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1500 questions
134
votes
9 answers
Is there an integral that proves $\pi > 333/106$?
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves $\pi > 333/106$?
anon
133
votes
12 answers
Modular exponentiation by hand ($a^b\bmod c$)
How do I efficiently compute $a^b\bmod c$:
When $b$ is huge, for instance $5^{844325}\bmod 21$?
When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$?
When $(a,c)\ne1$, for…
user7530
- 49,280
133
votes
20 answers
Real life applications of Topology
The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here.
What are the various real life applications of topology?
Bhargav
- 2,969
133
votes
2 answers
How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$
How can one prove that
$$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
larry
- 1,479
133
votes
10 answers
What's the difference between theorem, lemma and corollary?
Can anybody explain me what is the basic difference between theorem, lemma and corollary?
We have been using it for a long time but I never paid any attention. I am just curious to know.
monalisa
- 4,460
133
votes
13 answers
Why study linear algebra?
Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study linear algebra?
Aaron
- 1,369
133
votes
5 answers
Why does L'Hopital's rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?
$$\lim_{x \to \infty} \frac{x}{x+\sin(x)}$$
This is of the indeterminate form of type $\frac{\infty}{\infty}$, so we can apply l'Hopital's…
eMathHelp
- 2,409
132
votes
8 answers
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 define it to be $0$.
But why characteristic zero? Why…
Srivatsan
- 26,311
132
votes
3 answers
Why is an average of an average usually incorrect?
Can someone explain why taking an average of an average usually results in a wrong answer? Is there ever a case where the average of the average can be used correctly?
As an example, let's say that an assessment is given to three schools and I…
O.O
- 1,541
132
votes
8 answers
What is the difference between independent and mutually exclusive events?
Two events are mutually exclusive if they can't both happen.
Independent events are events where knowledge of the probability of one doesn't change the probability of the other.
Are these definitions correct? If possible, please give more than one…
Adnan Ali
- 1,493
132
votes
11 answers
How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$
Please help me to find a closed form for this integral:
$$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$
I suspect it might exist because there are similar integrals having closed forms:
$$\begin{align}\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm…
Oksana Gimmel
- 5,302
132
votes
5 answers
Getting better at proofs
So, I don't like proofs.
To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert.
Oftentimes the proof in the book is something that I get if I study, but hard to come up with on my own.…
bobobobo
- 9,502
132
votes
13 answers
How do I get the square root of a complex number?
If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
Macha
- 1,503
132
votes
5 answers
How to find solutions of linear Diophantine ax + by = c?
I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?
I tried to play around with it:
$x = (c - by)/a$, hence $a|(c - by)$.
$a$, $c$…
roxrook
- 12,081
132
votes
8 answers
What exactly is the difference between a derivative and a total derivative?
I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w / \partial r$ and $\partial w / \partial…
Chibueze Opata
- 1,455