All Questions
1,606,450
questions
1595
votes
87
answers
595k
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 ...
1274
votes
26
answers
138k
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 ...
- 15.9k
1084
votes
32
answers
152k
views
How long will it take Marie to saw another board 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 ...
- 10.3k
920
votes
29
answers
98k
views
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 ...
878
votes
22
answers
111k
views
843
votes
27
answers
200k
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 ...
833
votes
52
answers
135k
views
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 ...
815
votes
1
answer
45k
views
A proof for $\dim(R[T])=\dim(R)+1$ without prime ideals?
Please read this first before answering. This is not the right place for you to advertise your favorite proof of the dimension formula. This question is only concerned with a proof using the Coquand-...
- 155k
811
votes
17
answers
163k
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 ...
- 12.8k
774
votes
12
answers
223k
views
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 ...
- 7,741
689
votes
25
answers
71k
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 ...
- 15.8k
660
votes
164
answers
56k
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,...
630
votes
8
answers
44k
views
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 ...
- 5,890
623
votes
6
answers
85k
views
Why can you turn clothing right-side-out?
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the ...
- 6,233
618
votes
43
answers
56k
views
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 ...
596
votes
0
answers
23k
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, $f(S)\subset Y$ ...
- 71.9k
590
votes
21
answers
93k
views
Mathematical difference between white and black notes in a piano
The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me.
Apparently, adjacent notes in a piano (including white or ...
- 5,787
567
votes
14
answers
322k
views
Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
I need help with this integral:
$$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$
The integrand graph looks like this:
$\hspace{1in}$
The ...
- 12.9k
546
votes
29
answers
218k
views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my math ...
- 9,099
534
votes
37
answers
73k
views
Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
517
votes
22
answers
85k
views
What are imaginary numbers?
At school, I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$. When I ...
- 6,493
513
votes
7
answers
22k
views
"The Egg:" Bizarre behavior of the roots of a family of polynomials.
In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$
In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
- 26.5k
511
votes
10
answers
69k
views
Best Sets of Lecture Notes and Articles
Let me start by apologizing if there is another thread on math.se that subsumes this.
I was updating my answer to the question here during which I made the claim that "I spend a lot of time ...
465
votes
10
answers
518k
views
Is this Batman equation for real? [closed]
HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-...
- 5,491
453
votes
18
answers
62k
views
To sum $1+2+3+\cdots$ to $-\frac1{12}$
$$\sum_{n=1}^\infty\frac1{n^s}$$
only converges to $\zeta(s)$ if $\text{Re}(s)>1$.
Why should analytically continuing to $\zeta(-1)$ give the right answer?
perplexed
450
votes
23
answers
89k
views
Proofs that every mathematician should know. [closed]
There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc.
So I'd like to ...
447
votes
24
answers
83k
views
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
- 4,685
446
votes
4
answers
249k
views
What is the intuitive relationship between SVD and PCA?
Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important ...
- 4,571
444
votes
10
answers
30k
views
My son's Sum of Some is beautiful! But what is the proof or explanation?
My youngest son is in $6$th grade. He likes to play with numbers. Today, he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, ...
- 3,421
442
votes
14
answers
366k
views
Fourier transform for dummies
What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)?
This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. ...
435
votes
22
answers
28k
views
On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)
Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole."
For example, suppose you come across the novel term vector space, and want to learn more ...
- 14.1k
395
votes
33
answers
51k
views
Pedagogy: How to cure students of the "law of universal linearity"?
One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:
$$ \frac{1}{a+b} \mathrel{\...
393
votes
1
answer
16k
views
The Ring Game on $K[x,y,z]$
I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
- 60k
385
votes
37
answers
131k
views
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
Elementary row operations.
Linear ...
- 6,035
378
votes
15
answers
27k
views
Can every proof by contradiction also be shown without contradiction?
Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving ...
- 9,803
377
votes
20
answers
46k
views
Find five positive integers whose reciprocals sum to $1$
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was ...
- 4,545
376
votes
73
answers
82k
views
'Obvious' theorems that are actually false
It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".
Now, I know a fair few examples of things that are obviously true and that can be proved ...
372
votes
23
answers
49k
views
Zero to the zero power – is $0^0=1$?
Could someone provide me with a good explanation of why $0^0=1$?
My train of thought:
$$x>0\\
0^x=0^{x-0}=\frac{0^x}{0^0}$$
so
$$0^0=\frac{0^x}{0^x}=\,?$$
Possible answers:
$0^0\cdot0^x=1\cdot0^0$,...
- 3,979
360
votes
111
answers
36k
views
Collection of surprising identities and equations.
What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, ...
356
votes
31
answers
58k
views
Is it true that $0.999999999\ldots=1$?
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
355
votes
23
answers
29k
views
Why don't we define "imaginary" numbers for every "impossibility"?
Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do ...
- 3,638
351
votes
8
answers
54k
views
Calculating the length of the paper on a toilet paper roll
Fun with Math time.
My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
- 25.9k
351
votes
7
answers
50k
views
How can you prove that a function has no closed form integral?
In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/...
- 5,501
351
votes
0
answers
21k
views
Limit of sequence of growing matrices [closed]
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \\
1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 0 & 1/2\\
0 & 1/2 & 1/2 & 0
\end{array}\right),
$$
$K_1=\left(\...
- 7,705
349
votes
11
answers
227k
views
What is the importance of eigenvalues/eigenvectors?
What is the importance of eigenvalues/eigenvectors?
- 5,369
341
votes
36
answers
32k
views
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
The following is a quote from Surely you're joking, Mr. Feynman. The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
330
votes
31
answers
38k
views
Nice examples of groups which are not obviously groups
I am searching for some groups, where it is not so obvious that they are groups.
In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't ...
318
votes
8
answers
25k
views
Intuition for the definition of the Gamma function?
In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity
$$n! = \int_{0}^{\infty} t^n e^{-t} dt$$
, coming from ...
- 409k
312
votes
6
answers
91k
views
Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer:
Question: If you choose an answer to this question at random, what is the probability that you will be correct?
...
user11088
309
votes
9
answers
661k
views
Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is $A A^\mathrm T$ something special for any matrix $A$?
- 4,443