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1500 questions
101
votes
4 answers
A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational
Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational.
What I have tried:
Denote $x^n=r$ and $(x+1)^n=s$ with $r$, $s$ rationals. For each $k=0,1,\ldots,…
Dan Ismailescu
- 1,193
101
votes
10 answers
Given a die, what is the probability that the second roll of a die will be less than the first roll?
If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?
Salman Paracha
- 1,161
100
votes
28 answers
"Simple" beautiful math proof
Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
user138276
- 177
100
votes
8 answers
Union of two vector subspaces not a subspace?
I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
NSjonas
- 1,167
100
votes
15 answers
math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?
I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is:
$$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
newcomer
- 973
100
votes
18 answers
Can the golden ratio accurately be expressed in terms of $e$ and $\pi$
I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$
And so, I took it upon myself to try to find a way to express the golden ratio in terms of the infamous values, $\large\pi$ and $\large e$
The closest…
Nick
- 6,804
100
votes
14 answers
What's the difference between $\mathbb{R}^2$ and the complex plane?
I haven't taken any complex analysis course yet, but now I have this question that relates to it.
Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and we have a function $z=f(x,y)=\cos(x)+\sin(y)$.…
Cancan
- 2,747
100
votes
4 answers
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
It's not difficult to show that
$$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$
On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing terms gives us
$$\sum_{k=0}^n…
Skatche
- 1,522
100
votes
11 answers
What's so special about standard deviation?
Equivalently, about variance?
I realize it measures the spread of a distribution, but many other metrics could do the same (e.g., the average absolute deviation). What is its deeper significance? Does it have
a particular geometric interpretation…
aris
- 1,335
100
votes
19 answers
Anecdotes about famous mathematicians or physicists
I'm not sure whether this question suits this website, however, I don't know where else I could ask it. It is no mathematical problem or something similar, still I hope it won't be closed.
A few weeks ago, our assistant professor in physics told us…
Huy
- 6,674
100
votes
11 answers
Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?
In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object, such as a number, function, algorithm, or even…
lone student
- 14,709
100
votes
5 answers
Why are gauge integrals not more popular?
A recent answer reminded me of the gauge integral, which you can read about here.
It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue integrable, it is gauge integrable. (EDIT - as Qiaochu Yuan…
Chris Brooks
- 7,424
100
votes
7 answers
Is the "determinant" that shows up accidental?
Consider the class of rational functions that are the result of dividing one linear function by another:
$$\frac{a + bx}{c + dx}$$
One can easily compute that, for $\displaystyle x \neq \frac cd$
$$\frac{\mathrm d}{\mathrm dx}\left(\frac{a + bx}{c +…
stats_model
- 1,071
100
votes
3 answers
How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?
We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given
$$\int\limits_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1}…
user505183
100
votes
31 answers
Non-textbook Math book recommendation to read to my kids
I'm looking for a book to read to my kids.
NOT a kids book, but not too mature for a kid. My youngest kid that reads with me is 6 and the eldest is 10.
I'm looking for a book that is good literature, and is hard to put down.
Also maybe a book that…
Jeremy Campbell
- 1,097