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1500 questions
122
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
- 1,317
122
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
- 2,205
122
votes
11 answers
Is zero odd or even?
Some books say that even numbers start from $2$ but if you consider the number line concept, I think zero($0$) should be even because it is in between $-1$ and $+1$ (i.e in between two odd numbers). What is the real answer?
mvar2011
- 1,371
122
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
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
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
- 14,298
121
votes
21 answers
How do you explain to a 5th grader why division by zero is meaningless?
I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing division by zero?
Shubh Khandelwal
- 1,275
121
votes
1 answer
Lebesgue measure theory vs differential forms?
I am currently reading various differential geometry books. From what I understand differential forms allow us to generalize calculus to manifolds and thus perform integration on manifolds. I gather that it is, in general, completely distinct from…
sonicboom
- 10,273
- 15
- 54
- 87
121
votes
10 answers
Good 1st PDE book for self study
What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic solutions techniques as well as some basic theory.
Mykie
- 7,245
121
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…
Shuheng Zheng
- 977
121
votes
6 answers
How Do You Actually Do Your Mathematics?
Better yet, what I'm asking is how do you actually write your mathematics?
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed…
Uticensis
- 3,381
121
votes
39 answers
What's your favorite proof accessible to a general audience?
What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly $5 \pm\epsilon$ minutes.
Let's define…
userX
- 2,039
120
votes
2 answers
Determinant of a non-square matrix
I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby undermined the entire answer. However, it can be…
goblin GONE
- 69,385
120
votes
7 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
- 1,209
120
votes
15 answers
The math behind Warren Buffett's famous rule – never lose money
This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first.
Warren Buffett famously provided 2 rules to investing:
Rule No. 1: Never lose money. Rule No. 2: Never…
