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12
votes
6answers
584 views

Read old articles instead books.

I'd like to know if there is a site, or maybe a collection of books, where I can read old articles in mathematics in order to study topics directly from the source, instead reading books in the field. ...
10
votes
5answers
998 views

The status of high school geometry

Okay, so we've all seen Euclidean geometry in primary and high school. Back then, I really thought of points as indivisible entities in space and lines as 'breadthless lengths'. As far as I could ...
1
vote
1answer
342 views

Infinite induction “valid”

As you may know, induction works only when we have a statement involving natural numbers. For instance, For every $n$, the intersection of $n$ open sets is open. Now, the corresponding statement for ...
8
votes
2answers
430 views

What should be the next step?

This is a soft/educational question and I'll flag it to be made community wiki. A little bit of background, first. I am in my last undergraduate year, and I took a graduate course in category theory; ...
3
votes
1answer
134 views

summing series using circles inside curves

After watching the infinity elephants video http://www.youtube.com/watch?v=DK5Z709J2eo and seeing how a geometric series could be represented by drawing a circle between a pair of lines, then the ...
49
votes
7answers
2k views

How to explain to the layperson what mathematics is, why it's important, and why it's interesting [closed]

A mathematician walks into a party. No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. ...
40
votes
17answers
4k views

What's the goal of mathematics?

Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what? I've already read G. H. Hardy Apology but I didn't get an answer from it.
11
votes
3answers
2k views

Best way to learn Algebraic Geometry?

I've been reading the book Commutative Algebra with a view towards Algebraic Geometry. I was wondering is the best way to learn algebraic geometry through commutative algebra? As the book I'm ...
4
votes
2answers
633 views

How should I teach a high school student about inverse functions?

Today I tried to teach a high school student about inverse functions. I gave him this problem and defined the parts that he didn't understand: Let $f: \mathbb{R} \to \mathbb{R}$, $x \mapsto ...
7
votes
2answers
130 views

Why is $G(k)$ “more fundamental” than the Hilbert-Waring function $g(k)$?

In the Wikipedia entry for Waring's problem, the section on $G(k)$ starts as: “From the work of Hardy and Littlewood, more fundamental than $g(k)$ turned out to be $G(k)$, which is defined...” There ...
3
votes
3answers
469 views

What are some good examples for suggestive notation?

Motivation: Today I first wondered about and later remembered why the set of all functions from a set $X$ to $Y$ is denoted $Y^X$. They wikipedia page gives the explaination "The latter notation is ...
3
votes
1answer
147 views

Heuristics suggesting a unit interval is uncountable

This is maybe a soft question, I am not sure yet. Anyway, I am delivering a 8 (+ 4 supervisions) hour course on 'basic set theory' for undergraduates : set notation, bijections, functions, ...
9
votes
2answers
709 views

Motivating algebra from quadratic equations

This question gave me pause for thought. We have a quadratic equation $ax^2+bx+c=0$. How much algebra can be motivated from the standard solution. Comments point out that the formula does not apply in ...
12
votes
2answers
621 views

How rare is it that a theorem with published proof turns out to be wrong?

There is a story I read about tiling the plane with convex pentagons. You can read about it in this article on pages 1 and 2. Summary of the story: A guy showed in his doctorate work all classes of ...
25
votes
3answers
905 views

When to skip sections of a book when self-studying?

When you're taking a mathematics class, you usually know exactly what sections of a book you need to know, and you can focus your time on these important sections. However, when studying by myself, ...
4
votes
3answers
209 views

Why $2\sqrt{x} + \sqrt{3}$ can’t be simplified any further?

In How do you simplify the addition of square root, it has been pointed out that it cannot be simplified any further. As obvious as it seems, the following questions seemed need to be addressed: ...
1
vote
0answers
101 views

What is the convention for using results of theorems left as exercise in the text?

I'm working on an exam, and have a solid proof for one of the problems, but it's reliant on a number of theorems left exercises in the textbook which were not assigned as coursework. What, if any ...
9
votes
5answers
707 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
2
votes
2answers
261 views

Where can I get the OLD (1940-1970) high school scholarship question papers?

Today matematics are moving away from understanding basics & core methodology. Level of the difficulties are reducing. Questions are becoming more simpler. Hence I would like to see the difficulty ...
4
votes
1answer
184 views

Found a simpler proof, now how do I know if it's original?

I've found a simpler proof for some identity/theorem, hypothetically speaking, of course ;) How do I know if it hasn't been done before? For important results it's fairly easy to find. By the way, I ...
7
votes
2answers
924 views

Learning roadmap for Class Field Theory and more

I consider to study Class Field Theory and Advance Number Theory by my self this next semester. However, there are many books, lecture notes... I would like to choose the most comprehensive book to ...
6
votes
1answer
231 views

calculus textbook avoiding “nice” numbers: all numbers are decimals with 2 or 3 sig figs

Many years ago, my father had a large number of older used textbooks. I seem to remember a calculus textbook with a somewhat unusual feature, and I am wondering if the description rings a bell with ...
6
votes
3answers
316 views

Foreign undergraduate study possibilities for a student in Southeastern Europe

In the (non-EU) country I live in, the main problem with undergraduate education is that it's awfully constrained. I have only a minimal choice in choosing my courses, I cannot take graduate courses, ...
19
votes
5answers
2k views

Is it worth it to get better at contest math?

