Most Popular
1500 questions
139
votes
9 answers
Continuity of the roots of a polynomial in terms of its coefficients
It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed degree n (maybe monic? seems like that shouldn't…
Harry Altman
- 4,993
139
votes
10 answers
What's the difference between theorem, lemma and corollary?
Can anybody explain me what is the basic difference between theorem, lemma and corollary?
We have been using it for a long time but I never paid any attention. I am just curious to know.
HPS
- 4,706
139
votes
6 answers
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$, but in $\mathbb H$, ${\bf i}\cos x + {\bf…
MJD
- 67,568
- 43
- 308
- 617
138
votes
10 answers
Learning Lambda Calculus
What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus?
Specifically, I am interested in the following areas:
Untyped lambda calculus
Simply-typed lambda calculus
Other typed lambda…
Noldorin
- 6,788
138
votes
30 answers
Proofs of AM-GM inequality
The arithmetic - geometric mean inequality states that
$$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$
I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone…
Michiel Van Couwenberghe
- 2,847
138
votes
2 answers
How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$
How can one prove that
$$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
larry
- 1,529
137
votes
8 answers
What is the difference between independent and mutually exclusive events?
Two events are mutually exclusive if they can't both happen.
Independent events are events where knowledge of the probability of one doesn't change the probability of the other.
Are these definitions correct? If possible, please give more than one…
Adnan Ali
- 1,543
137
votes
35 answers
Examples of mathematical results discovered "late"
What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple that it's surprising no one thought of it…
Hrothgar
- 639
137
votes
28 answers
What are some examples of notation that really improved mathematics?
I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational conventions that simplify problem statements and ideas,…
Patrick
- 2,126
137
votes
26 answers
In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?
I am studying entailment in classical first-order logic.
The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is:
$$\begin{array}{|c|c|c|}
\hline
p&q&p\Rightarrow q\\…
Ethan
- 1,489
137
votes
14 answers
How do I get the square root of a complex number?
If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
Macha
- 1,563
137
votes
7 answers
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?
Are there some…
AlexBrand
- 345
137
votes
3 answers
Expected time to roll all $1$ through $6$ on a die
What is the average number of times it would it take to roll a fair $6$-sided die and get all numbers on the die? The order in which the numbers appear does not matter.
I had this questions explained to me by a professor (not math professor), but…
eternalmatt
- 1,535
137
votes
9 answers
Is there an integral that proves $\pi > 333/106$?
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves $\pi > 333/106$?
anon
137
votes
5 answers
Help find hard integrals that evaluate to $59$?
My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$.
I want to try and make a definite integral that equals $59$. So far I can only think of ones that are easy to…
Argon
- 25,971