Linked Questions

1
vote
0answers
66 views

Things you've believed for a long time were true, but are false in reality [duplicate]

Do you have any things (mathematical statements, statements about mathematics) you've believed for a long time were true, but now with enough mathematical knowledge you realize were wrong? For ...
446
votes
138answers
28k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
14
votes
3answers
689 views

How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
7
votes
5answers
5k views

Help: rules of a game whose details I don't remember!

In a probability course, a game was introduced which a logical approach won't yield a strategy for winning, but a probabilistic one will. My problem is that I don't remember the details (the rules of ...
16
votes
1answer
7k views

Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
10
votes
8answers
253 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
7
votes
4answers
998 views

How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
3
votes
3answers
3k views

an example of a continuous function whose Fourier series diverges at a dense set of points

Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
4
votes
3answers
211 views

The limit of the derivative of an increasing and bounded function is always $0$?

Let $\,f : \mathbb{R} \rightarrow \mathbb{R}$ be a infinitely differentiable function that is increasing and bounded. Then is it true that $\lim_{x\to \infty}f'(x)=0$?
2
votes
2answers
989 views

Graph of discontinuous linear function is dense

$f:\mathbb{R}\rightarrow\mathbb{R}$ is a function such that for all $x,y$ in $\mathbb{R}$, $f(x+y)=f(x)+f(y)$. If $f$ is cont, then of course it has to be linear. But here $f$ is NOT cont. Then show ...
5
votes
1answer
104 views

Chain of length $2^{\aleph_0}$in $ (P(\mathbb{N}),\subseteq)$

How can I find a chain of length $2^{\aleph_0}$ in $ (P(\mathbb{N}), \subseteq )$. The only chain I have in mind is $$\{\{0 \},\{0,1 \},\{0,1,2 \},\{ 0,1,2,3\},...,\{\mathbb{N} \} \}$$ But the ...
1
vote
2answers
153 views

Mathematical statements that cannot be proved or disproved [closed]

I've recently been reading about the continuum hypothesis and am fascinated by the fact that it cannot be proved or disproved, despite the fact that the statement itself is either true or false. What ...
1
vote
1answer
286 views

Is there a compact set which is not Jordan measurable?

Is there a compact set which is not Jordan measurable? Intuitively, the answer seems like there is no such set but I could not find a proof anywhere. Does anyone know of a proof or a counter example?
3
votes
2answers
99 views

Generating function: Probability regarding coin toss

If a coin is flipped 25 times with eight tails occurring, what is the probability that no run of six (or more) consecutive heads occur? Wasn't sure how to approach this and am quite positive my ...
1
vote
3answers
103 views

Why do we formalize conceptions?

Why do we always try to formalize conceptions? Let's take the naive conception of sets, why do we try to write down a list of axioms? what do we earn in doing so? I'm looking especially for ...

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