Hot answers tagged

29

Set theory and category theory are both foundational theories of mathematics (they explain basics), but they attack different aspects of foundations. Set theory is largely concerned with "how do we build mathematical objects (or what could we build)" while category theory is largely concerned with "what structure to mathematical objects have (or could ...


20

$e$ appears in the number of derangements The formula for the number of derangements of length $n$ turns out to be $$n! \cdot \sum_{j=0}^n \frac{(-1)^j}{j!}$$ Since the second part is just the standard series for $e^{-1}$ this can also be written as $$\bigl[ \frac{n!}{e} \bigr]$$ where $[ . ]$ denotes the closest integer. This also implies that the ...


15

Mathematics expropriates many terms of ordinary language. Different branches of mathematics expropriate the same term in different ways. And where confusion really arises is where one branch of mathematics --- say, topology --- depends on another branch of mathematics --- say, set theory --- but those two branches use the term in different ways. Your word ...


13

We have $$e=\lim_{n\to\infty}\sqrt[\large^n]{\text{LCM}[1,2,3,\ldots,n]},$$ where LCM stands for least common multiple.


12

My favourite, also in the area of probability, is the secretary problem. Copied (with editing) from the Wikipedia site: The task is to hire the best of $n$ applicants for a position. The applicants are interviewed one by one in random order. A decision about each one must be made immediately after the interview. Once rejected, an applicant cannot be ...


9

An almost magical appearance of $e$ comes from Pascal's Triangle. Let $s_n$ be the product of the terms on the $n$-th row of the Pascal's Triangle, that is: $$s_n=\prod_{k=0}^n\binom{n}{k}$$ Then $$\lim_{n\to \infty}\frac{s_{n-1}s_{n+1}}{s_n^2}=e$$ A proof of this fact can be found here. I think it's one of the things that struck me the most about ...


8

Here's a nice (longish) one. A sequence of numbers $x_1,x_2,...$ is generated randomly from $[0,1]$. This process is continued so long as the sequence is monotonically increasing or monotonically decreasing. Q: What is the the expected length of the monotonic sequence? The probability that the length $L$ of the monotonic sequence is greater than $k$ is ...


7

I kind of think this is cheating, but Euler's Identity comes to mind: $$e^{i\pi}+1=0$$ This is a specific case of $e^{ix}=\cos x+i\sin x$ when $x=\pi$. Deriving the formula requires only a knowledge of the Taylor expansions of $e^x$, $\sin x$, and $\cos x$. I suppose this is not particularly surprising, but it reveals a very deep connection between ...


6

$e$ finds itself in formulas involving $\pi$. Ramanujan's constant $$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925\ldots \approx 640320^3+744$$ is related to Heegner numbers and has deep connections to number theory. The Gaussian integral $$\int_{-\infty}^{+\infty} e^{-x^2}\,dx = \sqrt{\pi}$$ is related to polar coordinates and thus Euler's ...


6

Here is a simple answer (perhaps too simple). Inside and outside are partitions of a space of some number of dimensions. The informal idea is that any two points "inside" a shape can be joined by a continuous (not necessarily straight line) that does not intersect with the shape's boundary. A point is said to be "outside" the shape if it cannot be connected ...


4

I strongly agree with Karl's and Björn's comments regarding the Latin orgin: spatium. See: Leonhard Euler, Mechanica sive motus scientia analytice exposita, Tomus I, Petropoli, 1736 : Propositio 4 [ page 13 ] Sit spatium $AM$, sive sit linea recta sive curva, $=s$, et celeritas, quam corpus habet in M sit $c$, quae erit functio quaedam ipsius $s$. Ab ...


4

In any field (well, at least in math and physics, the two fields I am interested in) one of the skills acquired over time by a serious researcher is a feel for what problems are open, what problems are interesting enough to be published, who is currently working on what, and so forth. In general, one doesn't have this fell until at least the second year of ...


3

The whole Section 1.2 of Hatcher's book is dedicated to van Kampen's theorem, where nonabelian fundamental groups are abundant. The more-or-less canonical example of the wedge sum $S^1 \vee S^1$ is in particular described, and I find the explanation in the introduction of the section rather clear. You have two loops, $a$ and $b$, and it's not possible to ...


3

As noted in other answers, mathematics use different notions of ''inside/outside'' or ''interior/exterior''. And probably none of them completely capture the meaning of the usual language. So, instead of starting from mathematical definitions, I try starting from the intuitive meaning of '' inside/outside''. It seems to me that the idea of being inside or ...


