All Questions
45
votes
0answers
817 views
+50
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
44
votes
13answers
3k views
Why is the volume of a sphere $\frac{4}{3}\pi r^3$?
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! ...
44
votes
3answers
3k views
$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
44
votes
3answers
2k views
Cute Determinant Question
I stumbled across the following problem and found it cute.
Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant:
...
44
votes
19answers
14k views
Software for drawing geometry diagrams
What software do you use to accurately draw geometry diagrams?
44
votes
4answers
2k views
What are the Axiom of Choice and Axiom of Determinacy?
Would someone please explain:
What does the Axiom of Choice mean, intuitively?
What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice?
as simple ...
44
votes
6answers
2k views
What made you choose your research field?
I was wondering how come modern mathematicians do not seem to discover as many theorems as the older mathematicians seem to have done. Have we reached some kind of saturation limit where all commonly ...
44
votes
6answers
1k views
How far can one get in analysis without leaving $\mathbb{Q}$?
Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
44
votes
6answers
1k views
How Do You Actually Do Your Mathematics?
Better yet, what I'm asking is how do you actually write your mathematics?
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up ...
44
votes
1answer
2k views
How do we know an $ \aleph_1 $ exists at all?
I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have ...
44
votes
3answers
1k views
Why does the Mandelbrot set contain (slightly deformed) copies of itself?
The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.
The shape of this set is ...
43
votes
10answers
4k views
Complex math problem that is easy to solve
I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math ...
43
votes
10answers
5k views
What's a proof that the angles of a triangle add up to 180°?
Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point.
However, now that I'm in university, I'm not ...
43
votes
12answers
3k views
Conjectures that have been disproved with extremely large counterexamples?
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a ...
43
votes
7answers
5k views
What makes $9$ special?
I don't know if this is a well know fact but I have observed that every number no matter how large that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there ...
43
votes
6answers
3k views
Ways to evaluate $\int \sec \theta \, d \theta$
The standard approach for showing $\int \sec \theta \, d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a ...
43
votes
7answers
1k views
What's so “natural” about the base of natural logarithms?
There are so many available bases. Why is the strange number $e$ preferred over all else?
Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
43
votes
17answers
1k views
The Best of Dover Books (a.k.a the best cheap mathematical texts)
Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books ...
43
votes
3answers
1k views
Why is $i! = 0.498015668 - 0.154949828i$?
While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!.
Curiously, Google's ...
43
votes
7answers
1k views
Math and mental fatigue
Just a soft-question that has been bugging me for a long time:
How does one deal with mental fatigue when studying math?
I am interested in Mathematics, but when studying say Galois Theory and ...
43
votes
1answer
810 views
A fleshed-out version of the Noncommutative Geometry proof of the Gauss-Bonnet Theorem?
In Connes's book on noncommutative geometry, he outlines a rather short "algebraic" proof of the Gauss-Bonnet theorem that uses multilinear forms. (Start reading on page 19 of the book) This is given ...
42
votes
7answers
3k views
Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$
I was wondering the following. And I probably know the answer already: NO.
Is there another number with similar properties as $e$. So that the derivative of $\exp(x)$ is the same as the function ...
42
votes
12answers
3k views
All natural numbers are equal.
I saw the following "theorem" and its "proof".
I can't explain well why the argument is wrong. Could you give me clear explanation so that kids can understand.
Theorem: All natural numbers are ...
42
votes
6answers
2k views
Why are all the interesting constants so small?
A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is ...
42
votes
7answers
2k views
Surprise exam paradox?
I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows:
A maths teacher says to the class ...
42
votes
7answers
1k views
What exactly are eigen-things?
Wikipedia defines an eigenvector like this:
An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at most ...
42
votes
8answers
2k views
Intuition in algebra?
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) ...
42
votes
2answers
1k views
Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on
Popular mathematics folklore provides some simple tools
enabling us compactly to describe some truly enormous
numbers. For example, the number $10^{100}$ is commonly
known as a googol,
and a googol
...
42
votes
1answer
2k views
Quadratic reciprocity via generalized Fibonacci numbers?
This is a pet idea of mine which I thought I'd share. Fix a prime $q$ congruent to $1 \bmod 4$ and define a sequence $F_n$ by $F_0 = 0, F_1 = 1$, and
$\displaystyle F_{n+2} = F_{n+1} + \frac{q-1}{4} ...
42
votes
2answers
830 views
+500
Complexity class of comparison of power towers
Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
41
votes
4answers
1k views
What would base $1$ be?
Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be?
Let's say we define Base $1$ to use: $\{0\}$.
Because $10_2$ is equal to $010_2$, ...
41
votes
3answers
1k views
Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).
One can use interpolation to show that ...
41
votes
3answers
1k views
Getting Students to Not Fear Confusion
I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro ...
41
votes
3answers
568 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
...
41
votes
4answers
2k views
Is there a function with a removable discontinuity at every point?
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions ...
41
votes
3answers
736 views
Alice and Bob matrix problem.
Alice and Bob play the following game with an $n*n$ matrix, where $n$ is odd.
Alice fills in one of the entries of the matrix with a real number. then Bob fills one. Then Alice and so on so forth ...
40
votes
15answers
3k views
Why can't you flatten a sphere?
It's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old?
As soon as an explanation starts using terms like "Gaussian ...
40
votes
13answers
2k views
Pseudo Proofs that are intuitively reasonable
What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples ...
40
votes
5answers
2k views
Why is 987654321/123456789 = 8.0000000729?
Many years ago,
I noticed that $987654321/123456789 = 8.0000000729\ldots$.
I sent it in to Martin Gardner at Scientific American
and he published it in his column!!!
My life has gone downhill since ...
40
votes
26answers
2k views
Big List of Fun Math Books
To be on this list the book must satisfy the following conditions:
It doesn't require an enormous amount of background material to understand.
It must be a fun book, either in recreational math (or ...
40
votes
11answers
4k views
Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞?
I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old.
After several more mundane questions he asked his daughter what 1/0 ...
40
votes
14answers
3k views
How to round 0.4999… ? Is it 0 or 1?
If you want to round the repeating decimal 0.4999... to a whole number what is the right answer? Is it 0 or 1?
On the one hand you only look at the first digit when you round numbers which in this ...
40
votes
7answers
2k views
Is it possible to represent every huge number in abbreviated form?
Consider the following expression.
16313107343153908912074032799466965289077771751767944648966669091376847859711382649033004075188224
This is a 98 decimal digit number.
This can be represented as ...
40
votes
6answers
2k views
Why is $\omega$ the smallest $\infty$?
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
40
votes
4answers
2k views
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
40
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
40
votes
8answers
2k views
Are the “proofs by contradiction” weaker than other proofs?
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the ...
40
votes
3answers
712 views
Trying to define $\mathbb{R}^{0.5}$ topologically [duplicate]
A few days ago, I was trying to generalize the defintion of Euclidean spaces by trying to define $\mathbb{R}^{0.5}$.
Question: Is there a metric space $A$ such that $A\times A$ is homeomorphic to ...
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 ...
40
votes
5answers
2k views
In what sense are math axioms true?
Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers.
The kid asks: why?
Well, it's an axiom. It's called commutativity (which is not even true for most groups).
How do I ...
