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1500 questions
151
votes
15 answers
Why is the volume of a cone one third of the volume of a cylinder?
The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$ , which is exactly one third the volume of the smallest cylinder that it fits inside. This can be proved easily by considering a cone as a solid of revolution , but I…
bryn
- 10,135
151
votes
15 answers
Are "if" and "iff" interchangeable in definitions?
In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions means "iff". For example I am reading "Essential…
fiftyeight
- 2,757
151
votes
4 answers
Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that if a function is in two $L^p$ spaces, (e.g. $p_1$…
user1736
- 9,083
151
votes
6 answers
Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?
I was looking at a list of primes. I noticed that $ \frac{AM (p_1, p_2, \ldots, p_n)}{p_n}$ seemed to converge. This led me to try $ \frac{GM (p_1, p_2, \ldots, p_n)}{p_n}$ which also seemed to converge. I did a quick Excel graph and regression and…
Soham
- 1,573
151
votes
8 answers
Apparently sometimes $1/2 < 1/4$?
My son brought this home today from his 3rd-grade class. It is from an official Montgomery County, Maryland mathematics assessment test: True or false? $1/2$ is always greater than $1/4$ . Official answer: false Where has he gone wrong? Addendum ,…
SDiv
- 2,684
151
votes
12 answers
Why do we use the word "scalar" and not "number" in Linear Algebra?
During a year and half of studying Linear Algebra in academy, I have never questioned why we use the word "scalar" and not "number". When I started the course our professor said we would use "scalar" but he never said why. So, why do we use the word…
LiziPizi
- 2,915
151
votes
6 answers
Why does mathematical convention deal so ineptly with multisets?
Many statements of mathematics are phrased most naturally in terms of multisets. For example: Every positive integer can be uniquely expressed as the product of a multiset of primes. But this theorem is usually phrased more clumsily, without…
MJD
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150
votes
6 answers
What's 4 times more likely than 80%?
There's an 80% probability of a certain outcome, we get some new information that means that outcome is 4 times more likely to occur. What's the new probability as a percentage and how do you work it out? As I remember it the question was posed like…
Jim
- 1,251
150
votes
14 answers
Why do units (from physics) behave like numbers?
What are units (like meters $m$, seconds $s$, kilogram $kg$, …) from a mathematical point of view? I've made the observation that units "behave like numbers". For example, we can divide them (as in $m/s$, which is a unit of speed), and also square…
user377104
150
votes
12 answers
Are we allowed to compare infinities?
I'm in middle school and had a question (my dad is helping me with formatting). We're learning about infinity in math class and there are a lot of problems like how it's not a number and how if you add one to infinity it doesn't change value. But…
Alice
- 1,347
148
votes
6 answers
Can someone explain the math behind tessellation?
Tessellation is fascinating to me, and I've always been amazed by the drawings of M.C.Escher, particularly interesting to me, is how he would've gone about calculating tessellating shapes. In my spare time, I'm playing a lot with a series of…
ocodo
- 1,785
148
votes
23 answers
The Best of Dover Books (a.k.a the best cheap mathematical texts)
Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books are not: For example, while something like…
Three
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148
votes
11 answers
Physical meaning of the null space of a matrix
What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it. E.g.: I think of the rank $r$ of a matrix as the minimum number…
user541686
- 14,366
148
votes
0 answers
If polynomials are almost surjective over a field, is the field algebraically closed?
Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. That is, for all but finitely many $a\in K$,…
Eric Wofsey
- 344,856
148
votes
9 answers
Normal subgroup of prime index
Generalizing the case $p=2$ we would like to know if the statement below is true. Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Sigur
- 6,680