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1500 questions
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
7 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
- 2,610
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
- 2,905
151
votes
8 answers
When is matrix multiplication commutative?
I know that matrix multiplication in general is not commutative. So, in general:
$A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$
But for some matrices, this equations holds, e.g. A = Identity or A = Null-matrix $\forall B \in…
Martin Thoma
- 10,157
151
votes
7 answers
lim sup and lim inf of sequence of sets.
I was wondering if someone would be so kind to provide a very simple explanation of $\limsup$ and $\liminf$ of a sequence of sets. For a sequence of subsets $A_n$ of a set $X$ we have $$\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n…
Comic Book Guy
- 5,898
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
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
149
votes
5 answers
On Ph.D. Qualifying Exams
Where can I find Ph.D. qualifying exams questions? Is there any website that keeps a collection of such problems?
I need it for doing some revision of the basic topics. I know of a book but that doesn't have the full collection.
Koushik
- 4,832
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
- 14,298
148
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
- 67,568
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- 617
147
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
146
votes
5 answers
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$?
This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I am unsure how they are relevant.
jaynp
- 2,241
146
votes
2 answers
Is there any mathematical reason for this "digit-repetition-show"?
The number $$\sqrt{308642}$$ has a crazy decimal representation : $$555.5555777777773333333511111102222222719999970133335210666544640008\cdots $$
Is there any mathematical reason for so many repetitions of the digits ?
A long block containing only…
Peter
- 86,576