Mathematics Stack Exchange
Mathematics Stack Exchange

Most Popular

more tags
1500 questions
84
votes
5 answers

Are all eigenvectors, of any matrix, always orthogonal?

I have a very simple question that can be stated without any proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand principal components and it is crucial for me to see the basis of eigenvectors.
Bober02
  • 2,546
84
votes
19 answers

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\mathbf{j}$, but could not solve if I have $3$…
niko
  • 959
84
votes
5 answers

Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
thyrgle
83
votes
13 answers

The Monty Hall problem

I was watching the movie $21$ yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around…
Avicinnian
  • 941
83
votes
13 answers

Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqrt{2 \pi}$. What is a good way of doing this? Could…
James
  • 1,029
  • 2
  • 9
  • 5
83
votes
0 answers

Complete, Finitely Axiomatizable, Theory with 3 Countable Models

Does there exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory with exacly $3$ models. This theory is not…
Primo Petri
  • 5,174
83
votes
2 answers

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} \sqrt{ \frac{1}{2}+\frac{1}{2}\sqrt{ \frac{1}{2}+…
Olivier Oloa
  • 120,989
83
votes
6 answers

Are there any examples of non-computable real numbers?

Is this true, that if we can describe any (real) number somehow, then it is computable? For example, $\pi$ is computable although it is irrational, i.e. endless decimal fraction. It was just a luck, that there are some simple periodic formulas to…
Dims
  • 1,149
83
votes
6 answers

Why is the determinant the volume of a parallelepiped in any dimensions?

For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true for any dimensions?
ahala
  • 3,020
83
votes
3 answers

Is there a function with a removable discontinuity at every point?

If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the…
Pete L. Clark
  • 97,892
83
votes
29 answers

General request for a book on mathematical history, for a VERY advanced reader.

I am aware that there are answered similar questions on here, however I am specifically after a text that would be engaging for a professor of mathematics, also Fellow of the Royal Society (FRS). He is unwell and in the hospital, and I would like to…
Kino
  • 979
  • 1
  • 7
  • 6
83
votes
8 answers

Finding a point along a line a certain distance away from another point!

Let's say you have two points, $(x_0, y_0)$ and $(x_1, y_1)$. The gradient of the line between them is: $$m = (y_1 - y_0)/(x_1 - x_0)$$ And therefore the equation of the line between them is: $$y = m (x - x_0) + y_0$$ Now, since I want another point…
Kel196
  • 999
83
votes
3 answers

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same degree as $f(x)$, the polynomial $f(x)$ is…
spin
  • 11,956
83
votes
12 answers

What are the applications of functional analysis?

I recently had a course on functional analysis. I was thinking of studying the mathematical applications of functional analysis. I came to know it had some applications on calculus of variations. I am not specifically interested in applications of…
gamma
  • 924
83
votes
2 answers

Is there really no way to integrate $e^{-x^2}$?

Today in my calculus class, we encountered the function $e^{-x^2}$, and I was told that it was not integrable. I was very surprised. Is there really no way to find the integral of $e^{-x^2}$? Graphing $e^{-x^2}$, it appears as though it should be.…
Zolani13
  • 1,761
1 2 3
…
99 100