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284
votes
35answers
31k 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 ...
101
votes
29answers
31k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
27
votes
10answers
33k views

Why is Infinity multiplied by Zero not an easy Zero answer?

I did a bit of math at school and it seems like an easy one - what am I missing? $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$ $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ ...
74
votes
4answers
7k 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 ...
356
votes
25answers
67k 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 ...
21
votes
2answers
6k 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 ...
66
votes
6answers
34k views

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
65
votes
21answers
14k views
43
votes
17answers
21k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
483
votes
144answers
30k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
705
votes
28answers
47k 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 ...
13
votes
7answers
3k views

Choosing a text for a First Course in Topology

Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
89
votes
7answers
12k 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 ...
45
votes
4answers
5k 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 ...
29
votes
3answers
895 views

Create a Huge Problem

I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at ...
12
votes
8answers
18k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
506
votes
46answers
312k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
162
votes
62answers
34k views

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved ...
190
votes
30answers
14k 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 ...
130
votes
27answers
18k views

Best Fake Proofs? (A M.SE April Fools Day collection) [closed]

In honor of April Fools Day 2013, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a ...
195
votes
20answers
6k views

Why don't we define “imaginary” numbers for every “impossibility”?

Before the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had, so we said $i$ to be a new type of number. How come we don't do the ...
151
votes
22answers
8k 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. ...
89
votes
9answers
20k views

What is the importance of the Collatz conjecture?

I have been fascinated by this problem since I first heard about it in high school. From the Wikipedia article http://en.wikipedia.org/wiki/Collatz_problem: Take any natural number $n$. If $n$ is ...
53
votes
9answers
9k 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 ...
54
votes
9answers
4k 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, ...
7
votes
4answers
1k 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 ...
171
votes
87answers
14k views

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
136
votes
28answers
19k 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 ...
42
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 ...
45
votes
14answers
9k 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 ...
23
votes
9answers
2k 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
18answers
6k 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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
22
votes
3answers
2k 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 $y\rightarrow 0^+$. ...
6
votes
6answers
2k 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 ...
15
votes
6answers
8k views

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to ...
6
votes
9answers
999 views

what is the definition of Mathematics ?

we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically but the question is, what is mathematics ? how can we ...
9
votes
5answers
323 views

Constructing a degree 4 rational polynomial satisfying $f(\sqrt{2}+\sqrt{3}) = 0$

Goal: Find $f \in \mathbb{Q}[x]$ such that $f(\sqrt{2}+\sqrt{3}) = 0$. A direct approach is to look at the following $$ \begin{align} (\sqrt{2}+\sqrt{3})^2 &= 5+2\sqrt{6} \\ ...
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 ...
164
votes
28answers
18k views

Too old to start math

I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some ...
95
votes
9answers
8k views

How do I sell out with abstract algebra?

My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into ...
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 ...
68
votes
16answers
11k 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?
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. ...
44
votes
10answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
36
votes
10answers
4k 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 ...
26
votes
3answers
2k views

Subjects studied in number theory

I know little about number theory even after reading its Wikipedia article. I was wondering if the subjects studied by number theory is only about integers or even more restrictedly natural numbers ...
28
votes
2answers
2k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
18
votes
4answers
725 views

Advice for writing good mathematics?

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical ...
13
votes
2answers
863 views

Theorems with the greatest impact on group theory as a whole

In his Contemporary Abstract Algebra text, Gallian asserts that Sylow's Theorem(s) and Lagrange's Theorem are the two most important results in finite group theory. He also provides this quote by ...
16
votes
3answers
801 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, $S\subset{\bf C}$, ...