For questions that don't admit a definitive answer. Please do not ask too many of these.
206
votes
33answers
17k views
Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
16
votes
2answers
1k views
What does closed form solution usually mean?
This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form ...
47
votes
24answers
10k views
Best book ever on Number Theory
Which is the single best book for Number Theory that everyone who loves Mathematics should read?
62
votes
7answers
5k views
Tablet for reading textbooks and writing math by hand?
I am a math student. I'd like to find out if tablets (iPads, Galaxy Note 10.1) are worth the cost.
How good are tablets for the purposes of reading textbooks as PDF and writing mathematics with a ...
189
votes
21answers
17k views
How to study math to really understand it and have a healthy lifestyle with free time?
Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical ...
115
votes
28answers
8k views
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
11
votes
7answers
2k views
Choosing a topology text
Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
30
votes
3answers
2k views
Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
81
votes
22answers
5k views
List of Interesting Math Videos/ Documentaries
This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list.
Story of ...
115
votes
20answers
5k views
Why do mathematicians use single-letter variables?
I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated trying to follow mathematical notation. ...
34
votes
6answers
3k views
How do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
14
votes
3answers
620 views
Can spectrum “specify” an operator?
Given a bounded operator $A$ on a Banach space $X$, one may find the spectrum $\sigma(A)\subset{\bf C}$.
Here are my questions:
Given some set in the complex plane, say, $S\subset{\bf C}$, ...
30
votes
9answers
3k 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 ...
30
votes
19answers
5k views
Good book/lecture notes about category theory
what are the best books or lecture notes on category theory?
33
votes
5answers
3k views
Getting better at proofs
So, I don't like proofs.
To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert.
Oftentimes the proof in the book is something that I ...
29
votes
12answers
3k views
How to effectively and efficiently learn mathematics
How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours?
(Apologies in advance if the question is ill-posed or too subjective in its current ...
36
votes
13answers
1k 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 ...
5
votes
4answers
423 views
Expanding problem solving skill
I have a great passion for Math but my lack in problem solving skill always keeps me away from the "good stuff". I always wanted to be better at Math and one of the things I figured out was to keep ...
1
vote
2answers
293 views
Advantage of accepting non-measurable sets
What would be the advantage of accepting non-measurable sets?
I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
377
votes
25answers
24k views
Can I use my powers for good?
I hesitate to ask this question, but I read a lot of the career advice from mathOverflow and math.stackexchange, and I couldn't find anything similar.
Four years after the PhD, I am pretty sure that ...
32
votes
7answers
4k views
Advantages of IMO students in Mathematical Research
Everyone in this community i think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries ...
63
votes
7answers
2k views
Why do people use “it is easy to prove”?
Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
25
votes
17answers
3k views
Examples of mathematical induction
What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
17
votes
5answers
1k 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 ...
12
votes
1answer
524 views
Proofs given in undergrad degree that need Continuum hypothesis?
Or alternative you need to assume CH is false.
I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
7
votes
4answers
656 views
What is the name of the vertical bar?
I've always wanted to know what the name of the vertical bar in these examples was:
$f(x)=x^2+1\mid_{x = 4}$ (I know this means evaluate x at 4)
$\int_0^4 (x^2+1) dx = (\frac{x^3}{3}+x+c) ...
39
votes
20answers
2k views
Your favourite maths puzzles
Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly...
We all love a good puzzle
To a certain extent, any piece of mathematics is a ...
40
votes
2answers
3k views
Why study Algebraic Geometry?
I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are ...
29
votes
6answers
3k 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 ...
32
votes
5answers
7k views
What is the importance of eigenvalues/eigenvectors?
What is the importance of eigenvalues/eigenvectors?
29
votes
4answers
2k views
Why do we care about dual spaces?
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
26
votes
6answers
991 views
What are some good math specific study habits?
What are/ were some of your good mathematician's study habits that you found really worked for you? I'm a CS major at a respected school and have a solid GPA... However, I definitely lack when it ...
59
votes
16answers
6k views
How do you explain the concept of logarithm to a five year old?
Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
16
votes
5answers
1k views
Chatting about mathematics (with real-time LaTeX rendering)
Do you know about some tools which can be used for online chat about mathematics? In particular, I am interested in software which would be able to render LaTeX formulas. (Since LaTeX is probably the ...
20
votes
15answers
2k views
Useful examples of pathological functions
What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples?
For instance, if $f'(a) = b$, then $f(a)$ ...
14
votes
8answers
867 views
Where to begin with foundations of mathematics
I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
33
votes
8answers
1k views
What makes elementary functions elementary?
Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
30
votes
11answers
2k views
Paradox: increasing sequence that goes to $0$?
It is $10$ o'clock, and I have a box. Inside the box is a ball marked $1$.
At $10$:$30$, I will remove the ball marked $1$, and add two balls, labeled $2$ and $3$. At $10$:$45$, I will remove the ...
12
votes
3answers
915 views
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus says the following:
Theorem. $f$ is the derivative of $F$ at every point on $[a,b]$, then under suitable hypotheses we have that $$\int_{a}^{b} f(t) \ dt = ...
8
votes
5answers
2k views
Companions to Rudin?
I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I ...
30
votes
5answers
2k 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 ...
16
votes
7answers
2k views
How can I learn to “read maths” at a University level?
When I look at math, it's like my mind goes fuzzy.
The only way to describe it is in terms of what I can relate it to. You know how when you read, you see the letters and words, but your brain picks ...
25
votes
7answers
926 views
Must we use induction to prove a statement for all integers
This question is prompted by a remark from Bill Dubuque in his answer to this
question on proving a particular sum without using mathematical induction.
From Bill's answer:
A proof that a ...
18
votes
3answers
875 views
Math without infinity
Does math require a concept of infinity?
For instance if I wanted to take the limit of $f(x)$ as $x \rightarrow \infty$, I could use the substitution $x=1/y$ and take the limit as $y\rightarrow 0^+$.
...
17
votes
4answers
550 views
A question about the definition of a neighborhood in topology
Let $X$ be a topological space, and $x \in X$ be a point. There are two prevalent conventions on how to define a neighborhood of $x$:
1) A neighborhood of $x$ is any open subset $W \subset X$ such ...
3
votes
5answers
359 views
How to interpret material conditional and explain it to freshmen?
After studying mathematics for some time, I am still confused.
The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
4
votes
1answer
316 views
Motivation for the term “separable” in topology
A topological space is called separable if contains a countable dense subset. This is a standard terminology, but I find it hard to associate the term to its definition. What is the motivation for ...
2
votes
10answers
2k 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 ...
2
votes
5answers
664 views
Where can I find a review of discrete math
I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
14
votes
8answers
3k views
Why does “convex function” mean “concave *up*”?
A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$,
$$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$
Equivalently, a line ...