3

I don't know if this is relevant, but the fact that both $e$ and $\pi$ appear in the Gaussian function defining the normal distribution (which is very important in probability and in statistics) is something I find beautiful : $$f(x) = \frac{1}{\sqrt{2π}}e^{-\frac{x^2}{2}}$$ Another example : $e$ appears in the definition of the moment-generating function ...


3

I'm not in this area but I can say you that the main issue here is the application of this kind of Mathemathics to Quantum Mechanics. Indeed, even if Hilbert didn't started studying the argument with this in mind it was soon finded that this branch of mathemathics was really suitable to modelize Quantum phenomenas. Indeed what happened is that soon after ...


3

Pick non-overlapping pairs $(x,x+1)$ of integers from $[1,n]$ until no more pairs can be picked (i.e., until no more consecutive integers remain), and let $p$ be the number of integers that were picked (i.e. $w$ is twice the number of picked intervals). Then you have for the expected value $\mathbb{E}\ p$ of picked integers that $$ \lim_{n\to\infty} ...


2

You could look up some interesting topics such as Bernoulli trials, which use Euler's number to approximate probabilities involving large numbers, and Stirling's approximation which provides an approximation for factorials. I always liked the inequality rule that e is the only real number for which the following is true: $$\left(1+\frac{1}{x}\right)^x < ...


2

The graph of $$y=x^x, x>0$$ has minimum value $$y=\left(\frac 1e\right)^{\frac 1e}$$ when $x=\frac 1e$


2

Here's an idea based on the definition of roundness. Let $C$ be a simple closed curve in $\mathbb{A}^2$, let $E_{in}$ be the set of ellipses contained in the interior of $C$, and let $E_{out}$ be the set of ellipses contained in the exterior of $C$. Then you can define your 'ellipseness' to be $$ \sup\left\{\frac{\operatorname{area} ...


2

As already said in comments, it does not seem useful at all to memorize this. Using it does not prove that you know how to make it and there so many more important things to memorize in mathematics ! Let us do it simple $$k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}=\frac{u(x)}{j(x)}$$ $$k'(x)=\frac{u'(x)j(x)-u(x)j'(x)}{j^2(x)}$$ ...


2

To expand on Tomi's answer: when considering e.g. a general non-commutative binary operation, the concepts of left- and right-composition arise naturally from the asymmetry of the mathematical structure. The names we give to those concepts are dependent on language, writing conventions, accidents of history and a mathematical culture where new things are ...


2

With regard to intuition, imagine tying together two broom handles A,B by a rope going once round A and then once round B; this is not the same as, i.e. cannot be deformed into, going once round B and then round A. With regard to proof, I prefer to use the van Kampen theorem for the fundamental groupoid $\pi_1(X,S)$ on a set $S$ of base points found in ...


2

Your computation doesn't make any sense. Assuming that $n$ of the tosses were rigged doesn't mean that you're assigning a prior probability of $n/6$ to "not fair". If you're saying the only possibilities are "fair" and "not fair" and "not fair" means a 100% of Clinton winning all $6$ tosses (which is what your computation of "$P(6H)$" implies), then that ...


2

I agree with you that one good reason for using $\cos$ is that it corresponds well with the initial conditions for harmonic motion. Another good reason is experimental observation. If I come across a physical system that is oscillating, I will want to measure its amplitude and period. Examples include a swinging pendulum, a planet observed rotating around ...


2

Here are a few that I know of (with overlap of course). It's unclear how up-to-date they are. Douglas West's page: http://www.math.illinois.edu/~dwest/openp/ The Open Problem Garden: http://www.openproblemgarden.org/category/graph_theory Erdos' Problems on Graphs: http://www.math.ucsd.edu/~erdosproblems/


2

To some extent I think so. You don't have to love every sub-discipline in order to excel in some. However I don't think you should assume that you will suck on analysis for that reason. First of all you're probably required to take the Real analysis course and also some multi-dimensional analysis. Second there are parts of analysis that are very ...


1

Using the 5 constants of Euler's identity $ e^{i\pi} + 1 = 0 $ it is possible to include $ \varphi $ into an equation to give an identity containing six constants as follows: $$ e^{\frac{i\pi}{1+\varphi}} + e^{-\frac{i\pi}{1+\varphi}} + e^{\frac{i\pi}{\varphi}} + e^{-\frac{i\pi}{\varphi}} = 0 $$ See article and OEIS Sequence A193537


1

Wikipedia has a listing of open problems in graph theory.


1

Another reason is to use the initial position as a parameter. Compare $z=z_0\lambda^t$ and $h=h_0\cos \omega t$ You can't do the same with $\sin$ - at best you can say perhaps $d=d_{max}\sin \omega t$ By the way, this argument fails once you introduce a phase shift!



Only top voted, non community-wiki answers of a minimum length are eligible