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1500 questions
167
votes
26 answers
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
Can someone give a simple explanation as to why the harmonic series
$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$
doesn't converge, on the other hand it grows very slowly?
I'd prefer an easily comprehensible explanation…
bryn
- 10,045
167
votes
3 answers
Why is the eigenvector of a covariance matrix equal to a principal component?
If I have a covariance matrix for a data set and I multiply it times one of it's eigenvectors. Let's say the eigenvector with the highest eigenvalue. The result is the eigenvector or a scaled version of the eigenvector.
What does this really…
Ryan
- 5,639
167
votes
8 answers
What's the point in being a "skeptical" learner
I have a big problem:
When I read any mathematical text I'm very skeptical. I feel the need to check every detail of proofs and I ask myself very dumb questions like the following: "is the map well defined?", "is the definition independent from the…
Dubious
- 14,048
167
votes
22 answers
Examples of mathematical discoveries which were kept as a secret
There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret.
For example it is completely expected that if some mathematician find a proof of $P=NP$, he is not allowed by the…
user180918
166
votes
1 answer
What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?
I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the Picard group continues to elude me.
One of the…
topspin1617
- 1,693
166
votes
20 answers
How to distinguish between walking on a sphere and walking on a torus?
Imagine that you're a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can't see the difference from this point of view.
If you are interested, this question arose while I was…
Julien__
- 2,485
166
votes
33 answers
What are the most overpowered theorems in mathematics?
What are the most overpowered theorems in mathematics?
By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I'm looking for the biggest guns a research…
Samuel Handwich
- 2,831
166
votes
5 answers
Can you raise a number to an irrational exponent?
The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\sqrt[y]{a^x}$. So what happens when you raise a…
tel
- 1,913
166
votes
19 answers
Is there another simpler method to solve this elementary school math problem?
I am teaching an elementary student. He has a homework as follows.
There are $16$ students who use either bicycles or tricycles. The total
number of wheels is $38$. Find the number of students using bicycles.
I have $3$ solutions as…
kiss my armpit
- 3,963
166
votes
4 answers
What happens when we (incorrectly) make improper fractions proper again?
Many folks avoid the "mixed number" notation such as $4\frac{2}{3}$ due to its ambiguity. The example could mean "$4$ and two thirds", i.e. $4+\frac{2}{3}$, but one may also be tempted to multiply, resulting in $\frac{8}{3}$.
My questions pertain to…
Zim
- 4,623
166
votes
1 answer
Pythagorean triples that "survive" Euler's totient function
Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple:
\begin{equation}
a^2 + b^2 = c^2. \tag{1}\label{1}
\end{equation}
Additionally, suppose that when you apply Euler's totient function to each term, the equation…
Misha Lavrov
- 159,700
165
votes
6 answers
What are the numbers before and after the decimal point referred to in mathematics?
Is there an actual term for the numbers that appear before and after the decimal point?
For example:
25.18
I know the 1 is in the tenths position, the 8 is in the hundredths position but I am seeking singular terms which apply to all of the numbers…
Calvin Froedge
- 1,861
165
votes
16 answers
What's new in higher dimensions?
This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3".
What I am wondering about is what new geometrical phenomena are there in higher dimensions. When I say new I mean phenomena…
Martin Hurtado
- 1,843
165
votes
20 answers
Online tool for making graphs (vertices and edges)?
Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw.
(why do I have so many? It's for pathing in a game)
Only tool…
f20k
- 1,753
164
votes
19 answers
What is the difference between a point and a vector?
I understand that a vector has direction and magnitude whereas a point doesn't.
However, in the course notes that I am using, it is stated that a point is the same as a vector.
Also, can you do cross product and dot product using two points instead…
6609081
- 1,745