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1500 questions
73
votes
13 answers
What is the definition of a set?
From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition . My question is what the definition of a set is? I have…
John Doe
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73
votes
3 answers
Is $ 0.112123123412345123456\dots $ algebraic or transcendental?
Let $$x=0.112123123412345123456\dots $$ Since the decimal expansion of $x$ is non-terminating and non-repeating, clearly $x$ is an irrational number. Can it be shown whether $x$ is algebraic or transcendental over $\mathbb{Q}$ ? I think $x$ is…
ASB
- 4,157
72
votes
2 answers
Differentiating an Inner Product
If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components shows that $$ \frac{d}{dt} \langle f, g \rangle =…
ItsNotObvious
- 11,361
72
votes
2 answers
When does L' Hopital's rule fail?
This thought jumped out of me during my calculus teaching seminar. It is well known that the classical L'Hospital rule claims that for the $\frac{0}{0}$ indeterminate case, we have: $$ \lim_{x\rightarrow A}\frac{f(x)}{g(x)}=\lim_{x\rightarrow…
Bombyx mori
- 20,238
72
votes
4 answers
Learning Roadmap for Algebraic Topology
I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, and also have knowledge on Homological Algebra(in…
Arsenaler
- 4,090
72
votes
18 answers
Why should we prove obvious things?
Obviously, there are obvious things in mathematics. Why we should prove them? Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$? Prove that $f(x)=x$ is continuous on $\mathbb{R}$? $\dotsc$ Just to list few examples.
x.y.z...
- 1,150
72
votes
4 answers
Proof that the irrational numbers are uncountable
Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable?
nkassis
- 871
72
votes
13 answers
What Is Exponentiation?
Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on this was expanded to include rational exponents, so…
baum
- 1,541
72
votes
4 answers
Areas versus volumes of revolution: why does the area require approximation by a cone?
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. However, if we want to find the surface area, then…
Eric O. Korman
- 20,165
72
votes
3 answers
Set of continuity points of a real function
I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable or…
Daniel
- 3,163
72
votes
11 answers
Why do we use a Least Squares fit?
I've been wondering for a while now if there's any deep mathematical or statistical significance to finding the line that minimizes the square of the errors between the line and the data points. If we use a less common method like LAD, where we just…
tom
- 3,297
72
votes
6 answers
Origin of the dot and cross product?
Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product…
Curiosity
- 1,638
72
votes
4 answers
Symbol for elementwise multiplication of vectors
This is a notation question. Assume one is given two vector $\mathbf{a}$ and $\mathbf{b}$, and one constructs a third vector $\mathbf{c}$ whose elements are given by $$c_k=a_k b_k$$ Is there any standard notation for this simple operation? Is the…
D R
- 1,168
72
votes
6 answers
What is the general equation of the ellipse that is not in the origin and rotated by an angle?
I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
andikat dennis
- 999
72
votes
7 answers
How to straighten a parabola?
Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$ . Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$ , how does one find the point $(x_1,y_1)$ so…
sam wolfe
- 3,583