# Linked Questions

**100**

votes

**32**answers

20k views

### Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?

**71**

votes

**32**answers

48k views

### Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...

**70**

votes

**21**answers

18k views

### Examples of math hoaxes/interesting jokes published on April Fool's day?

What are examples of math hoaxes/interesting jokes published on April Fool's day?
For a start P=NP.

**56**

votes

**7**answers

4k views

### When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...

**30**

votes

**12**answers

9k views

### Elementary / Interesting proofs of the Nullstellensatz

Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques?
One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...

**55**

votes

**2**answers

16k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**41**

votes

**8**answers

8k views

### Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$.
Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...

**45**

votes

**2**answers

4k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

**17**

votes

**5**answers

3k views

### Is there a version of inclusion/exclusion for vector spaces?

I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...

**24**

votes

**4**answers

6k views

### When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in ...

**18**

votes

**7**answers

1k views

### Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts:
$$ \int_a^b f(x)g'(x)ds = \...

**9**

votes

**5**answers

3k views

### Projection of Borel set from $R^2$ to $R^1$

Hello
This should be easy to prove but i have no idea how to do it:
If $X \subseteq \mathbb{R}^2$ is borel then $f(X)$ is borel where $f(x,y) = x$
Thanks
Tobias

**36**

votes

**4**answers

3k views

### Cocomplete but not complete abelian category

This is a duplicate of the following question to which I did not receive any answer: http://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category
Let $\mathfrak C$ be an ...

**39**

votes

**1**answer

3k views

### What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...

**8**

votes

**2**answers

2k views

### When is sin(r \pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary.
As the question suggests, I would like to know when $\sin(...