Most Popular
1500 questions
125
votes
5 answers
How were 'old-school' mathematics graphics created?
I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today:
Does anybody know how this graphic was created? Were…
TSGM
- 1,263
125
votes
1 answer
How do we know an $ \aleph_1 $ exists at all?
I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exist no other cardinals between it and $ \aleph_0 $? (We would have to assume or derive the existence of such an object…
anon
- 155,259
125
votes
10 answers
How can I find the surface area of a normal chicken egg?
This morning, I had eggs for breakfast, and I was looking at the pieces of broken shells and thought "What is the surface area of this egg?" The problem is that I have no real idea about how to find the surface area.
I have learned formulas for…
yiyi
- 7,458
125
votes
11 answers
What does it mean to have a determinant equal to zero?
After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.
I hope someone can explain this to me in plain English.
user2171775
- 1,393
125
votes
13 answers
Do factorials really grow faster than exponential functions?
Having trouble understanding this. Is there anyway to prove it?
Billy Thompson
- 1,900
- 7
- 23
- 24
125
votes
26 answers
Examples of problems that are easier in the infinite case than in the finite case.
I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.
Asinomás
- 107,565
125
votes
13 answers
Why can't calculus be done on the rational numbers?
I was once told that one must have a notion of the reals to take limits of functions. I don't see how this is true since it can be written for all functions from the rationals to the rationals, which I will denote $f$, that
$$\forall…
Praise Existence
- 1,485
125
votes
7 answers
What remains in a student's mind
I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts…
Dubious
- 14,048
125
votes
13 answers
What are some interpretations of Von Neumann's quote?
John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics.
Did he mean that with experience and practice,…
124
votes
4 answers
Expectation of the maximum of gaussian random variables
Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large?
If $F$ is the cumulative distribution function…
Chris Taylor
- 29,755
124
votes
4 answers
What is the difference and relationship between the binomial and Bernoulli distributions?
How should I understand the difference or relationship between binomial and Bernoulli distribution?
user122358
- 2,832
124
votes
8 answers
probability $2/4$ vs $3/6$
Recently I was asked the following in an interview:
If you are a pretty good basketball player, and were betting on whether you could make $2$ out of $4$ or $3$ out of $6$ baskets, which would you take?
I said anyone since ratio is same. Any…
zephyr
- 1,069
124
votes
8 answers
Probability that a stick randomly broken in five places can form a tetrahedron
Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here.
Randomly break a stick in five places.
Question: What is the probability that the resulting…
Benjamin Dickman
- 16,038
124
votes
20 answers
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive…
marty cohen
- 110,450
124
votes
18 answers
How can I understand and prove the "sum and difference formulas" in trigonometry?
The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos(\alpha \pm \beta) &= \cos \alpha \cos…
Tyler
- 3,147