Linked Questions

1
vote
0answers
136 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 example,...
663
votes
164answers
52k views

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

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 Lament,...
225
votes
36answers
26k views

What are some counter-intuitive results in mathematics that involve only finite objects?

There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem ...
125
votes
44answers
23k views

What are some examples of a mathematical result being counterintuitive?

As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition. My first and favorite experience of this is Gabriel's Horn that you see in ...
53
votes
7answers
3k views

False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
26
votes
10answers
4k views

Explain non-convergent sums to a bright high schooler.

I know the higher-math answer to this question, but I'm asking how on Earth to explain it to a bright high school student. Here's the question, paraphrased: “I can see that $\left(1+x+x^{2}+x^{3}+x^{...
31
votes
6answers
12k 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 ...
21
votes
3answers
3k 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 ...
41
votes
3answers
14k 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), ...
8
votes
6answers
8k views

Help to understand material implication [closed]

This question comes from from my algebra paper: $(p \rightarrow q)$ is logically equivalent to ... (then four options are given). The module states that the correct option is $(\sim p \lor q)$. ...
16
votes
3answers
2k 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 $\lim_{x\to \infty}f'(x)=0$?
14
votes
4answers
4k 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 ...
11
votes
2answers
3k views

Must the (continuous) image of a null set be null?

Say $E \subset [0,1]$ is a null set. Let $f: [0,1] \rightarrow [0,1] $. Do you think $f(E)$ is a null set or not? Just being curious. (DEF): A set $A$ is null if given any $\epsilon > 0$, there ...
6
votes
3answers
10k 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.
13
votes
2answers
4k views

Graph of discontinuous additive function is dense

$f:\mathbb{R}\to\mathbb{R}$ is a function such that for all $x,y$ in $\mathbb{R}$, $f(x+y)=f(x)+f(y)$. If $f$ is continuous, then of course it has to be linear. But here $f$ is NOT continuous. Then ...

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