# All Questions

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### 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 ...
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### 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! ...
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### $\int_{-\infty}^{+\infty} e^{-x^2} dx$ 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 ...
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### 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: ...
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### Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 $010_2$, ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...