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1500 questions
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
- 4,139
72
votes
9 answers
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$
I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$…
user365
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
$$
\frac{d}{dt} \langle f, g \rangle…
ItsNotObvious
- 11,263
72
votes
2 answers
When does L' Hopital's rule fail?
This thought jumped out of me during my calculus teaching seminar.
It is well known that the classical L'Hospital rule claims that for the $\frac{0}{0}$ indeterminate case, we have:
$$
\lim_{x\rightarrow A}\frac{f(x)}{g(x)}=\lim_{x\rightarrow…
Bombyx mori
- 20,152
72
votes
1 answer
Fractal behavior along the boundary of convergence?
The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, emeritus of Reed College) showed me 15 years ago,…
2'5 9'2
- 56,991
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
11 answers
What is the difference between only if and iff?
I have read this question. I am now stuck with the difference between "if and only if" and "only if". Please help me out.
Thanks
user2857
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…
Daniel
- 3,133
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…
tom
- 3,277
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…
D R
- 1,168
72
votes
4 answers
Can there be two distinct, continuous functions that are equal at all rationals?
Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the…
Charles Stewart
- 4,697
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 that…
sam wolfe
- 3,465