I have never done well in math competitions, and am now past the point at which I can participate in them. I am asking if it is worth it to go back and practice such types of problems until I gain ...
8
votes
1answer
580 views

Old French papers which haven't been translated into English

So I've been studying French for a few years now and I've decided that translating an old French math paper into English would be a good exercise to further improve my French competency. I would also ...
4
votes
2answers
192 views

Counting and Ordering of Numbers

Are there differences between 'counting' and 'ordering'? As such, the whole of rational number is countable, or they order-able too? In what cases counting and ordering are same or not?
2
votes
1answer
130 views

Mathematical Results from Counting Points in Lattices

I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the ...
3
votes
2answers
239 views

Introduction to proofs with a fair amount of hand-holding?

Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff ...
5
votes
2answers
461 views

Programs for precocious prodigies

I am the director of my university's mathematics honors program, and we just had an inquiry from the parent of a 15 year old who has already completed most of the math courses for a standard ...
11
votes
5answers
1k views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
8
votes
3answers
591 views

Research in plane geometry or euclidean geometry

I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic ...
1
vote
1answer
195 views

Algebraic Topology in Simple Terms

I just wanted to clarify a few basic concepts in algebraic topology. Suppose one space is my room ($\text{Room} \ A$). Suppose the other space is another room in my house ($\text{Room} \ B$). So ...
5
votes
10answers
4k views

Does a negative number really exist?

Second Update: I see that some answers that reference my image are more closely answering my question. Here is a second image to clarify my point. Take this image representing a checkerboard like ...
12
votes
1answer
627 views

Compact = Closed + Bounded + (?)

In $\mathbb{R}^n$ we know (Heine-Borel Theorem) that a set is compact if and only if it is closed and bounded. In $C(X)$ for a compact metric space $X$, we know (corollary of Ascoli-Arzela Theorem) ...
4
votes
1answer
327 views

Looking for an “arrows-only” intro to category theory

I have often seen it remarked in passing that the "collection of objects" that appears in the standard definition of a category is, strictly speaking, superfluous, and that it is possible to give an ...
1
vote
2answers
248 views

Accessible topics with a background of linear Algebra and Calculus

I have a background of a one year course in linear algebra (covering most of K&H) and two years of calculus (the first year was a one real variable course at the level of Spivak's Calculus book ...
12
votes
2answers
894 views

Math Notes and Knowledge Organization Methodology

I, like many of you I suspect, take copious notes when reading and working through math exercises/theorems/constructions. I have stacks and stacks of notes ranging from one day old to several years. ...
5
votes
2answers
270 views

Group actions in towers of Galois extensions

Assume we are given an extension of number fields or $\mathfrak{p}$-adic number fields $L/E/K$ where each extension is abelian and $L/K$ is only assumed Galois. Now take any element $\sigma\in ...
1
vote
3answers
770 views

Why does higher level mathematics more often than not use Greek lettering?

In high school, at least from what I've seen, mathematics courses never use Greek lettering in their description of concepts, with the notable exceptions of $\Sigma$ for summations, $\Delta$ for ...
23
votes
5answers
741 views

relaxing maths papers

I am looking for some relaxing, funny maths article that one can read when tired. For instance, I found Bouton's paper "Nim, a game with a complete mathematical theory" very funny. If you know ...
6
votes
2answers
394 views

Greens theorem: why does path orientation matter?

$$\oint_{\partial D} P\;dx + Q\;dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\;dA$$ Is it correct to interpret this equation as relating the surface area of ...
5
votes
7answers
2k views

Best Cities for Mathematical Study

This may sound silly, but... Suppose an aspiring amateur mathematician wanted to plan to move to another city... What are some cities that are home to some of the largest number of the brightest ...
7
votes
2answers
246 views

Is this a bad way of approaching math problems?

Teacher assigns a problem. I work on it for thirty minutes, then check the textbook to see if it has similar problems. Often the textbook's problems have hints that I can use to solve the original ...
7
votes
1answer
285 views

Finding 'verbally smallest' element of a finitely generated group

Let $G = ({\Large\ast}^n\mathbb{Z})/K$ be a group, and for each $g \in G$ define $l(g)$ as the smallest positive integer $m$ such that $g = g_1 \ldots g_m$, where each $g_i$ is a generator of $G$. Now ...
7
votes
4answers
211 views

Why are convex sets assumed / defined to be subsets of vector spaces (and not of more general spaces)?

Roughly speaking, my question is: Why are convex sets assumed to be subsets of vector spaces? Well, one might answer that convex sets are defined to be subsets of vector spaces, but this is not the ...
40
votes
6answers
8k views

How Do You Go About Learning Mathematics?

I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a ...
2
votes
1answer
156 views

The reason for different terminologies

Different authors seem to have different conventions when they define the term affine variety (similarly projective variety). For the purposes of this question let us stick with the affine case, and ...
13
votes
2answers
1k views

Who are some forgotten mathematicians? [closed]

In Thomas' Calculus, he presents ''Nicole Oresme's Theorem'': $$ \sum_{n=1}^\infty {n\over 2^{n-1}}=4. $$ My first reaction was "who is this person?''. As it turns out, he was a Frenchman from the ...
3
votes
2answers
132 views

Advice for Calculus Tutoring

I am tutoring a friend in calculus. Right now, she is working on finding relative maxima and minima as well as Rolle's theorem. While she gets how to find relative maxima and minima she does not get ...
22
votes
2answers
552 views

Qual question archives?

Qual questions seem like a great way to study for a new topic, since they usually test slightly deeper understanding than typical questions in a textbook. Princeton has this great archive of questions ...