matt grime

yes: andrew wiles, ribet and a handful of others understand it, though no one would be able to reproduce it on demand, and no one here will understand it.

Perhaps it is just my preference then, but I would prefer to distinguish between the exact form [math](x+y)^n=\sum_{r=0}^n x^ry^{n-r}\binom{n}{r}[/math] that is valid for all x and y. the series expansion [math] (1+x)^t = 1+ tx+\frac{t(t-1)}{2!}x^2+ \frac{t(t-1)(t-2)}{3!}x^3 + \ldots[/math] [math] + \frac{t(t-1)\ldots (t-r+1)}{r!}x^r + \ldots[/math] whcih is not valid unless |x|<1 (though it may converge if |x|=1) and is an analytic statement (it involves infintie sums). The decision to declare the non-integer component the binomial expansion is one fraught with difficulty. Even that repository of knowledge Wolfram makes a mistake in this (that is an internal inconsistency) in that it states the binomial expansion is valid for all exponents (including complex ones) but fails to explain how to define the "binomial cofficients" in this case. Perhaps the absolute is to call the first the binomial theorem, then.

yes it is just linearization (of the expression (1-t)^{-1/2}) though it is quadratic in beta admittedly. it is just the first two terms of the taylor series. it isn't a binomial expansion since the exponent is not a natural number.

Define practical. Why should it be practical? Maths is more than its mere applcitaions just as English is more than a mere language to communicate the fact that we want food. But the point was that it in one simple line links the most important mathematical objects there are. it doesn't need a practical application anymore than the Mona Lisa. You are entitled not to find it interesting or beautiful, but it is certainly succinct not to say elegant.

it depends on what you take as your definiton of exp(x) i suppose, but it is a standard exercise in anaylsis to demonstrate that (1+1/n)^n converges to e (where we define e in some other equivalent way), and not too hard if you are familiar with analysis.

for the first one start one the right with the definition of sinh and vosh in terms of exponetials, the answer is then one step away. recall tanh = sinh/cosh and that you can differentiate sinh and cosh easily from the defintions as exponentials, so now apply the usual rules of differntiation. is the third supposed to be sin and cos? again just start from the right and simplify cos(2x) in terms of sin

can i move for an instant deletion of any thread in mathematics that starts off by asking philosophical questions about infinity? not that i think it is a bad thing per se, but that it doesn't half attract some crack pot rambling.

"So if the universe has a limit than so does the number system." what has the physical property of finiteness or otherwise of the universe got to do with the fact that the natural numbers are not a finite set?

did you think of googling for "equation of a torus"? cos it'll show you are correct and that is about as simple as it gets in cartesian co-ords. but then there is no reason why it should be simple is there?

what do you meant by "qualified" as infinity?

Still waiting to hear what "primiordial chaos theory" is. It seems that your issue is that you write in an odd way (using your own terminology and addendums) about ordinary mathematics. Incidentally, it isn't ad hominem to criticize the content of your post, criticism you are free to refute, though you haven't. It would be ad hominem if i insulted you about you personallyy (eg calling you an idiot. i don't think you are that). i do think that you are confusing and mixing up terminolgy, as one can easily see by looking at your response to being asked what you meant by "significance". Your response here was to say "ah, you criticized my use of signficance with respect to infinity, go on, show me that signficance has nothing to do with infinity" and then to post a completely unrelated link (a non sequitur, is my english that difficult to parse? how is it impossible to simultaneously think that you have some good topics but that your not posting well on them?). We wanted to know what *your* particular use of the word was in reference to railway lines meeting at infinity. THat was all. Your response was a non sequitur.

I think you'll find I said that some of your post was garbagr. post a link to "primordial chaos theory" (or stop changing you mind about what yopu are complaining about, i take it you are now not accusing me of misquoting you). those links certainly contain the word signficant, but none of them are related to the first instance of your use of the word (railway lines meeting at points beyond the horizon at a point of insignificance). you cannot just randomly apply words as you deem fit. I think some of the things you have mentioned are good. But I also think that a lot of it is garbage and a series of non sequiturs. That is only my opinion and only of *some* parts of what you have chosen to write in a way that implies you are knowledgable about these things, things that are nothing to do with any part of known mathematics. I am perfectly happy to discuss chaos theory (or non linear dynamics, or dynamical systems) as necessary to any debate about infinity, if there is any. that still doesn't remove the fact that you talked about "primordial chaos theory" and its links to chaos theory (a branch of mathematics of less importance than many think). As you say, you are a biochemist by training and presumably that is what you are an "X-professor" of. You may want to pay attention to what a mathematician has to say about your opinions of mathematics.

