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1500 questions
63
votes
4 answers
Soviet Russian mathematics books
The introductory part of this book briefly describes the popularity of mathematics in Soviet Russia. It touches on Russian mathematical circles and generally how society in Russia took to mathematics in a good way. A particular passage caught my…
seeker
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63
votes
4 answers
Parabola is an ellipse, but with one focal point at infinity
While I was reading about conic sections, I came across the following statement:
A parabola is an ellipse, but with one focal point at infinity.
But it is not clear to me. Can someone explain it clearly?
Kumar
- 2,319
63
votes
6 answers
Area covered by a constant length segment rotating around the center of a square.
This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it's somehow relevant to anything or useless.
I describe my thoughts with the following image:
What would the area of…
user136800
63
votes
16 answers
What is a good book for learning math, from middle school level?
Which books are recommended for learning math from the ground up and review the basics - from middle school to graduate school math?
I am about to finish my masters of science in computer science and I can use and understand complex math, but I…
user16974
63
votes
20 answers
Coin flipping probability game ; 7 flips vs 8 flips
Your friend flips a coin 7 times and you flip a coin 8 times; the person who got the most tails wins. If you get an equal amount, your friend wins.
There is a 50% chance of you winning the game and a 50% chance of your friend winning.
How can I…
Jason
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63
votes
3 answers
Every power series is the Taylor series of some $C^{\infty}$ function
Do you have some reference to a proof of the so-called Borel theorem, i.e. every power series is the Taylor series of some $C^{\infty}$ function?
user365
63
votes
5 answers
Every nonzero element in a finite ring is either a unit or a zero divisor
Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
rupa
- 713
63
votes
4 answers
What's the probability that a sequence of coin flips never has twice as many heads as tails?
I gave my friend this problem as a brainteaser; while her attempted solution didn't work, it raised an interesting question.
I flip a fair coin repeatedly and record the results. I stop as soon as the number of heads is equal to twice the number of…
Elliott
- 4,244
63
votes
18 answers
Which is larger? $20!$ or $2^{40}$?
Someone asked me this question, and it bothers the hell out of me that I can't prove either way.
I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas…
Alec
- 4,174
63
votes
14 answers
How do I motivate myself to do math again?
I have been thinking of asking for help for a few months now but posting in a public forum like this is intimidating.
Still, I am currently in a university studying mathematics as an undergrad. I took quite a few knocks a few months back when I…
Dust
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63
votes
4 answers
A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$
How can we prove that:
$$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function.
The best I could do was to express it in terms of Euler Sums. Let $I$ denote…
Shobhit
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63
votes
23 answers
Interesting results easily achieved using complex numbers
I was just looking at a calculus textbook preparing my class for next week on complex numbers. I found it interesting to see as an exercise a way to calculate the usual freshman calculus integrals $\int e^{ax}\cos{bx}\ dx$ and $\int e^{ax}\sin{bx}\…
Adrián Barquero
- 15,454
63
votes
4 answers
Are there open problems in Linear Algebra?
I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory.
There are a lot of open problems and conjectures in…
Mebat
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63
votes
5 answers
Is there a rational number between any two irrationals?
Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such a rational number?
[I posted this only so that the…
MJD
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63
votes
3 answers
$5^n+n$ is never prime?
In the comments to the question: If $(a^{n}+n ) \mid (b^{n}+n)$ for all $n$, then $ a=b$, there was a claim that $5^n+n$ is never prime (for integer $n>0$).
It does not look obvious to prove, nor have I found a counterexample.
Is this really…
Aryabhata
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