. The ant will crawl from point $A$ to point $B$ on the sphere by following the shortest path (the queen can't wait) which is always an arc of great circle. An additional point that students find illuminating is the phenomenon that a ...


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This is an interesting question, thoughtfully asked. I really like @SSS 's answer, and have upvoted it. You are right that mathematics is much more than being able to pass the tests you get in school. One way to learn what mathematics is in life after school is to learn more - but not just what's in the next course If you're curious and determined enough ...


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This is in no way a complete answer, but I want to touch on something really important that you said. The only thing that differentiated an easy problem from a difficult one was the fact that the person solving the problem did not know the trick it required to solve the problem. I've done a lot of competition math, and I've also taken the IIT/JEE exam ...



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