matt grime

It is a well known identity: the sum of the first n squares. You can prove it by induction. It is in general true that the sum of the first r powers of n is a polynomial in n of degree r+1. Another, way to prove this result is to divide top and bottom by n, so we are investigating [math] \frac{1}{\sqrt{\sum\frac{1}{n}}}[/math] and that tends to zero since harmonic series [math]\sum \frac{1}{n}[/math] diverges.

Can you draw two non-parallel lines in parallel planes? Of course you can. (Note, in 3-d the definition of parallel is that the minimal distance from a point on one line to the other line is constant; it is not that the lines do not intersect.)

Draw a picture. Hint, you may as well suppose that A and B are in fact the same plane.

Since we're supposed to elaborate on the 'other reason' option, here it is. This pertains purely to my experience of mathematics articles, though that is not to say it is not also true of other subjects. I do not believe that this website has enough people with anywhere near enough knowledge, or mathematical writing experience, to create a mathematics wiki. The original articles there were very poor indeed, both in presentation and content. The manner of writing, however, seemed to imply that the articles were in some way authoritative, when nothing could be further from the truth. I edited some of them to dispel such things as the opinion that algebra is the study of linear algebra, and so forth, but I have neither the time nor inclination to correct articles (by people writing apparently from the experience of having taken some maths courses whilst majoring in engineering) so that they bear some relation to how mathematics might be more properly viewed from the larger perspective, and I am not sufficiently knowledgeable about mathematics to do that for all, or indeed many, articles. Planetmath, and wolfram, are far better reference sources for mathematics than WiSci can ever hope to be. That is just realism. You should expunge the mathematics section: it is an embarrassment.

Just google for hypothesis testing, I"m sure there must be many many sites explaining this (high school in th UK) level of statistics.

I'm not in charge. It's good to have a discussion on this level in here for a change. Keep it up. My knowledge of string theory is slightly better than my knowledge of elementary properties of fibre bundles, although I am more of al algebraist than anything, so 1-d TFTs are the same as A-infinity algebras. I tend to come in to this subject as someone who understands (as much as is possible) triangulated categories (the subject of my PhD) and is curious as to what it is the string theorists are doing with them, or trying to do with them (D-, A-, and B-branes).

There is slightly more to a fibre bundle than that (such as transition maps and a structure group). The bundle space does not have only one element (not that I'm that sure exactly what you mean by bundle space: X is the total space, Y the base space in your notation). The base space is homotopy equivalent to a one point set, obviously. The point is to show that, for contractible spaces, homotopy equivalence is the same as homeomorphism or whatever.

I'd prefer for you to call it realism. The level of mathematics questions asked (and answered) here is relatively low. If we need a scale, let's use the US one:normally the hardest questions are 'easy' freshman or sophmore calc. questions. This question is more 'grad school qualifier' standard. I.e. probably the question has come up to a good final year undergrad student or later. I think it only reasonable to point the OP at a better (i.e. wider) source of knowledge than we have here. I'd be interested to know how many users here can eve define what a fibre bundle is, for instance. From the answers I've observed here in the past I would suggest at most 3 others, all of whom also post at physicsforums, maybe one of whom actually knows the answer. I've never really studied fubre bundles, and though the question, or rather what the question is getting at, seems sort of clear, I can't provide an answer other than to say he/she should ask somewhere else.

I think this question goes (far) beyond the average knowledge of the user base of this forum. Try posting it at http://www.physicsforums.com instead where there are more likely to be people who can answer it. My knowledge of fibre bundles is sketchy at best. I don't particularly see why arguing aobut bundles over R^n says anything bout bundles over other spaces which are not R^n, for instance. Presumably, the argument is to show that any bundle of degree m over your base B is homotopic to the space {pt}xR^m, and then invoke this theorem you stated to deduce the final result. Not knowing what the homotopy axiom for bundles is I couldn't say with any confidence this is what you're supposed to do. I imagine mathwonk at physicsforums know the best argument.

