# Tag Info

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I have an analogy that I use with my students, and it is applicable here. Mathematics is like literature. Things like elementary arithmetics (grade school $+$,$-$,$\cdot$, etc. on real numbers) are like the "abc"s. Things like algebraic manipulation (but not Algebra) (e.g. $log_2 (4 x)=9$, solve for $x$) are like words. (A lot of Americans reach only ...

4

When working on my bachelor's I didn't like computations much and what appealed to me about math was the concepts. Senior year that changed. I'm not sure what to attribute it to, but my perspective shifted. Computations, done well, build the intuition that motivates all the nice concepts. The nasty technical stuff of cutting edge research gets sifted, the ...

1

First, I want to say that I think this is an excellent question (both in content and in the manner it was posed). I'm looking forward to reading others' answers, but here's mine. In general, I don't really feel there is a lot to be gained by learning mathematics in this way. As Kevin Carlson said in the comments, you first have to overcome the differences ...

1

That would depend on what you mean by "applied." To give you an example, consider quantitative finance, a field of applied math that studies things like pricing of derivatives/contingent claims, hedging & portfolio risk management, statistical arbitrage, etc. and more academic concepts like completeness of markets, models for evolution of asset prices, ...

1

You aren't really going to get much by reading a third through eight grade math book. If you think there are some aspects of arithmetic you don't really understand, then grab an introductory number theory text. It will be far more interesting to you. I would suggest getting a good calculus book (maybe Apostle Volume I) and really working through it. If ...

-1

There are two reasons for this prejudice Many great mathematicians have produced their greatest works at a young age. For example J.F. Nash received a Nobel prize for actually publishing 3 articles at an age of 25-30. His career after 30 was at best mediocre. Similarly Galois (who unfortunately died at a very joung age) and many others. On the contrary, ...

0

I have compiled a list of undergraduate level university Math textbooks. Hopefully it is of help to you! http://mathtuition88.com/2014/10/19/undergraduate-level-math-book-recommendations/ I would like to suggest the above books (mainly for Pure Mathematics). Ideally, the motivated student is able to self study and obtain the knowledge equivalent to a 4 ...

0

I am a double major in mathematics and economics. Here's what helps: Undergraduate degree in economics: calculus and an upper-level statistics course. If you really want to impress your professors with research, I highly recommend taking multivariate calculus and differential equations. Linear algebra is not necessary, but it will make life a lot easier. ...

2

Something to understand is that although it may all look the same from the outside, each branch of mathematics is quite different. They all have a different feeling, different ways of approaching problems, different levels of rigor. My personal story is that I've always been one of those people who just "gets" math. I'd be able to look at problems and either ...

1

At one time, if not now, Chinn/Steenrod's First Concepts of Topology (in the MAA's New Mathematical Library series, see here) was probably the standard book for what you're looking for. You might also want to look at Mitch Struble's Stretching a Point and Donovan Johnson's Topology. The Rubber-Sheet Geometry, although I think the connections to algebraic ...

1

Apart from suggestions given above, you can have a look at $"From \ Geometry\ to \ Topology"$ by H.Graham Flegg ( http://www.amazon.com/Geometry-Topology-Dover-Books-Mathematics/dp/0486419614 ) ... it contains a lot of topics like Euler Characteristic, idea of Simply Connected Spaces, Homotopy, Colouring of Graphs, Jordan Curve Theorem etc. and that too ...

2

The late great Paul Sally told me a story about his days as a post-doc. He was struggling with his research, and complained to his mentor, "I'm busting my a** and I still can't get a theorem!" The mentor replied that yes, hard work is necessary for good results. In consolation, he replied that "In time, you'll find it's sufficient." Apologies for the ...

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I suggest William S. Massey, A Basic Course in Algebraic Topology, and the Brouwer fixed-point theorem in dimension 2, if it hasn't to be a very original topic.

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What you can do is read good books to revise those topics. Doing some exercises in those books leads to further mastery and also helps increase your confidence. There are too many good books on basic undergraduate algebra and real analysis, for example Cohn's Classic Algebra and Rudin's Principles of Mathematical Analysis. Find one that contains the topics ...

2

Have a look at the articles, including "Making a mathematical Exhibition", on my Popularisation and Teaching web page. The actual exhibition on knots is part of this web page. The intention was mainly to use the idea of knots to present the methods of mathematics to a general audience. See also this article on knots. I have given masterclasses to 13 ...

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Write it down here as some kind of question. But be careful! Read all about posting questions $-$ else it will be closed in a few minutes. Think about it for a while and make no mistake. But of cause, the more obvious importance of your material, the less is the risk to be rejected. If you are obvious wrong, however, a lot of talents will be more than glad ...

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When you say that you're experiencing some difficulties understanding intuitively some elementary things, there are a couple of possibilities: 1 - By "understanding intuitively", you actually mean the point in which you have devoted enough time and effort to certain topic that everything becomes clear and straightforward. That is what understanding ...

4

If you are smart enough to be in number theory, then you are smart enough to understand it. If you are smart enough to choose number theory as your field of study, then you are smart enough to find your way to your goal. I am not just saying that. Considering the amount of time you have been spending studying number theory, your brain has been taking in ...

