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1500 questions
71
votes
18 answers
Unsolved Problems due to Lack of Computational Power
I was recently reading up about computational power and its uses in maths particularly to find counterexamples to conjectures. I was wondering are there any current mathematical problems which we are unable to solve due to our lack of computational…
user671231
71
votes
4 answers
Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$
One of fundamental inequalities on logarithm is:
$$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$
which you may prefer write in the form of
$$ \frac{x}{1+x} \leq \log{(1+x)} \leq x \quad\text{for all $x > -1$}.$$
The upper bound is…
Federico Magallanez
- 1,988
71
votes
3 answers
Where does the word "torsion" in algebra come from?
Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry (which relates back to the ordinary use of the…
Dan Piponi
- 4,566
71
votes
28 answers
Non-associative operations
There are lots of operations that are not commutative.
I'm looking for striking counter-examples of operations that are not associative.
Or may associativity be genuinely built-in the concept of an operation? May non-associative operations be of…
Hans-Peter Stricker
- 18,719
71
votes
10 answers
There are 4 cups of liquid. Three are water and one is poison. If you were to drink 3 of the 4 cups, what is the probability of being poisoned?
In Season 5 Episode 16 of Agents of Shield, one of the characters decides to prove she can't die by pouring three glasses of water and one of poison; she then randomly drinks three of the four cups. I was wondering how to compute the probability of…
student
- 952
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71
votes
29 answers
Understandable questions which are hard for non-mathematicians but easy for mathematicians
A friend of mine has set me the challenge of finding an example of the following:
Is there a question, that everyone (both mathematicians and non-mathematicians) can understand, that most mathematicians would answer correctly, instantly, but that…
Zestylemonzi
- 4,205
71
votes
4 answers
A nasty integral of a rational function
I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$
Mathematica has no problem evaluating it while I haven't the slightest…
user54031
71
votes
4 answers
Center-commutator duality
I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6:
Any subgroup of $G$ which contains $[G,G]$ is normal in $G$.
He says this as evidence that commutator and center play dual roles,…
Bruno Stonek
- 12,973
71
votes
12 answers
Will assuming the existence of a solution ever lead to a contradiction?
I'm reading Manfredo Do Carmo's differential geometry book. In section 1-7, he discusses the "Isoperimetric Inequality" which is related to the question of what 2-dimensional shape maximizes the enclosed area for a closed curve of constant length.…
Geoffrey
- 2,415
71
votes
3 answers
Recognizable vs Decidable
What is difference between "recognizable" and "decidable" in context of Turing machines?
metdos
- 1,007
71
votes
11 answers
Is it technically incorrect to write proofs forward?
A question on an assignment was similar to prove:
$$2a^2-7ab+2b^2 \geq -3ab.$$
and my proof was:
$$2a^2-4ab+2b^2\geq0$$
$$a^2-2ab+b^2\geq0$$
$$(a-b)^2\geq0$$
which is true.
However, my professor marked this as incorrect and the "correct" way to do…
mtheorylord
- 4,340
71
votes
1 answer
Difference between simplicial and singular homology?
I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and maybe this doesnt't help my intuition), but I am…
user48168
- 1,345
71
votes
8 answers
Dominoes and Induction, or How Does Induction Work?
I've never really understood how math induction is supposed to work.
You have these 3 steps:
Prove true for base case ($n=0$ or $n=1$ or whatever)
Assume true for $n=k$. Call this the induction hypothesis.
Prove true for $n=k+1$, somewhere using…
bobobobo
- 9,782
71
votes
11 answers
Why is the complex plane shaped like it is?
It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this?…
user64742
- 2,195
71
votes
5 answers
What is so interesting about the zeroes of the Riemann $\zeta$ function?
The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$
The Riemann hypothesis asserts that all the…
Karna
- 729