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1500 questions
71
votes
7 answers
Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens)
I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there is $1$ digit.
A quick example of what I…
JD Isaacks
- 873
71
votes
12 answers
Why do we not have to prove definitions?
I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? It seems like some definitions should have a…
Zduff
- 4,252
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
71
votes
5 answers
Nice expression for minimum of three variables?
As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the…
Oscar Cunningham
- 16,299
71
votes
1 answer
Is this determinant identity known?
Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the main diagonal all equal -1 and entries on even lower…
Vincent
- 10,614
71
votes
4 answers
Are there any differences between tensors and multidimensional arrays?
I see lots of references saying things like
A tensor is a multidimensional or N-way array
But others that say things like
it should be remarked that other mathematical entities occur in physics that, like tensors, generally consist of…
rhombidodecahedron
- 1,532
71
votes
17 answers
Why is a circle 1-dimensional?
In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be…
mr eyeglasses
- 5,539
71
votes
5 answers
How can a probability density be greater than one and integrate to one
Wikipedia says:
The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
and it also says.
Unlike a probability, a probability density function can take on values greater than one; for…
zenna
- 1,479
70
votes
1 answer
Evaluating sums and integrals using Taylor's Theorem
Taylor's theorem states that
$$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$
We can use this to evaluate integrals. For example, consider $f(x)=\frac{b!x^{b+n+1}}{(b+n+1)!}$. This has…
Pauly B
- 5,272
70
votes
7 answers
What’s the difference between analytical and numerical approaches to problems?
I don't have much (good) math education beyond some basic university-level calculus.
What do "analytical" and "numerical" mean? How are they different?
jbrennan
- 819
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- 7
- 6
70
votes
2 answers
When does L' Hopital's rule fail?
This thought jumped out of me during my calculus teaching seminar.
It is well known that the classical L'Hospital rule claims that for the $\frac{0}{0}$ indeterminate case, we have:
$$
\lim_{x\rightarrow A}\frac{f(x)}{g(x)}=\lim_{x\rightarrow…
Bombyx mori
- 19,638
- 6
- 52
- 112
70
votes
4 answers
Learning Roadmap for Algebraic Topology
I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them.
I have a solid background in Abstract Algebra, and also have knowledge on Homological Algebra(in…
Arsenaler
- 3,930
70
votes
7 answers
What is the oldest open problem in geometry?
Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history.
What I would like to know is: What is the oldest open problem in geometry?
Also (soft questions): Why is it so hard?…
Sudoku Polo
- 693
70
votes
2 answers
Integration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}}$ be a non-negative bounded measurable…
user54992
70
votes
4 answers
Proof that the irrational numbers are uncountable
Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable?
nkassis
- 851