Most Popular
1500 questions
95
votes
4 answers
How do we know that Cantor's diagonalization isn't creating a different decimal of the same number?
Edit: As the comments mention, I misunderstood how to use the diagonalization method. However, the issue I'm trying to understand is a potential problem with diagonalization and it is addressed in the answers so I will not delete the question.…
Hugh
- 2,351
95
votes
24 answers
What is a real-world metaphor for irrational numbers?
In an effort to develop better number sense (and to create my own journey from fish to infinity ), I have been going through Khan Academy's math material from the very beginning. So far, I have been able to develop strong intuitions and metaphors…
jds
- 2,336
95
votes
2 answers
A goat tied to a corner of a rectangle
A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze? Well, it seems like the goat can turn a full circle of radius 8…
space
- 4,899
95
votes
14 answers
I roll a die repeatedly until I get 6, and then count the number of 3s I got. What's my expected number of 3s?
Consider the following experiment. I roll a die repeatedly until the die returns 6, then I count the number of times 3 appeared in the random variable $X$. What is $E[X]$? Thoughts: I expect to roll the die 6 times before 6 appears (this part is…
nettle
- 1,289
- 1
- 12
- 14
95
votes
7 answers
Intuition in algebra?
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) MacLane and Birkhoff's Algebra on my own. The…
Michael Benfield
- 1,875
95
votes
1 answer
Is there an easy way to show which spheres can be Lie groups?
I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it involved de Rham cohomology, but I don't really know…
Aaron Mazel-Gee
- 8,369
94
votes
10 answers
Surprise exam paradox?
I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows: A maths teacher says to the class that during the year he'll give a surprise exam, so…
sashoalm
- 1,419
94
votes
4 answers
Does every Abelian group admit a ring structure?
Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive: \begin{align*} &a * (b * c) = (a*b) * c \\ &a * (b + c) = a * b + a * c \\ &(a + b) * c…
Mikko Korhonen
- 26,406
94
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
- 5,354
94
votes
1 answer
The Right Triangle Game
I am looking for a deeper understanding, particularly the optimum strategy and the maximum score as a function of grid size, of the following (single-player) game played with an $n$ by $m$ grid : ($6 \times 6$ example) Rules Start with a grid made…
user139000
94
votes
6 answers
Why are differentiable complex functions infinitely differentiable?
When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn't hold for functions from $\mathbb R ^2$ to $\mathbb R ^2$. Can anyone explain…
Casebash
- 9,497
94
votes
4 answers
A continuous, nowhere differentiable but invertible function?
I am aware of a few example of continuous, nowhere differentiable functions. The most famous is perhaps the Weierstrass functions $$W(t)=\sum_k^{\infty} a^k\cos\left(b^k t\right)$$ but there are other examples, like the van der Waerden functions, or…
levitopher
- 2,705
94
votes
7 answers
Compact sets are closed?
I feel really ignorant in asking this question but I am really just don't understand how a compact set can be considered closed. By definition of a compact set it means that given an open cover we can find a finite subcover the covers the…
InsigMath
- 2,153
94
votes
8 answers
Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?
Circular manholes are great because the cover can not fall down the hole. If the hole were square, the heavy metal cover could fall down the hole and kill some man working down there. Circular manhole: Can manholes be made in other shapes than…
hpekristiansen
- 860
94
votes
5 answers
Derivative of the inverse of a matrix
In a scientific paper, I've seen the following $$\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$$ where $K$ is a $n \times n$ matrix that depends on $p$ . In my calculations I would have done the following $$\frac{\delta…
Sara
- 1,157