I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does anyone have any ideas on ways to focus the essay on a section of the topic infinity.
122791
I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.  Are there any practical applications for the concept of infinity? Is it a useful concept in maths at all?  I know that Donald E. Knuth has argued that for all practical purposes, a very, very large number has the same effect as infinity, in his book "Things a Computer Scientist Rarely Talks About" (can't remember the exact quote, nor find it online, unfortunately).  Examples are appreciated.
76438
I was wondering if some one here would please recommend an abstract algebra text book. In particular, I am seeking a text from among one of these:  1) Abstract Algebra by Dummit and Foote  2) Algebra by Michael Artin  3) Topics in Algebra by Herstein.  I was wondering which text is best for a high school student and has great exposition and good problems. My goal is study some algebraic number theory some time later in the near future as I liked elementary number theory.
99113
Consider the question   Asymptotic notation questions have been pretty standard for Stack Overflow.  This fairly interesting beginner question is up, but has already accumulated a couple of 'close' votes.  One of them is for 'off topic' but, possibly because I'm a theory guy, it seems very much on topic for programming and related topics.  You could argue for "computer science" potentially, but it would seem this, like questions on, oh, NP-complete problems and such, are things serious professional programmers should know and are interested in.  You could also make a case for mathematics and event the theory stack, although it's probably a little light for theory.  But with all of these separate categories, there seems to me to be a real risk that interesting questions will be modded/closed/moved until the OP loses interest.  Perhaps we've gotten a little too compartmentalized?
86426
The title says it all (I'm referring to the case, when writing for example an article to be published or something similar - something where the writing itself should also be of quality).  Example: "Let $A_1,A_2,\ldots,A_n$ be subsets of a set $L$ and $f:L\rightarrow L$. Then define $B=\cup_{i=0}^n A_i$ and $C=\cup_{i=1}^{\infty} f^i (A_2)$, where $f^i$ is $f$ iterated with itself $i$ times."  So (in this example), would it be a problem to use $i$ twice as an index variable the ranges over different sets ? Or is it ok to think of $i$ as being define like a (lets do a computer-science analogy) "local variable", that is only used to create the object "in" which it is defined ($B$ for example) and after the object is created, the variable is "freed" again (so it can be used again to define $C$) ?  If it is NOT okay to do that, when is it allowed to use the same symbol again, after its first use ? After half a page ? After two pages ? In index-heavy proofs one would very soon have finished all letters commonly used as indices, of one would always have to use a different letter. And indexing sets with letters like $x$ or $f$ seems even worse style to me (ok, maybe there are areas in math, in which this would be common practice, but I don't know any).
61538
From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one...
54904
I've been lurking on this site for several months, and as someone studying higher mathematics independently (i.e., outside of a college/institutional setting), this forum has probably been the best reference for gleaning general information about college-level math.  However, recently I've come to realize that few of the major/popular math textbooks tend to have solutions to the exercises contained within them or available for free/online. As someone without access to a professor or peers who can check the accuracy of my answers, working through book solutions becomes very tedious. How do I know if I am getting the right answers and thus understanding the material? I know people are encouraged to post here for help with challenging problems, but it doesn't seem appropriate to post in a forum such as this to simply check the veracity of answers. In addition, certain books (e.g., Pugh's Real Mathematical Analysis) contain hundreds of exercises throughout the text, and it would be neither plausible nor acceptable to constantly ask for help with problem after problem for an entire textbook.  I am not in school nor do I plan to enroll anytime soon, so there's not really an option of waiting until college to talk things over with someone more knowledgeable about mathematics. I am learning math out of pure curiosity and not in preparation for future classes or career goals, and while there are a plethora of resources available to the independent student of mathematics now more than ever before, being able to assess how well the material is being absorbed remains a difficult obstacle to overcome.
9936
Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet."  I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got me thinking... Is this really true?  Of course, I've learned that dealing with infinity can be tricky, but my intuition says that time is countably infinite while the number of works the monkeys could produce is uncountably infinite.  Therefore, it isn't necessarily given that the monkeys would write Hamlet.  Could someone who's better at this kind of math than me tell me if this is correct?  Or is there more to it than I'm thinking?
99994
I'm writing an article on  and some people pointed out that this is completely useless.  My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.  I came with some ideas like the Turing machine but I think I'm not grasping the right examples.  Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'?
108212
I'm having a look at analysis right now, and I just thought up this question after reading about the comparison test.  Does there exist a "critical" infinite sum of real numbers (which is divergent) such that if $S_n$ is any infinite sum for which the terms in $S_n$ are all less than the terms in this critical sum for $n &gt; N_0$ for some finite $N_0$, then $S_n$ converges?   I know my terminology is somewhat incorrect.
76183
Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me.  I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - \frac{1}{\infty} = 1$ as well? Because $\frac{1}{\infty} $ would be infinitely close to $0$, perhaps as $1^{-\infty}$?  So $1 - 1^{-\infty}$, or $\frac{1}{\infty}$ would be equivalent to $0.999 \cdots$? Or am I missing something? Is infinity something that can even be used in this sort of mathematics?
