Highest scored questions
1,698,195 questions
1663
votes
90
answers
615k
views
Visually stunning math concepts which are easy to explain
Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are ...
1331
votes
27
answers
156k
views
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
- 16.6k
1100
votes
32
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If it took 10 minutes to saw a board into 2 pieces, how long will it take to saw another into 3 pieces?
So this is supposed to be really simple, and it's taken from the following picture:
Text-only:
It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it ...
- 10.5k
920
votes
29
answers
104k
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Can I use my powers for good? [closed]
I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar.
Four years after the PhD, I am pretty sure that ...
913
votes
24
answers
123k
views
890
votes
1
answer
54k
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A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Background. If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'...
- 184k
885
votes
60
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Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler ...
866
votes
27
answers
206k
views
How to study math to really understand it and have a healthy lifestyle with free time? [closed]
Here's my issue I faced;
I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost:
In the last few years, I had ...
848
votes
20
answers
189k
views
What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
- 13.5k
810
votes
12
answers
249k
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Does $\pi$ contain all possible number combinations?
$\pi$ Pi
Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that
every possible number combination exists somewhere in pi. Converted
into ASCII text, somewhere in that infinite string of ...
- 8,119
697
votes
26
answers
74k
views
Splitting a sandwich and not feeling deceived
This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
- 16.3k
660
votes
164
answers
59k
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,...
650
votes
8
answers
46k
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Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
So we all know that the continued fraction containing all $1$s...
$$
x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}}.
$$
yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x ...
- 6,130
648
votes
0
answers
27k
views
Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
- 75.7k
643
votes
46
answers
63k
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Examples of patterns that eventually fail
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up ...