1

It is often a matter of technique. Once you master the technique you advance easy and fast, but before that it is many long stops. Those who dwells in this first state may also achieve more insight into the next stage? And previous knowledge really means a lot (that's why it take such a long time breaking new grounds).

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Listen! You don't need to ask others for whether you want to continue in math or have potential. Based on what you said, which is thinking about Math in all your spare time, I'd say you could be a genius already, just not fully developed! Take you time and enjoy the process. Many people seem to think they are good at math, but all they are doing is ...

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In my mind, "understanding intuitively" means that you understand something in terms of things you know really well from your past experiences. This means that you need to gather experiences to have a lot of intuition. That is, mathematical experiences to build mathematical intuition, and mathematical intuition to understand mathematics. Other kinds of ...

54

Let me give you a personal story. As a young kid, I was always very strong in math but was pretty hampered by one of the worst educational environments in the USA. I ended up entering a magnet school for junior high and had to take a math placement exam to determine which of three math classes I would join: regular math, pre-algebra and algebra. I didn't do ...

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Think of mathematics as a language. Some people are native speakers or have a gift for speech, some have to work at it, some people stutter. Replace "math" with "French" in your question above. It will get easier the more you do it, even if you don't become a fluent speaker.

3

Take a look at A Combinatorial Introduction to Topology by Michael Henle. The point-set topology is done gently, he uses combinatorial methods to bypass some otherwise complicated proofs (to get the Brouwer fixed point theorem, for example), and he uses what is essentially mod 2 homology (if I remember right) to prove some other results, like the Jordan ...

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When I learned number theory, I found that I had no intuition for anything about the proofs, where my classmates seemed to pull the things from thin air; in math, there's always that little (or not-so-little) brilliant leap required for the proof. When I just started learning number theory, I had no idea how people were figuring these things out - even ...

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I don't think the time it takes to learn a concept is necessarily an indication of ones intelligence. Some people are quick learners and some aren't. Of course there are advantages to learning quickly; however, perhaps when you do finally understand a concept, you understand it better than those quick learners! With this in mind, some people just aren't so ...

0

Many things in calculus can be derived easily just by using the approximation $$f(x + \Delta x) \approx f(x) + f'(x) \Delta x.$$ For example, the chain rule. Let $h(x) = f(g(x))$. Then \begin{align} h(x + \Delta x) &= f(g(x + \Delta x)) \\ &\approx f(g(x) + g'(x) \Delta x) \\ &\approx f(g(x)) + ...

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If you want to do well on the Putnam, I think you'd do well to look over the books that Graphth suggested. However, to dismiss real analysis, abstract algebra, and topology, could be a pretty grave mistake- there are often at least 2-3 questions that cover those topics. Fortunately, such questions really only require basic knowledge of basic principles in ...

0

Well I love algebra, and I read Gallian anf Fraleigh as introductory books, which both I love, I won't replace them. But I guess starting your mathematical dreams in college, $\textbf{Elementary Number Theory}$ by $\textbf{BURTON}$ must be read by everyone and also should be taught in the very $1$st year of college, as it is so Inspirational, with so ...

2

At Dal's express request in a comment to this post I'll answer and elect Lang's Undergraduate Analysis. The book is elementary and nevertheless contains non-trivial and not so often taught topics like the Fejér kernel, the heat operator or Schwartz spaces. But the main attraction to me is the relentless geometric point of view: there are many fine, simple ...

1

In fact, you can say nothing if the contours are not real, and you mean "integration with respect to dz". Take for example the function $f(z) = 1$, and let $I_1$ and $I_2$ be the semi-circle and circle, parametrized by $z(t) = e^{it}$, where $0 \le t \le \pi$ and $0 \le t \le 2\pi$ respectively. Then $$\int_{I_1} 1\,dz = \int_0^\pi ie^{it}\,dt = -2$$ and ...

2

The probabilistic method. $~~~$

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Consider the integrand $f(z)=(\delta(z-z_1)+\delta(z-z_2))g(z)$ If the integration contour of $I_2$ contains $z_1$ and $z_2$. The integration contour of $I_1$ only $z_1$ then $I_2=g(z_1)+g(z_2)$ while $I_1=g(z_1)$ which could be both arbitrary. But if $g$ is real and $\forall z: g(z)\ge0$ you can see that $I_1\leq I_2$.

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I'd be quite careful about your background assumptions. Very many undergraduates, at least in the United States, don't learn the definition of a manifold during their degree-so your assumptions are only accurate if they've all taken an advanced geometry course. As for advice, a first idea is to talk about the theorema egregium. The independence of a ...

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It's not a book, but I found Timothy Gowers' address "The Importance of Mathematics" quite inspiring as a graduate student. I wish I had watched it earlier in my career.

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This question has been answered in comments: You should become more comfortable with linear algebra. The rest of abstract algebra is less necessary. – Qiaochu Yuan Jul 19 '13 at 9:56 and Some Linear Algebra texts: Anton; Strang; Noble & Daniel. – Gerry Myerson Jul 19 '13 at 10:06

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