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1500 questions
66
votes
16 answers

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in particular "zeroes" and "vertical asymptotes", I…
user641
66
votes
2 answers

Conway's "Murder Weapon"

The following quote is an excerpt from an interview with John Conway: Coxeter came to Cambridge and he gave a lecture, then he had this problem for which he gave proofs for selected examples, and he asked for a unified proof. I left the…
Oiler
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66
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7 answers

$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee

So one of the question on the MIT Integration Bee has baffled me all day today $$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$$ I have tried a variety of things to do this, starting with Integration By Parts Part 1 $$\frac{\tan…
Teh Rod
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66
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
  • 2,152
66
votes
18 answers

Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every 3rd term as positive, thus it would…
Kristaps John Balodis
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66
votes
2 answers

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty p_k = 4\pi^2$$ where $n\#$ is a primorial, and $p_k$…
Timmy Turner
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66
votes
6 answers

Fourier transform of function composition

Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually? I know you can do this for the sum, the product and the convolution of two functions. But I…
MathematicalOrchid
  • 6,185
66
votes
10 answers

Symbol for "probably equal to" (barring pathology)?

I am writing lecture notes for an applied statistical mechanics course and often need to express the notion that something is very probably true for functional forms found in the wild, without launching into a full digression for pathological…
mewahl
  • 719
66
votes
13 answers

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the…
wythagoras
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66
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
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65
votes
4 answers

taylor series of $\ln(1+x)$?

Compute the taylor series of $\ln(1+x)$ I've first computed derivatives (up to the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ $f^{''''}(x) = \frac{-6}{(1+x)^4}$ Therefore the…
Minu
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65
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18 answers

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend these numbers to instances of owing other people money…
user164061
  • 831
65
votes
4 answers

Similar matrices have the same eigenvalues with the same geometric multiplicity

Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities. Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb R$ or $\mathbb C$. We say $A$ and $B$ are…
user2723
65
votes
5 answers

How to prove that a compact set in a Hausdorff topological space is closed?

How to prove that a compact set $K$ in a Hausdorff topological space $\mathbb{X}$ is closed? I seek a proof that is as self contained as possible. Thank you.
Elias Costa
  • 14,658
65
votes
7 answers

Why does the volume of the unit sphere go to zero?

The volume of a $d$ dimensional hypersphere of radius $r$ is given by: $$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$$ What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any fixed $r$. How can this be? For fixed…
probabilityislogic
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