Depends what you want to do with it. The standard axioms of set theory are ZF (Zermelo Frankel), and you might want to throw in the continuum hypothesis, the generalized continuum hypothesis and definitely the axiom of choice, though there is no real need to do this in any depth. Just be aware of the fact that it is forbidden to have a set of sets that (do not) contain themselves (Barber's paradox or Russell's paradox), e.g. so that you're happy with the idea the class of all sets is not a set. You also ought to know the axioms of the three geometries (Euclidean, hyperbolic, and spherical). It might be a good idea to get to grips with the fact that 1. axioms are not self evidently true; they are not in themselves true at all, nor are they false, they are just strings of symbols. 2. it is models of axioms that count - this is where we actually do maths.

There is no (finite) set of axioms from which all of mathematics is deduced. The axioms you choose either explcitly or implicitly are those which seem most reasonable for the task at hand.

How can it be your defintion? Think about your definition - one to one means given an input there is one output. That is jsut the definition of a function. Consider my definition, given the output there is exactly one input that gets that output. e.g. x^2 is not 1-1 on R since if x^2=1 then x could be 1 or -1. It is one to one on the range x=>k whenever k is greater than or equal to 0 (which gives you your answer after translation).

debatable. If I hit your thumb with a hammer, you know it hurts, you're not going to debate what 'hurt' means before screaming. no, the root of the problem is that people do not know the definitions. what does that even mean? from what? and what does this have to do with the problem? define 'process' and 'complete' define construct, and what it means for something to 'be' constructed so tthat we can get into context the next line: how is this true? Perhaps if you offer a definition of construct then we can agree or disagree that a continuous object can be constructed. This is moot. I agree it would be infinitely more illustrative to use the terminology of 'equivalent', however, utility, and consistency dictates that we use equal just as we do when we say 1/2 = 2/4 what process? it ain't approaching anything. it is a number. again, it is not 'reaching' or tending towards anything, it is a number. what process? and? the same surely holds for dividing by 3 (or any number divisible by any prime not equal to 2 or 5) nonsense. the integral of exp{-x^2} is sqrt(2pi), a perfectly exact representation of the answer. just cos you might want it in decimal notation rather than a symbolic one means nothing. i'll leave the rest (which still uses the word 'process' without defining it properly)

You have never been using the idea of 1-many. SUch things are not functions. one to one means that f(x)=f(y) implies x=y. For these purposes it is sufficient to think of drawing a horizontal line - the function will be one-one if it intersects the graph only once.

Irrationals do not have infinite numbers after their decimal places (there is no such thing as an infinite number in a decimal representation system). They *all* have non-repeating decimal expansions, however. There is no problem writing an exact solution to any equation, by the way. It is just your preference for thinking that 2 is exact but sqrt(2) is not. The point is that rational numbers, and integers are more natural than the real numbers, they are what has fascinated mankind throughout the centuries.

6^{-n}+3^{-n} rarely (never, apart from n=-1) equals 9^{-n}, KLB.

OH dear. It is somewhat depressing that people think that is proof. It is not. It is at best a justification for a definition. Did none of you stop to consider if the operations you performed are actually valid on such symbols? No, you didn't, you just accepted they were, and two wrongs do not make a right. Still, if no one is prepared to consider the *definitions* of these things, I fail to see what you can do.....

Yes, you have, but you don't realize it: any proof that uses the fact that the reals are theC completion of the rationals will almost certainly be correct. I believe that I've even posted the formal proof here in some form, it is after all equivalent to the assertion that, in the real numbers, 1/n converges to zero. I think you mean the let x=0.99..., then 10x-9=x, hence x=1. I agree, that is not a correct proof, but it is a justification of the definition. It says that if we want to treat infintely long strings of decimals as representatives of real numbers in the obvious way then we have to declare 0.99... to be equal to 1. That implicitly states that we *are* pulling the wool of your eyes. We are not. Again, who here has proclaimed themselves an expert (and bragged about it)? who here as appealed to a proof by authority? Again, who's debating? We're not, you might be, but we're not. The problem is that too many people (all those in the disbeliever camp and many in the believer camp) do not understand the definitions of the things they are playing around with. The conclusion that you can write the same thing in two different ways is forced on us by the definitions, just like 1/2=2/4. This is not a subjective thing to debate at all. It is certainly frustrating that people refuse to look up the definitions and suggested further reading. Mathematics is a language, in some sense, and these 'debates' are a little like someone attempting to argue in mandarin whilst refusing to speak anything other than english.