56789
I recently reviewed a paper (general applied mathematics context). During the review, I realized that the fairly general ideas introduced in the paper can potentially be applied to a specific problem in a similar context. The authors did not mention this application in their paper, so I suggested to add a short comment about it in my review.  I later started thinking about the application in more detail and tried out some things, which actually seem to work quite well for this application. Now, the paper is still in the review process and might take some time until it gets published. What is the proper way how to proceed here in order to use the ideas and results which I in my exclusive position as a reviewer saw? How can I refer to the paper that is still under review? How can I best discuss the ideas with my co-workers (and potential co-authors) without disclosing that I reviewed the paper?  The paper under review is not available online on any preprint servers such as arXiv etc
109706
I always thought that the phrase "note that" in a mathematical writing is simply something you say to express that what comes after "note that" is a fact. Due to its neutrality and versatility, I use it a lot, without even thinking the possible sentiments that might possibly make some people get me wrong. Moreover, many times we have to prevent symbols to appear directly after a comma or a period, so "note that" is convenient in this aspect too.   However, some incidents lead me to presume that perhaps those readers who are not that mathematically experienced could take "note that" as an imperative or even a condescending sign? This is much to my surprise. So, if I am writing something to someone that is senior than me in academia and that is not that mathematically experienced (say an applied engineering scholar whose math background is all from the school curriculum), would it be suggested that I avoid using "note that"? Thank you.
20567
There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are there more numbers in one range as opposed to the other?  What about $[0,1]$ and $[0, \infty)$ ?  I'd assume this is akin to how limits work in that the limit as $x\to\infty$ of $\frac{x^x}{\log(x)}$ is $\infty$, even though both the numerator and denominator approach infinity.
15479
It seems there are a lot of respected physicists appearing on pop-sci programs (discovery channel, science channel, etc.) these days spreading the gospel of   Three examples, quickly: 1) Many programs feature Michio Kaku saying that he is on a quest to find an equation, "perhaps just one inch long," which will "describe the whole universe."  2) Max Tegmark has come out with  in which he expresses the gut feeling that "nothing is off-limits" to science.  The subtitle of this book is My Quest for the Ultimate Nature of Reality.  3) In the series  there is talk about a search for the "God equation."  (A good counter-example would be Feynman, but his self-described "non-axiomatic" or "Babylonian" approach does not seem popular with physicists today.)  Is there any sense among physicists that it might be impossible to articulate the "ultimate nature of reality" in equations and formal logic?  It seems to me that physicists are following in the footsteps of the 19th century mathematicians (led by Hilbert) who were on a similar quest which was put to rest by Gödel's incompleteness theorems in 1931.  Is there any appreciation for how the Incompleteness Theorems might apply to physics?  Has any progress been made on  for the 20th century?  Shouldn't this be addressed before getting all worked up about a "God equation?"
98783
If any number times zero is zero and any number time infinity is infinity, then what do you get when you multiply zero times infinity? Do they cancel one another out and equal any number since any number = 0 to infinity) and any number infinity to 0? "also can 0 and infinity be one an others inverse?
43832
I've been browsing through Jech's and Levy's texts on set theory, and the ideas of ordinals come up fairly quickly. The idea of a limit ordinal is introduced, which is an ordinal with no maximum element. My question is, can any infinite ordinal be written as the sum of a limit ordinal and a finite ordinal, possibly unique?  My thinking was, if $\alpha$ is an infinite ordinal with no maximum, it is a limit ordinal, so $\alpha=\alpha+0$. Otherwise, suppose $\alpha$ has some order type $\{a_0,a_1,\ldots, b\}$, so $\alpha=\omega+1$. Similarly, if $\alpha$ has order type $\{a_0,a_1,\ldots,b,c\}$, we could write it as $\omega+2$.     (Thanks to Arturo Magidin, for pointing out that the following example I gave is not an ordinal.) But what about an order type like $\{a_0,a_1,a_2,\ldots, b_2,b_1,b_0\}$, this has order type $\omega+\omega^*$, would it still be possible to write is a sum of a limiting ordinal and a finite ordinal? Thanks.
87125
I can't wrap my mind around this: An accumulation point is a point of a set, which in every yet so small neighborhood (of itself) contains infinitely many points of the set, right?  So if (in my case) $-\infty$ is an accumulation point, then we could find a point of the set, that is finitely distant from infintity?!  Here's the task: Let $(a_n)_{n∈\mathbb N}$ be a set in $\mathbb R$. For every $r ∈ \mathbb R$ exits an accumulation point $b ∈ R$ ∪ {−∞} of $(a_n)_{n∈\mathbb N}$ (+∞ und −∞ are allowed as accumulation points) with b &lt; r. Show, that −∞ is an accumulation point of $(a_n)_{n∈\mathbb N}$.