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1500 questions
297
votes
8 answers
Please explain the intuition behind the dual problem in optimization.
I've studied convex optimization pretty carefully, but don't feel that I have yet "grokked" the dual problem. Here are some questions I would like to understand more deeply/clearly/simply:
How would somebody think of the dual problem? What…
littleO
- 51,938
295
votes
27 answers
Why do mathematicians use single-letter variables?
I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated when I try to follow mathematical notation. Specifically, I get frustrated trying to keep track…
eater
- 3,083
292
votes
24 answers
Is mathematics one big tautology?
Is mathematics one big tautology? Let me put the question in clearer terms:
Mathematics is a deductive system; it works by starting with arbitrary axioms, and deriving therefrom "new" properties through the process of deduction. As such, it would…
Coffee_Table
- 2,909
292
votes
41 answers
Can't argue with success? Looking for "bad math" that "gets away with it"
I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare").
One example would be "cancelling" the 6's in
$$\frac{64}{16}.$$
Another one would be something like
$$\frac{9}{2}…
kjo
- 14,334
292
votes
11 answers
Is '$10$' a magical number or I am missing something?
It's a hilarious witty joke that points out how every base is '$10$' in its base. Like,
\begin{align}
2 &= 10\ \text{(base 2)} \\
8 &= 10\ \text{(base 8)}
\end{align}
My question is if whoever invented the decimal system had chosen $9$…
Shubham
- 2,513
288
votes
63 answers
Funny identities
Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?
AD - Stop Putin -
- 10,970
287
votes
14 answers
Help with a prime number spiral which turns 90 degrees at each prime
I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the…
Karl
- 4,693
286
votes
5 answers
Is $7$ the only prime followed by a cube?
I discovered this site which claims that "$7$ is the only prime followed by a cube". I find this statement rather surprising. Is this true? Where might I find a proof that shows this?
In my searching, I found this question, which is similar but…
David Starkey
- 2,363
280
votes
3 answers
How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?
Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \det \left(A^2+B^2+C^2\right)$$
This problem is…
math110
- 93,304
280
votes
4 answers
The Mathematics of Tetris
I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game.
Background: The Tetris…
Eric Naslund
- 72,099
280
votes
20 answers
Conjectures that have been disproved with extremely large counterexamples?
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is…
Justin L.
- 14,532
278
votes
5 answers
Evaluate $\int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx$
Evaluate the following integral
$$
\tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx
$$
My Attempt:
Letting $x=\frac{\pi}{2}-x$ and using the property that
$$
\int_{0}^{a}f(x)\,\Bbb dx = \int_{0}^{a}f(a-x)\,\Bbb dx
$$
we…
juantheron
- 53,015
275
votes
13 answers
Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?
Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet."
I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got…
Jason
- 1,405
274
votes
5 answers
A 1400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I
The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician.
I wondered how much this could be improved using our…
Claude Leibovici
- 260,315
274
votes
32 answers
Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:
$$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$
Well, can anyone prove this without using Residue theory? …
user9413