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1500 questions
133
votes
4 answers
Why, historically, do we multiply matrices as we do?
Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very intuitive operation: if you were to ask someone how to mutliply two…
msh210
- 3,948
133
votes
16 answers
Division by zero
I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.
$\dfrac {x}{0}$ is Impossible ( If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are…
danielsyn
- 1,575
133
votes
19 answers
Past open problems with sudden and easy-to-understand solutions
What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius,…
Damian Reding
- 8,894
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133
votes
12 answers
Can you be 1/12th Cherokee?
I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "$1/16$th", or generally $1/2^n, n \in \mathbb{N}$.
Obviously you could also be some summation of same…
Nick T
- 1,803
132
votes
26 answers
Simplest or nicest proof that $1+x \le e^x$
The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which…
Ashley Montanaro
- 1,437
132
votes
13 answers
Advantages of Mathematics competition/olympiad students in Mathematical Research
Everyone in this community I think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries participating from around the world.
What's…
user9413
132
votes
3 answers
Are all limits solvable without L'Hôpital Rule or Series Expansion
Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion?
For example,
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$
$$\lim_{x\to0}\frac{\sin…
lab bhattacharjee
- 279,016
132
votes
7 answers
Can an irrational number raised to an irrational power be rational?
Can an irrational number raised to an irrational power be rational?
If it can be rational, how can one prove it?
John Hoffman
- 2,774
132
votes
8 answers
Are half of all numbers odd?
Plato puts the following words in Socrates' mouth in the Phaedo dialogue:
I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called…
Jon Ericson
- 1,449
131
votes
8 answers
What are differences between affine space and vector space?
I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by Paul Bamberg and Shlomo Sternberg. In Chapter 1…
user41451
- 1,415
131
votes
9 answers
Produce an explicit bijection between rationals and naturals
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such…
Alex Basson
- 4,331
131
votes
1 answer
Can $x^{x^{x^x}}$ be a rational number?
If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ?
We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not be rational:
Denote…
lsr314
- 16,048
131
votes
11 answers
Prove that every convex function is continuous
A function $f : (a,b) \to \Bbb R$ is said to be convex if
$$f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$$
whenever $a < x, y < b$ and $0 < \lambda <1$. Prove that every convex function is continuous.
Usually it uses the fact:
If $a…
cowik
- 1,353
131
votes
4 answers
Could someone explain conditional independence?
My understanding right now is that an example of conditional independence would be:
If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are…
Ryan
- 1,721
131
votes
1 answer
Application of Hilbert's basis theorem in representation theory
In Smalø: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand:
Two orders are defined on the set of $d$-dimensional modules over an algebra…
Julian Kuelshammer
- 9,750