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1500 questions
331
votes
31 answers
Nice examples of groups which are not obviously groups
I am searching for some groups, where it is not so obvious that they are groups.
In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't think that these examples are helpful to…
Dominic Michaelis
- 19,935
323
votes
8 answers
Intuition for the definition of the Gamma function?
In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity
$$n! = \int_{0}^{\infty} t^n e^{-t} dt$$
, coming from the Gamma function. I have a mathematical…
Qiaochu Yuan
- 419,620
317
votes
9 answers
Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is $A A^\mathrm T$ something special for any matrix $A$?
Martin Ueding
- 4,523
315
votes
6 answers
Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer:
Question: If you choose an answer to this question at random, what is the probability that you will be correct?
a. $25\%$
b. $50\%$
c. $0\%$
d. $25\%$
Does this…
user11088
313
votes
21 answers
Really advanced techniques of integration (definite or indefinite)
Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every time I search for "Advanced Techniques of…
user3002473
- 8,943
310
votes
10 answers
V.I. Arnold says Russian students can't solve this problem, but American students can -- why?
In a book of word problems by V.I Arnold, the following appears:
The hypotenuse of a right-angled triangle (in a standard American examination) is $10$ inches, the altitude dropped onto it is 6 inches. Find the area of the triangle.
American…
Eli Rose
- 8,141
306
votes
5 answers
In Russian roulette, is it best to go first?
Assume that we are playing a game of Russian roulette (6 chambers) and that there is no shuffling after the shot is fired.
I was wondering if you have an advantage in going first?
If so, how big of an advantage?
I was just debating this with…
nikkita
- 2,709
306
votes
16 answers
Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals.
This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many…
Orest Xherija
- 1,049
304
votes
1 answer
How discontinuous can a derivative be?
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "highly" discontinuous Darboux functions are known--the…
Chris Janjigian
- 9,438
303
votes
5 answers
Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$.
It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot…
Hans Parshall
- 6,246
303
votes
28 answers
In the history of mathematics, has there ever been a mistake?
I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument.…
Steven-Owen
- 5,556
302
votes
39 answers
One question to know if the number is 1, 2 or 3
I've recently heard a riddle, which looks quite simple, but I can't solve it.
A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "Yes", "No", or "I don't know," and…
Gintas K
- 765
301
votes
15 answers
Math without pencil and paper
For someone who is physically unable to use a pencil and paper, what would be the best way to do math?
In my case, I have only a little movement in my fingers. I can move a computer mouse and press the left button. Currently I do very little math…
Jeroen
- 2,569
299
votes
18 answers
Why does this innovative method of subtraction from a third grader always work?
My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand.
Here is an example: $61-17$
Instead of borrowing, making it…
user535429
- 2,157
297
votes
22 answers
Why can ALL quadratic equations be solved by the quadratic formula?
In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use it. I have tried to figure it out by proving…
idonno
- 3,909