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1500 questions
123
votes
8 answers
Finite subgroups of the multiplicative group of a field are cyclic
In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by saying "Since $G$ is the direct product of its Sylow…
QETU
- 1,247
123
votes
16 answers
Expected Number of Coin Tosses to Get Five Consecutive Heads
A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
leava_sinus
- 1,343
123
votes
12 answers
Is there an "inverted" dot product?
The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$ What about the quantity? $$\mathbf{a} \star \mathbf{b} = \prod_{i=1}^{n}…
doc
- 1,327
123
votes
10 answers
Motivation for the rigour of real analysis
I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far. One thing I feel I am lacking in…
1729
- 2,225
123
votes
5 answers
Finding a primitive root of a prime number
How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
user27617
123
votes
15 answers
Why rationalize the denominator?
In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there is a bias against roots in the denominator of a…
Reinstate Monica
- 5,377
122
votes
3 answers
Why are infinitely dimensional vector spaces not isomorphic to their duals?
Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the vector space $V=\mathbb F^{(\kappa)}$ (that is an…
Asaf Karagila
- 408,303
122
votes
16 answers
Is 10 closer to infinity than 1?
This may be considered a philosophical question but is the number "10" closer to infinity than the number "1"?
termsofservice
- 1,059
122
votes
7 answers
Strategies for Effective Self-Study
I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied mathematics. In particular, I am interested in…
ItsNotObvious
- 11,361
122
votes
10 answers
The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)
In general, the indefinite integral of $x^n$ has power $n+1$. This is the standard power rule. Why does it "break" for $n=-1$? In other words, the derivative rule $$\frac{d}{dx} x^{n} = nx^{n-1}$$ fails to hold for $n=0$. Is there some deep reason…
Shuheng Zheng
- 987
122
votes
12 answers
Is there a domain "larger" than (i.e., a supserset of) the complex number domain?
I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and multiplication always generate natural numbers,…
user1324
121
votes
18 answers
Good book for self study of functional analysis
I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic functional analysis so that I can be better prepared…
EVK
- 1,369
121
votes
9 answers
Is $[0,1]$ a countable disjoint union of closed sets?
Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
Kevin Ventullo
- 3,862
121
votes
4 answers
What are the Axiom of Choice and Axiom of Determinacy?
Would someone please explain: What does the Axiom of Choice mean, intuitively? What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice? as simple words as possible ? From what I've gathered from the…
user541686
- 14,366
121
votes
2 answers
$A$ and $B$ disjoint, $A$ compact, and $B$ closed implies there is positive distance between both sets.
Claim: Let $X$ be a metric space. If $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed, then there is $\delta>0$ so that $ |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$ . Proof . Assume the contrary. Let $\alpha_n\in…
Benji
- 6,230