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1500 questions
99
votes
8 answers
Why are groups more important than semigroups?
This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I'm not entirely certain…
user23211
98
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
- 15,910
98
votes
13 answers
How to show that $\det(AB) =\det(A) \det(B)$?
Given two square matrices $A$ and $B$ , how do you show that $$\det(AB) = \det(A) \det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
Learner
- 2,814
98
votes
11 answers
Demystify integration of $\int \frac{1}{x} \mathrm dx$
I've learned in my analysis class, that $$ \int \frac{1}{x} \mathrm dx = \ln(x). $$ I can live with that, and it's what I use when solving equations like that. But how can I solve this, without knowing that beforehand. Assuming the standard rule for…
polemon
- 1,071
98
votes
1 answer
Why does "Turn! Turn! Turn!" equal 241217.524881?
If you search for "Turn! Turn! Turn!" on Google , then the second result is this YouTube video of The Byrds performing the Pete Seeger song of that name. But the first result is Google's internal calculator displaying "241217.524881". With a bit of…
tparker
- 7,023
98
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
- 19,199
- 3
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- 112
98
votes
6 answers
Can the product of infinitely many elements from $\mathbb Q$ be irrational?
I know there are infinite sums of rational values, which are irrational (for example the Basel Problem). But I was wondering, whether the product of infinitely many rational numbers can be irrational. Thank you for your answers.
Mister Set
- 1,247
98
votes
4 answers
Do numbers get worse than transcendental?
Mathematicians have come up with many ways of classifying how "exotic" some numbers are. For example, the most ordinary numbers are the natural "counting" number, and the next most exotic numbers are zero and the negative integers. Next are the…
Franklin Pezzuti Dyer
- 41,890
98
votes
3 answers
Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture
In the most recent numberphile video , Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows: If the hypothesis is undecidable, there is no proof it is false. If we…
Sriotchilism O'Zaic
- 2,410
98
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 sure…
Arpon Basu
- 1,241
98
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
- 875
98
votes
2 answers
What is the relation between rank of a matrix, its eigenvalues and eigenvectors
I am quite confused about this. I know that zero eigenvalue means that null space has non zero dimension. And that the rank of matrix is not the whole space. But is the number of distinct eigenvalues ( thus independent eigenvectos ) is the rank of…
Shifu
- 1,133
97
votes
7 answers
A Case Against the "Math Gene"
I'm currently teaching a mathematics course for elementary educators (think of it as math methods, but with less focus on methods and more focus on content). In a student's essay, I encountered the phrase "one is either a 'math person' or not". That…
Austin Mohr
- 26,442
97
votes
19 answers
Should I put number combinations like 1111111 onto my lottery ticket?
Suppose the winning combination consists of $7$ digits, each digit randomly ranging from $0$ to $9$. So the probability of $1111111$, $3141592$ and $8174249$ are the same. But $1111111$ seems (to me) far less likely to be the lucky number than…
arax
- 2,815
97
votes
4 answers
Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
Is the following true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. I guess that it is true but I am not sure how to prove…
gulim
- 1,013