You aren't wrong, but you did miss a summation sign out in the opening line: [math] 0.9\ldots = 0.9+0.09+\cdots = \sum_{n=1}^{\infty} \frac{9}{10^n}[/math]

That is not the formal definition of the real numbers. The real numbers are the unique totally ordered complete real field. Complicated huh? Right, here's what they really are: start with the rational numbers. now, consider *all* possible sequences of rational numbers x_n that are Cauchy. The reals are their completion. OK, not helping huh? Right take the rationals and then append on every conceivable limit of sequences of rational numbers, if x is the limit of x_n and y is the limit of y_n then we declare x=y if x_n - y_n converges to zero. (I am being deliberately vague). 'Proposition': when done properly, the result is a decent object that contains the rational numbers, and all bounded increasing sequences converge to a *unique* limit (which is the whole point of the construction - to make some space where these things are true). Now, it took several centuries, if not milennia to come to decide that this was what we wanted to do. In any case, in this space 0.999.... and 1 are representations of the same object.

You are thinking far too restrictively. In particular you are free to explore, in maths, as many avenues as you wish, as long as you are consistent. Take one avenue, just for the sake of argument, where we wish to use a system with no infinitesimals (whatever an infinitesimal may be - do you not find the notion of something smaller than anything nonzero but larger than zero a little troubling because that is what you are demanding) and we wish to use strings of decimal digits to do this, then we have to declare 1=0.999.... to do this. If you want to do otherwise you're talking about a different system. But here's the nub: you can't talk about your 'new' system and claim it is the same as everyone elses because it is not. Mathematics is about universally agreed conventions and one of them is that the real numbers behave as we want, and 0.999...=1 in decimal represnetations of these things. That is a stone solid fact that follows directly from the definitions. At best you can argue that those definitions aren't what we ought to be using, but that does not change the validity of the argument. You are just arguing about how to use the symbols, not the symbols themselves, and in this argument you're wrong because you can't argue with a definition. You understand different bases, right, like binary. In binary 1+1=10. If you say this is wrong then you're arguing with definitions, and the 0.999... thing is just the same: these are just representations of numbers, not acutally the numbers themselves.

Let's put it another way: you are free to play around with infinite stings of decimal digits and declare that 0.999... and 1 are strictly different elements of this space of strings of decimal digits. That's fine, and no one would ever say that 0.999.... and 1 are the same string. But these things, whatever they are, cannot then be a model for the real numbers (in any natural sense). Equally, I am perfectly free to play around with the symbols a/b for a and b integers, and do things like a/b + c/d = (a+b)/(c+d), or even a/b + 1 = b/a, but what I've done is not give a model of the rational numbers with + meaning its usual thing. It is just a definition you're now arguing against. You can philosophize about that but it ain't really a mathematical problem now.

I did not declare myself an expert; I am not an expert in mathematics, no matter what any silly badge says. If you accept that the difference between these two strings is 'infitesimally small' then if we are treating them as representations of real numbers that means *by definition* that they are equal: the only real infinitesimal is 0. Ok, you've just proved that they are equal real numbers even though you don't realize or accept this, because you do not know what the real numbers are (and this is not an insult, I do not know what the chemical formula for acetone, however, I do know what methane is (hopefully), and if I were to say to you that I've found this chemical CH_4 and refuse to acknowledge it is methane even after everyone pointed the correct formula and so on, you'd be right for thinking ill of me for my obstinate refusal to learn a definition. These are just labels in maths too).

that is not a real number. infinity is not a number. and 1 to the -n where n is any integer is 1. they are equal owing to the properties that we declare they must have to be *representatives of real numbers*, it is nothing to do with rounding or measuring or phyisical niceties like that.

you agree 0.999... ought to be a real number, and you agree that 1 is a real number and that addition and subtraction are defined on real numbers, right? so i'll let e=1-0.999..., that's a real number. Now, what is 0.999..+e?