Top new questions this week:

In plain language, what's the significance of a field?

I just started Linear Algebra. Yesterday, I read about the ten properties of fields. As far as I can tell a field is a mathematical system that we can use to do common arithmetic. Is that correct?

(linear-algebra) (field-theory)

asked by Hal 27 votes answered by Jean-Claude Arbaut 28 votes

Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent?

As I was just checking this 'child prodigy' out on Youtube, I stumbled upon this video, in which Glenn Beck asks the kid to do the following proof: Further on, the kid starts sketching a proof …

(sequences-and-series) (divergent-series)

asked by Victor 26 votes answered by Peter 33 votes

How many lists of 100 numbers (1 to 10 only) add to 700?

Each number is from one to ten inclusive only. There are $100$ numbers in the ordered list. The total must be $700$. How many such lists? Note: if, as it happens, this is one of those math problems …

(permutations) (combinations)

asked by Joe Blow 23 votes answered by robjohn 34 votes

Can't find mistake in an easy proof.

Consider the following theorem. $\textbf{Theorem:}$ for any sets $A, B, C, D$, if $A \times B \subseteq C \times D$ then $A \subseteq C$ and $B \subseteq D$. Then the following proof is given. …

(proof-verification) (proof-writing)

asked by Fazzolini 23 votes answered by Frunobulax 33 votes

Demonstration that 0 = 1

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e …

(complex-analysis) (limits) (fake-proofs) (natural-numbers) (paradoxes)

asked by astabada 22 votes answered by Prometheus 38 votes

What is the least $n$ such that it is possible to embed $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?

Let $\operatorname{GL}_2(\mathbb{F}_5)$ be the group of invertible $2\times 2$ matrices over $\mathbb{F}_5$, and $S_n$ be the group of permutations of $n$ objects. What is the least …

(group-theory) (representation-theory)

asked by Jack D'Aurizio 22 votes answered by Derek Holt 12 votes

Can a function be applied to itself?

There are some functions f that can be composed with themselves. But is there a function which can be applied to itself? In other words, is there a function f such that f is an element of Domain(f)?

(functions)

asked by user107952 17 votes answered by Carl Mummert 2 votes

Greatest hits from previous weeks:

How to convince a math teacher of this simple and obvious fact?

I have in my presence a mathematics teacher, who asserts that $$ \frac{a}{b} = \frac{c}{d} $$ Implies: $$ a = c, \space b=d $$ She has been shown in multiple ways why this is not true: $$ …

(algebra-precalculus) (arithmetic) (education) (teaching)

asked by user86484 82 votes answered by dtldarek 97 votes

Why is the volume of a cone one third of the volume of a cylinder?

The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside. This can be proved easily by …

(geometry)

asked by bryn 33 votes answered by Larry Wang 35 votes

Can you answer these?

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing …

(calculus) (real-analysis) (integration) (definite-integrals) (approximation)

asked by sefi 6 votes

find these linear functionals

I'm trying to solve this question: My attempt of solution: If $x\in E$, see $x$ in the first $m$ coordinates of $\mathbb R^n$ (can we do this?). I know how to find linear functionals such that …

(linear-algebra) (abstract-algebra)

asked by Rob1.1 3 votes

Notation $(a,b)$ for $]a,b[$.

Is there any logic or justification for the notation $(a,b)$ to represent $]a,b[$? To me this notation is very ambiguous and confusing because it looks like a couple of numbers and not an interval. …

(notation)

asked by idm 3 votes