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If you would like a book giving challenging problems in linear algebra, modern algebra, and real analysis and complex analysis, which really put your mathematical reasoning skills to a good test and training, as well as forcing you to understand how/when certain classical theorems are used, I highly recommend "Berkeley problems in mathematics". It is a ...


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When I want to study a new subject in math, I find myself a good textbook on the subject, buy a new notebook and sit down at the desk. I work my way front to back through the book, taking very descriptive notes in the notebook. This allows me to have the exact balance of algebra and intuition in my notes. I also don't actually write anything down until I ...


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Analysis text of Terence Tao - Analysis I takes the axiomatic approach of first constructing natural numbers using Peano Axioms in a very intuitive manner. Maybe, you can try looking at that to see if you are comfortable with his approach.


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Conway, in On Numbers and Game recommended starting with $\mathbb N = \mathbb Z_{\ge 0}$, then going to $\mathbb Q_{\ge 0}$, then to $\mathbb R_{\ge 0}$, then to $\mathbb R$. The reason is that it reduces the number of cases, as opposed to going doing $\mathbb N \to \mathbb Z \to \mathbb Q \to \mathbb R$, which requires many special cases. You construct ...


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As littleO pointed out in the comments, I think its a great idea to build real analysis starting from an axiomatic description of the real numbers. The reals are decribed uniquely by a surprisingly short list of properties: every complete ordered field is functionally equivalent to $\mathbb{R}$. Two textbooks that follow this approach are Anbar Sengupta's ...


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To some extent, the "proper" answer depends on why you wish to do this. Starting with the axioms of ZF set theory is, in a sense, "minimalist"; however, this theory was developed with an aim to formalize a logical foundation of mathematics, and isn't the most intuitive way to proceed. A more intuitive approach might start with the Peano axioms for the ...


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Do not focus on constructing the real numbers. Just learn their properties, and move into doing analysis. Once you are comfortable with analysis: convergence, integration, derivatives, ect. Go back to the real numbers and see if you can construct them. You appreciate what is being done with Dedekind's construction more if you already done some analysis.


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Landau's beautiful "Foundations of Analysis" gives an excellent account of the standard way of progressing from the natural numbers to the real and complex numbers. There are lots of alternative routes, but you should understand the standard account first.


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My advice is that, if your goal is to study analysis, choose any construction of the real numbers that makes sense to you and move on to actually studying analysis.


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If you want to go from first principles, you can read Principia Mathematica in which it takes hundreds of pages to get to the point where they prove $1+1=2$. However for most people this is going too far. :-) From a practical standpoint, you can understand analysis fine if you start with $\mathbb{Q}$, as many textbooks do.


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It depends on your tastes. People assume that a theoretical approach is more difficult because "students don't like abstraction". I personally found it easier to study "theoretical calculus" than "computational calculus". That is, I like the theory because it is easier! The reason I prefer the abstract theoretical approach is that it simplifies things: you ...


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Rings and homomorphism are part of Abstract Algebra. A good introductory book to study for this might be C. Pinter's A Book on Abstract Algebra. Galois theory is a field of study within Abstract Algebra. Have you learned any Linear Algebra? Linear Algebra and Calculus are, in my opinion, two really standard solid subjects, and it's a good idea to be ...


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The answer to this question is: "How would anyone know it?" - Unless there are some people on MSE that do have a very sensitive extrasensory perception and could predict what happens with you$^{[1]}$ - A better question you could make$^{[2]}$ is "In a course of graph theory, what are the students expected to learn after $2$ months studying $2-3$ hours a ...


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You're asking two different questions: How to ask dumb questions? How to get the approval of your peers? Those two questions have little to do with each other. At most, you should follow the cues of the person speaking, not the cues of the other people listening. The better and the more practiced speakers will handle questions more easily ...


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This is a flowchart, which might help you. (Source)


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Metric spaces are so common place and so diverse that you can't really expect some general trick that will save you simply checking the property for any given particular definition of metric. There are some general tricks, like transforming one metric by a certain concave function always yields a new metric, but really, most often, you simply check the ...


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Have a go at the book Mathematical omnibus. Thirty lectures on classic mathematics by Dmitry Fuchs and Sergei Tabachnikov. This book will engage your mind and point you to very interesting mathematics. Another good book to try out is this one: Vladimir Arnold: Problems for children from 5 to 15.


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Here is the thing which many many people don't get (Understand). You don't need to have a degree in order to be good at something and I really mean this. For example, you would see many people who did undergraduate studies in Computer science or physics and they move on to do masters or even PHD's In mathematics.Most of these guys just self study a lot of ...


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The basic difference between the Mathematics that you already encountered and the higher mathematics taught in colleges will be rigor. It will be more of proofs and intuition and the development of ideas and so on. Teach yourself 1) Real Analysis 2) Abstract Algebra 3) Topology 4) Calculus as much as possible with a special focus on proofs and the ...


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This is partly what university is for: finding out what you are interested in. You are not obligated to take only courses in your major. I highly recommend people who are curious enough to take things well outside their wheelhouse and comfort zone. I believe we should be as well-rounded as possible and should challenge ourselves to take courses that we ...


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As a point of reference, every HW assignment I did in graduate school I typeset with Latex. This wasn't required, but it helped for a number of reasons. First, it is more difficult to draw pictures when you use latex. This makes you think twice before attempting to argue-by-picture... in my opinion, that is a big plus. Second, at my school at least, it ...


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I'd say that if you can finish a MSc or even a PhD in math, then you can learn the basics of LaTeX in one afternoon. Also, it seems you have used LaTeX in other questions on this site. Just keep going and learn whatever you need for the current project. From time to time read a tutorial to get inspired.



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