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1500 questions
96
votes
6 answers
Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational.
Problem. Let there be six irrational numbers. Prove that there exists three irrational numbers among them such that the sum of any two of those irrational numbers is also irrational.
I have tried to prove it in the following way, but I am not…
Arpon Basu
- 1,161
96
votes
11 answers
What is the best book for studying discrete mathematics?
As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms, so I want to know which Discrete Mathematics…
idonno
- 3,909
96
votes
21 answers
How do you explain the concept of logarithm to a five year old?
Okay, I understand that it cannot be explained to a 5 year old. But how do you explain the logarithm to primary school students?
Sandbox
- 1,265
96
votes
4 answers
Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$?
First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an eigenvalue of $BA$?
If it's not true, then under what…
dantswain
- 1,125
96
votes
3 answers
Complexity class of comparison of power towers
Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < b_1^{b_2^{\cdot^{\cdot^{\cdot^{b_m}}}}}$.
Is this…
Vladimir Reshetnikov
- 47,122
95
votes
4 answers
Space of bounded continuous functions is complete
I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete.
The lecturer then proved two things, (i) that $f(x) = \lim f_n (x)$ is bounded and (ii) that $\lim f_n \in…
Rudy the Reindeer
- 46,359
95
votes
15 answers
Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
I'm a second year math student. And I've the following problem.
When I prepare myself for an exam, I can distinguish two phases. First I'm mainly interested in whatever is necessary to pass the exam. This means that I do not always read the theory…
Kasper
- 13,528
95
votes
9 answers
What is a universal property?
Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia.
To make the following summary more readable I do equate "universal" with "initial" and omit the tedious details concerning duality.
Suppose that $U:…
Hans-Peter Stricker
- 18,159
95
votes
2 answers
$6!\cdot 7!=10!$. Is there a natural bijection between $S_6\times S_7$ and $S_{10}$?
Aside from $1!\cdot n!=n!$ and $(n!-1)!\cdot n! = (n!)!$, the only product of factorials known is $6!\cdot 7!=10!$.
One might naturally associate these numbers with the permutations on $6, 7,$ and $10$ objects, respectively, and hope that this…
RavenclawPrefect
- 17,299
95
votes
2 answers
Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing
Prove without calculus that the sequence
$$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$
is strictly decreasing.
user 1591719
- 44,216
- 12
- 105
- 255
95
votes
5 answers
Reversing an integer's digits is multiplicative for small digits
So my 7 year old son pointed out to me something neat about the number 12: if you multiply it by itself, the result is the same as if you took 12 backwards multiplied by itself, then flipped the result backwards. In other words: $$12 × 12 =…
BCA
- 793
95
votes
2 answers
Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic
what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.
So far, I have not found any precise definition in literature.
shuhalo
- 7,485
95
votes
7 answers
Where are the axioms?
It is said that our current basis for mathematics are the ZFC-axioms.
Question: Where are these axioms in our mathematics? When do we use them? I have now studied math for a year, and have yet to run into a single one of these ZFC axioms. How can…
Imean
- 845
94
votes
6 answers
$x^y = y^x$ for integers $x$ and $y$
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
Paulo Argolo
- 4,210
94
votes
23 answers
What is your favorite application of the Pigeonhole Principle?
The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item.
I'd like to see your favorite application of the pigeonhole principle, to prove some…
Álvaro Lozano-Robledo
- 15,390