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43
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7answers
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What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
43
votes
9answers
6k views

How to effectively study math?

Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble ...
43
votes
8answers
7k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
43
votes
14answers
3k views

Largest “leap-to-generality” in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that ...
43
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 ...
43
votes
3answers
8k views

What is the importance of Calculus in today's Mathematics?

For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
43
votes
8answers
3k views

Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock." There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational. ...
42
votes
15answers
4k views

How do I motivate myself to do math again?

I have been thinking of asking for help for a few months now but posting in a public forum like this is intimidating. Still, I am currently in a university studying mathematics as an undergrad. I took ...
42
votes
10answers
2k views

Have there been efforts to introduce non Greek or Latin alphabets into mathematics?

As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic ...
42
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3answers
1k views
42
votes
3answers
3k views

Why learning modern algebraic geometry is so complicated?

Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the ...
41
votes
26answers
17k views

Best Maths books for non-mathematicians

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
41
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.
41
votes
11answers
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What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
41
votes
14answers
2k views

Surprising Generalizations

I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same ...
41
votes
7answers
4k views

What does it take to get a job at a top 50 math program in the U.S.?

I'm a senior undergrad right at a small liberal arts college right now who is applying to math PhD programs in the U.S. I would like to eventually become a professor at a relatively good university ...
41
votes
10answers
1k views

Why does being holomorphic imply so much about a function?

I haven't yet started my complex analysis course (soon!), but recently (inspired by you guys) I've been looking into holomorphic functions. And wow, they're cool! There's so much stuff that's true ...
40
votes
25answers
7k views

“Negative” versus “Minus”

As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and not "minus $0.8$" to denote $-0.8$? The so called "textbook answer" regarding this question reads: ...
40
votes
13answers
3k views

Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
40
votes
7answers
4k views

How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?

I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by ...
40
votes
6answers
4k views

What should an amateur do with a proof of an open problem?

Assuming that somebody is not an employee of a university, just a math amateur, and makes a proper proof of some well known open math problem, what should he do with it? Publish on the internet for ...
40
votes
5answers
2k views

Why do we negate the imaginary part when conjugating?

For $z=x+iy \in \mathbb C$ we all know the definition for the "conjugate" of $z$, $\bar{z}=x-iy$. Geometrically this is the reflection of $z$ across the $y$ axis. My question is: couldn't we have ...
40
votes
6answers
5k views

What is mathematical research like?

I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail ...
40
votes
7answers
1k views

Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra

I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra. Now, this sort of ...
40
votes
5answers
3k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
39
votes
8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
39
votes
5answers
2k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
39
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7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
39
votes
4answers
2k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
38
votes
17answers
4k views

Nuking the Mosquito — ridiculously complicated ways to achieve very simple results [closed]

Here is a toned down example of what I'm looking for: Integration by solving for the unknown integral of $f(x)=x$: $$\int x \, dx=x^2-\int x \, dx$$ $$2\int x \, dx=x^2$$ $$\int x \, ...
38
votes
15answers
5k views

Do we really need reals?

It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions. ...
38
votes
7answers
6k views

How to tell if I'm good enough for graduate school?

I'm concerned about my level of preparation for graduate school. I have decent grades (3.64 GPA, 4.0 in math), great recommendations lined up, and some OK research experience (no publications), but I ...
38
votes
7answers
8k views

I can't do math?

So after failing another math test today I have realized that I can't do math on my own. For whatever reason when I get to a test I just can't do math without extra resources. I think this is because ...
38
votes
17answers
4k views

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend ...
38
votes
16answers
8k views

Pen, pencils and paper to write math [closed]

The question is inspired by this thread on MO. My father is a mathematician, so I remember seeing very nice pens all around. When I started to do math by myself I've realized how important can be a ...
38
votes
15answers
2k views

Nobody told me that self teaching could be so damaging…

Even though I've been teaching myself math for a couple of years now I only just started (a month ago) at the university. My experience is rather mixed. For starters, I'd like to mention that I'm 21 ...
38
votes
8answers
2k views

Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? ...
38
votes
5answers
6k views

What are the work habits of the best mathematicians? [closed]

Of course I would never compare myself to, and don't expect to be, one of the great mathematicians. However, I am curious as to how my work habits, dedication and passion differs from theirs. How ...
38
votes
6answers
2k views

Has any previously unknown result been proven by an automated theorem prover?

The Wikipedia page on automated theorem proving states: Despite these theoretical limits, in practice, theorem provers can solve many hard problems... However it is not clear whether these 'hard ...
38
votes
2answers
621 views

Effective Research Notes

Note-taking for research is vital to your success as a mathematician. As I look back at some of my handwritten notes, I realized how poor they were. I had thought to myself, "What happened?" I was ...
37
votes
15answers
4k views

Why are real numbers useful?

A question (by a fellow CS student taking a first course in calculus, presumably after the lecture in which continuity was introduced: was as follows. In the real, physical world, we deal with ...
37
votes
7answers
4k views

Open math problems which high school students can understand

I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on ...
37
votes
8answers
3k views

How important is programming for mathematicians?

I'm a freshman student in mathematics, and I'm considering whether or not to take a programming course. How important is programming for mathematicians? Do working mathematicians use programs to aid ...
37
votes
5answers
4k views

Soft Question - Intuition of the meaning of homology groups

I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
37
votes
3answers
687 views

Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
36
votes
10answers
2k views

Mind-blowing mathematics experiments

We've all heard of some mind-blowing phenomena involving the sciences, such as the double-slit experiment. I was wondering if there are similair experiments or phenomena which seem very ...
36
votes
6answers
1k views

Need a result of Euler that is simple enough for a child to understand

Talking to my 8 yr old about "the greatest mathematician of all time", I said it was probably Gauss in my opinion, but that Gauss was not very kind to his kids (for example, forbidding them to go into ...
36
votes
6answers
3k views

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or ...
36
votes
2answers
3k views

What does “communicated by” mean in math papers?

[This question involves mostly math papers, and may be relevant to graduate students learning to write and cite papers, although this is my only justification for this being a math question.] Usually ...
36
votes
6answers
3k views

How hard is the proof of $\pi$ or e being transcendental?

I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...