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1500 questions
116
votes
19 answers
Is there a "positive" definition for irrational numbers?
An irrational number is a real number that cannot be expressed as a ratio of integers.
Is it possible to formulate a "positive" definition for irrational numbers?
A few examples:
An irrational number is a real number that can be expressed as...
An…
barak manos
- 43,109
116
votes
6 answers
What did Alan Turing mean when he said he didn't fully understand dy/dx?
Alan Turing's notebook has recently been sold at an auction house in London. In it he says this:
Written out:
The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find
extremely difficult to understand in spite of it having been the one I
…
Matta
- 1,658
115
votes
13 answers
In calculus, which questions can the naive ask that the learned cannot answer?
Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them.
Calculus is not known to be such a field, as far as I know. (For…
115
votes
5 answers
How to sum this series for $\pi/2$ directly?
The sum of the series
$$
\frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1}
$$
can be derived by accelerating the Gregory Series
$$
\frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2}
$$
using Euler's Series Transformation.…
robjohn
- 345,667
115
votes
18 answers
A good way to retain mathematical understanding?
What is a good way to remember math concepts/definitions and commit them to long term memory?
Background:
In my current situation, I'm at an undergraduate institution where I have to take a lot of core classes and easy math classes because there are…
010110111
- 2,161
- 2
- 17
- 22
115
votes
12 answers
Why are mathematical proofs that rely on computers controversial?
There are many theorems in mathematics that have been proved with the assistance of computers, take the famous four color theorem for example. Such proofs are often controversial among some mathematicians. Why is it so?
I my opinion, shifting from…
Gerard
- 4,264
115
votes
1 answer
Gross-Zagier formulae outside of number theory
(Edit: I have asked this question on MO.)
The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the derivative…
Bruno Joyal
- 54,711
115
votes
7 answers
Is black hole pattern possible in Conway's Game of Life that eats/clears everything?
Is black hole pattern possible in Conway's Game of Life that eats/clears the infinite universe plane ?
Formally, is there a pattern that satifies following requirements?
The pattern has finite size.
The universe plane has infinite size.
Each cell…
VainMan
- 1,735
115
votes
2 answers
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: $$[x,y^{-1},z]^y[y,z^{-1},x]^z[z,x^{-1},y]^x=1,$$ which holds for any…
Alexander Gruber
- 26,963
115
votes
4 answers
Element-wise (or pointwise) operations notation?
Is there a notation for element-wise (or pointwise) operations?
For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same length z, where $z_i = x_i * y_i$ .
In mathematical…
levesque
- 1,429
115
votes
25 answers
Can an infinite sum of irrational numbers be rational?
Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear combination of the other ?
By linear combination, we…
User1
- 1,841
115
votes
13 answers
Why, intuitively, is the order reversed when taking the transpose of the product?
It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$
and a nice way for me to remember this is the following sentence:
The opposite of putting on socks and shoes is taking the shoes off, followed…
user1337
- 24,381
115
votes
18 answers
Fastest way to meet, without communication, on a sphere?
I was puzzled by a question my colleague asked me, and now seeking your help.
Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and the sphere is way bigger than you two. There are no…
Rob Audenaerde
- 1,185
115
votes
11 answers
Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
I've been looking at
$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$
It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:
$$\displaystyle \int\limits_0^\infty {\frac{{{x^1}}}{{1 +…
Pedro
- 122,002
114
votes
5 answers
What is the term for a factorial type operation, but with summation instead of products?
(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)
I'm aware of Sigma notation, but is there a function/name for e.g.
$$ 4 + 3 + 2 + 1 \longrightarrow 10…
barfoon
- 1,359