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1500 questions
92
votes
9 answers
Why is the construction of the real numbers important?
There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers as a complete ordered field?
What's the importance…
user42912
- 23,582
92
votes
8 answers
Will it become impossible to learn math?
I was thinking about this today and it seems like a good question. Assuming mathematics will keep on expanding, do you think it will ever become impossible for a beginner to learn all the known material on a subject (such as mechanics), simply…
Ninja Boy
- 3,133
91
votes
17 answers
In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?
Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$":
$$\begin{array}{|c|c|c|}
\hline
p&q&p\implies q\\ \hline
T&T&T\\
T&F&F\\
F&T&T\\
F&F&T\\\hline
\end{array}$$
My understanding is that "$p\implies q$" means "when there…
user701510
- 1,053
91
votes
4 answers
Intuition behind using complementary CDF to compute expectation for nonnegative random variables
I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here) and understand its mechanics, but I'm having trouble understanding the intuition behind this formula or why it should be the case at all. Does…
bouma
- 1,135
91
votes
2 answers
Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$
Let
$$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6\,=\,\big(2+\sqrt{3}\big) \big(\sqrt{2} \sqrt[4]{27}-3\big)\,=\,\frac{3\sqrt{3}}{3+\sqrt2\ \sqrt[4]{27}}.\tag1$$
Note that $\alpha$ is the unique positive root of the polynomial…
Vladimir Reshetnikov
- 47,122
91
votes
5 answers
Is it possible to represent every huge number in abbreviated form?
Consider the following expression.
$1631310734315390891207403279946696528907777175176794464896666909137684785971138$
$2649033004075188224$
This is a $98$ decimal digit number.
This can be represented as $424^{37}$ which has just 5 digits.
or…
endrendum
- 233
91
votes
7 answers
Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix
The rotation matrix
$$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$
has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$. The real eigenvector of a 3d rotation…
Alf
- 2,597
91
votes
3 answers
Given this transformation matrix, how do I decompose it into translation, rotation and scale matrices?
I have this problem from my Graphics course. Given this transformation matrix:
$$\begin{pmatrix}
-2 &-1& 2\\
-2 &1& -1\\
0 &0& 1\\
\end{pmatrix}$$
I need to extract translation, rotation and scale matrices.
I've also have the answer (which is…
metavers
- 911
91
votes
15 answers
What are the practical applications of the Taylor Series?
I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series.
Question: What are the practical applications of the Taylor…
Rivasa
- 1,575
91
votes
3 answers
Cute Determinant Question
I stumbled across the following problem and found it cute.
Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant:
$$\left|
\begin{matrix}
2 & 3&0&2&8 \\
3 &…
Potato
- 40,171
91
votes
6 answers
Mathematically, why was the Enigma machine so hard to crack?
Mathematically, why was the Enigma machine so hard to crack?
In laymen terms, what was it exactly that made cracking the Enigma machine such a formidable task? Everything I have seen about the Enigma machine, from a general article to information…
Daniel W. Farlow
- 22,531
91
votes
9 answers
Proof that a Combination is an integer
From its definition a combination $\binom{n}{k}$, is the number of distinct subsets of size $k$ from a set of $n$ elements.
This is clearly an integer, however I was curious as to why the expression
$$\frac{n!}{k!(n-k)!}$$ always evaluates to an…
Akusete
- 1,013
91
votes
4 answers
How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?
How can one prove this identity?
$$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$
There is a formula for $\zeta$ values at even integers, but it involves Bernoulli numbers; simply plugging it in…
E.H.E
- 23,280
91
votes
7 answers
What is special about the numbers 9801, 998001, 99980001 ..?
I just saw this post, and realized that
1/9801…
Lazer
- 1,535
90
votes
6 answers
Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$?
Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$ ?
$$\mathbb{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbb{Q}\}$$
$$\mathbb{Q}(\sqrt{2}+\sqrt{3}) = \lbrace a+b(\sqrt{2}+\sqrt{3}) \mid a,b \in…
Tashi
- 1,593