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1500 questions
69
votes
7 answers
Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?
Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
Clayton Kershaw
- 693
69
votes
11 answers
Average Distance Between Random Points on a Line Segment
Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?
Kenshin
- 2,240
69
votes
12 answers
Is it morally right and pedagogically right to google answers to homework?
This is a soft question that I have been struggling with lately. My professor sets tough questions for homework (around 10 per week). The difficulty is such that if I attempt the questions entirely on my own, I usually get stuck for over 2 hours per…
yoyostein
- 20,598
69
votes
12 answers
How is the derivative truly, literally the "best linear approximation" near a point?
I've read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving approximate way, but I've recently read that: " Some…
jeremy radcliff
- 4,973
69
votes
5 answers
Are there an infinite number of prime numbers where removing any number of digits leaves a prime?
Suppose for the purpose of this question that number $1$ is a prime number. Consider the prime number $311$. If we remove one $1$ from the number we arrive at the number $31$ which is also prime. If we removed $3$ instead of $1$ we would arrived at…
Farewell
- 5,046
69
votes
7 answers
How do we know the ratio between circumference and diameter is the same for all circles?
The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same for every circle? How do we know that $\pi = {C\over…
Happy
- 1,463
69
votes
6 answers
Proving the existence of a proof without actually giving a proof
In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. the celebrated probabilistic method and many things…
Damian Reding
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69
votes
12 answers
What are the Laws of Rational Exponents?
On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number seems to be shown as equal to its opposite…
Daniel R. Collins
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69
votes
6 answers
Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only "unlikely" or that the probability approaches zero or…
regularmike
- 703
69
votes
5 answers
Adding two polar vectors
Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?
lash
- 799
69
votes
17 answers
Why does the derivative of sine only work for radians?
I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm discovering that I've been wrong all along! I'm guessing…
Kyle Delaney
- 1,451
- 1
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69
votes
6 answers
Difference between topology and sigma-algebra axioms.
One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically…
Creator
- 3,228
69
votes
9 answers
Do commuting matrices share the same eigenvectors?
In one of my exams I'm asked to prove the following Suppose $A,B\in \mathbb R^{n\times n}$ , and $AB=BA$ , then $A,B$ share the same eigenvectors. My attempt is let $\xi$ be an eigenvector corresponding to $\lambda$ of $A$ , then $A\xi=\lambda\xi$ ,…
Vim
- 14,051
69
votes
3 answers
Is Stokes' Theorem natural in the sense of category theory?
Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = \int_{\partial M} \omega.$$ Doesn't that look like…
Martin Brandenburg
- 183,887
69
votes
6 answers
Chance of meeting in a bar
Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet? I am mainly interested to see how people would model this formally. I…
Beltrame
- 3,186