Read post 27 that you wrote that includes the phrase primordial chaos theory, which is teh only thing I recall quoting from you. "X-professor"? I presume you mean ex-professor, or exprofessor. Of what? If it's mathematics I will eat my hat. (Professor in England would mean someone of international award winning research history, professor in the US might mean anyone attached to a univeristy with tenure. Is either of those applicable or do you mean associate or assistant professor? And again, of mathematics?) As for the challenge, please feel free to post one mathematical link to an article containing the theory of "significance" to as related to infinity in any form that you may use to back up you assertions so that we may respond to said challenge with full information. I only as that if you post authoritaively on this particular subforum you stick to dsicussing mathematics. And that request is only from me in the capacity of being a mathematician and nothing to do with any administration of this forum of which I have no part.

i think if soemone posts a question in a mathematics forum about soemthing you should answer it mathematically and not spout guff about "primordial chaos theory" and other such things. and that means that you should explain the stone-cech compactifications, and infinite cardinals and ordinals. and when people ask for claridication then do so. if you want to explain something that isn't mathematics then say so and tell them where they should have posted the question if that was the material they were enquiring about. now, stop being part of the problem.

Your being wildly unmathematical and misleading and quite sepculative, EL, professionally speaking. None of this is helpng to explain the mathematics.

Err, well, not in mathematics. In mathematics there are objects with the label infinite, and there are points at infinity and we can work with them all quite easily if we stop trying to read real life into them.

well, whatever your question is it is not one about mathematics since we are prefectlyu happy with it here. i suggest you post your question in a physics or philosophy forum so you don't start another one in the mathematics forum where your question isn't really suited.

the cat is alive or dead... can you prove that? without opening the box? do you know? how do you know? think on it as a metaphor if you will. classically the cat is alive or dead, one or t'other though we know not. but not in the QM model. remember these are just models, and it turns out that the QM model is the correct (or the best we have) at the (sub)atomic scale. apologies if i'm offending the experts here; i am not one. take the two slits experiment. though i know not the details it is possible to fire a photon (or was it an electron) at two slits and to show it simultaneously passes through both slits which classically it couldn't do, since it must pass through one or the other. this, i am sure, is a gross misrepresentation of it. there are, if i recall correctly, two different interpretations of "the cat", the danish and the other one. try reading some essays on them.

as i told you, you can do this by elementary, though tedious, analysis, which is "algebraic" without being graphical, though there is no harm in using graphical means since it is just finding the turning points, and if the answer happens to be the symmetric one so much the better i suppose. it is a question of when your patience tuns out that is all.

right, that's better, so you want to maximize the sum of the two volumes. (saying maximise two things does not a priori have meaning, is (1,3) bigger than (3,1) or (2,2)? if you don't state the ordering it is meaningless). simple calculus will give you the answer, though symmetry makes the symmetric answer the favourite, but that is just a gut reaction and may turn out to be incorrect.

johnny was permantently banned for a reason, and it was little do with the proper way to do exponents, whatever that may even mean. can you take 5th roots manualy? and what about irrational powers?

what do you mean maximize the volume of the "two cones"? the sum of the volumes? the product of the volumes?

i don't like people posting authoritatively on things that they seemingly know little about. had he just stuck to the point rather than this "primordial chaos theory" i would'nt have said anything at all. there is far too much non-mathematical science (if it is even that and not a pseudo-science as i suspect) masquerading as mathematics anyway, to your point. division by zero is not a well defined operation within the real numbers so there the answer is not defined since division satisfies (a/b)*b=a, and thus (a/0)0=a, but we know that 0b=0 for any b, and we also know that x/x=1 when defined. thus there is no way to define it that is internally consistent. now, that doesn't stop us positing the existence of a diferent structure extending the real numbers, and indeed in the extended complex plane (which is the more widely used, so i'll talk complexes for now) a/0 is taken to be the point at infinity (as long as a is not zero or infinity itself, for even then 0/0 and infinity/0 are not defined).

that isn't what i said. what matirx formed from those 4 points? i didn't say anything about that. the matrix is a priori given to me (by you). a matrix is invertible if and only if its determinant is noty zero, which is why you need to know it, and to find the inverse (and thus solve the basic linear algebra exercises) it is necessary to know the determinant.

but again this all boils down to what you are attempting to describe by "infinity". mathematics is is simply about the deductions one can make about things that are defined to be mathematical, and there is no hard and fast rule about what is mathematical before you ask. i can only tell you about the things that mathematics has declared to be associated to the word infinty (or infinite). (a quick check will tell you that 'primordial chaos' has nothing to do with mathematics) perhaps you should step back and ask a metemathematical question: what do i mean when i use the word infinity? in mathematics we say that natural numbers (1,2,3 etc) are finite. that any element of the integers (..,-2,-1,0,1,2,...) the rationals or the reals are finite. the term infinite means "is in some sense bigger than something in one of these sets of finite objects". we can use the naturals to describe the number of elements of some sets, so that {1,2,4,7} has 4 elements. if we consider the set of all natural numbers it is clear that no natural number is able to describe the set of all natural numbers in this sense, so this leads us to one way of discussing infinite things. all of your questions are, to me, ill-defined. that is to say they introduce even more terms that have no meaning.