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1500 questions
102
votes
19 answers
Unconventional mathematics books
I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it demonstrates many of Euclid's proofs as lurid and lusciously coloured geometric figures.…
seeker
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102
votes
13 answers
Don't see the point of the Fundamental Theorem of Calculus.
$$\frac{d}{dx}\int_a^xf(t)\,dt$$
I would love to to understand what exactly is the point of FTC. I'm not interested in mechanically churning out solutions to problems. It doesn't state anything that isn't already known. Prior to reading about…
JackOfAll
- 4,701
101
votes
11 answers
Mathematician vs. Computer: A Game
A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same range. If the numbers chosen share a prime factor,…
user139000
101
votes
1 answer
What's the largest possible volume of a taco, and how do I make one that big?
Let $f$ be a continuous, even function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. [You might imagine something like $f(x)=x^2$.] View the graph of…
2'5 9'2
- 54,717
101
votes
14 answers
Problems that become easier in a more general form
When solving a problem, we often look at some special cases first, then try to work our way up to the general case.
It would be interesting to see some counterexamples to this mental process, i.e. problems that become easier when you formulate them…
MGA
- 9,636
101
votes
10 answers
Why can't you add apples and oranges, but you can multiply and divide them?
What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided?
This looks more like a question for Physics, but lengths and…
danza
- 1,205
101
votes
4 answers
Factorial, but with addition
Is there a notation for addition form of factorial?
$$5! = 5\times4\times3\times2\times1$$
That's pretty obvious. But I'm wondering what I'd need to use to describe
$$5+4+3+2+1$$
like the factorial $5!$ way.
EDIT: I know about the formula. I want to…
akinuri
- 1,353
101
votes
23 answers
What are Your Favourite Maths Puzzles?
We all love a good puzzle
To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element…
Tom Boardman
- 3,505
101
votes
0 answers
What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?
Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with a unit perimeter?
A reasonable first guess would be the regular tetrahedron of side length $1/6$, with volume…
RavenclawPrefect
- 17,299
101
votes
4 answers
Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer
Let $a,b$ be positive integers.
When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.
Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second…
quanta
- 12,425
101
votes
4 answers
Probability density function vs. probability mass function
I've a confession to make. I've been using PDF's and PMF's without actually knowing what they are. My understanding is that density equals area under the curve, but if I look at it that way, then it doesn't make sense to refer to the "mass" of a…
0x0
- 2,641
101
votes
4 answers
Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar?
I gave the following problem to students:
Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$.
Prove that if $A$ and $B$ are two similar $n\times n$ matrices, then they have the same…
Taladris
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101
votes
1 answer
Relations between p norms
The $p$-norm on $\mathbb R^n$ is given by $\|x\|_{p}=\big(\sum_{k=1}^n
|x_{k}|^p\big)^{1/p}$. For $0 < p < q$ it can be shown that $\|x\|_p\geq\|x\|_q$ (1, 2). It appears that in $\mathbb{R}^n$ a number of opposite inequalities can also be obtained.…
PianoEntropy
- 2,606
101
votes
9 answers
Why is the product of two rotation matrices not commutative?
Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?
Navin Prashath
- 1,145
101
votes
1 answer
Arithmetic-geometric mean of 3 numbers
The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows:
$$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,\quad b_{n+1}=\sqrt{a_n…
Vladimir Reshetnikov
- 47,122