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1500 questions
66
votes
12 answers
What are the Laws of Rational Exponents?
On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number seems to be shown as equal to its opposite…
Daniel R. Collins
- 9,177
66
votes
5 answers
Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure
Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables formula of the form:
$$
\int_{\phi(\Omega)} f…
Oyqcb
- 661
66
votes
10 answers
Symbol for "probably equal to" (barring pathology)?
I am writing lecture notes for an applied statistical mechanics course and often need to express the notion that something is very probably true for functional forms found in the wild, without launching into a full digression for pathological…
mewahl
- 719
66
votes
16 answers
Choice of $q$ in Baby Rudin's Example 1.1
First, my apologies if this has already been asked/answered. I wasn't able to find this question via search.
My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2. In the second version of…
Rachel
- 2,994
66
votes
6 answers
Difference between topology and sigma-algebra axioms.
One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically…
Creator
- 3,198
66
votes
4 answers
Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$
I'm interested in integrals of the form
$$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for }a>0,\,b>0}\tag1$$
It's…
Vladimir Reshetnikov
- 32,650
66
votes
10 answers
Why is $1/i$ equal to $-i$?
When I entered the value $$\frac{1}{i}$$ in my calculator, I received the answer as $-i$ whereas I was expecting the answer as $i^{-1}$. Even google calculator shows the same answer (Click here to check it out).
Is there a fault in my calculator or…
Arulx Z
- 1,051
66
votes
5 answers
Jacobi identity - intuitive explanation
I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as an axiom for defining a Lie algebra). Could…
aelguindy
- 2,718
66
votes
3 answers
"Every linear mapping on a finite dimensional space is continuous"
From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector spaces (not necessarily the same), as along as the domain is…
Tim
- 49,162
65
votes
1 answer
Why is the absolute value function not differentiable at $x=0$?
They say that the right and left limits do not approach the same value hence it does not satisfy the definition of derivative. But what does it mean verbally in terms of rate of change?
user187397
- 651
65
votes
2 answers
Why is $L^{\infty}$ not separable?
$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces.
What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$?
Our teacher gave us some hints that there exists an uncountable subset such that the distance of…
Andylang
- 1,743
65
votes
18 answers
What exactly is a number?
We've just been learning about complex numbers in class, and I don't really see why they're called numbers.
Originally, a number used to be a means of counting (natural numbers).
Then we extend these numbers to instances of owing other people money…
user164061
- 839
65
votes
18 answers
What are some good ways to get children excited about math?
I'm talking in the range of 10-12 years old, but this question isn't limited to only that range.
Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn…
David McGraw
- 885
65
votes
22 answers
Nuking the Mosquito — ridiculously complicated ways to achieve very simple results
Here is a toned down example of what I'm looking for:
Integration by solving for the unknown integral of $f(x)=x$:
$$\int x \, dx=x^2-\int x \, dx$$
$$2\int x \, dx=x^2$$
$$\int x \, dx=\frac{x^2}{2}$$
Can anyone think of any more examples?
P.S.…
Aidan F. Pierce
- 1,431
65
votes
17 answers
Interesting "real life" applications of serious theorems
As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like:
"You have a great circular pizza with $n$…
user149890