matt grime

what did you want to discuss about them? the first is easy by S-B, and the direct proof is the standard: if X is an infinite set there is an countable (infinite) subset, let x_1=0 , x_2=1/2 x_3=1/3, x_4 =1/4 etc and define bijection from [0,1] to (0,1] be y to y if y not one of the x_i x_i to x_{i+1} other wise. clearly a bijection. it is then trivial to show (0,1] ~ (0,1) by a similare argument. the second question is a fancy proof of S-B, but no different from the usual one really.

it would strongly depend on what they are rating it on. warwick is far superior to nottingham in its quality of research, and that isn't blowing your own trumpet in the slightest. teaching evaluations are slightly odd things.

An olive branch: yes, having to just deal in basic definitions is tedious and dull, no argument there, and often it is unnecessary to make explicit reference to them all. However *this* particular question exactly boils down to people not knowing what the definitions are (complete totally ordered field, or completion of the rationals in the euclidean norm, or the set of all dedekind cuts, whatever tickles your fancy).

Sorry, apologies for sounding insulting. I have no problem with you, though i have a problem with answers that don't use the properties of the objects in the question.

sorry, but being a research mathematician has nothing to do with it, being a mathematician does (i don't use real numbers in my research, none of my colleagues uses them in research either). if you don't actually pay attention to the definition of any of the terms in such a question as "why is 0.999.. equal to 1" then you've not answered the question since the truth of the statement *follows* from the definitions and the principal reason people do not understand this particular question is because they do not understand the definition of the terms. Those two objects are certainly different *representations* of the same real number but they are definitely the same *real* number. The reals are a basic object that we all use, though often without knowing what's going on. asking this question is a good sign of curiosity, understanding the (correct) answer a good sign of aptitude. http://www.dpmms.cam.ac.uk/~wtg10/decimals.html you do not need to discuss them, they are not up for discussion, they simply *are* if you will.

the distance between two points x and y in R^n is [MATH](\sum_1^n(x_i-y_i)^2)^{1/2}[/MATH] for x=(x_1,x_2,..x_n) etc why would you need to picture any of these things in your head though? it doesn't help particularly.

So what has an inductive statement about a finite decimal expansion (hence only apllicable to the rationals) got to do with the real numbers in generality. Oh, wait you're not about to conclude the "infinite" case follows from the finite cases inductively.... and 0.00...01, an infinite number of 0s then a 1 makes no sense as a decimal expansion for anyone still under the impression it was meaningful. there have been several worrying expressions of dislike of definitions (esp by a maths moderator). worrying because maths IS its definitions.

n=1 anybody?

my correction also contained an error, by the way. H should also be greater than any real number. i didn't post the corrected definition because i didn't think having the correct definition would add anything in this case.

No' date=' that isn't true. put x=2 in if you don't understand why. [MATH'] \sqrt{x}[/MATH] click to see code

you're implying there exists exactly one infinte number and one infinitesimal in non-standard analysis by saying it in that way. H, in your notation, is the smallest non-finite element in the extended system. also your definition of infintesimal is wrong.

Algebras are not fields, your implicit ordering fails at this point (i'm not sure what the "and so on"might be about)

A ring is a set that has an additely written abelian group structure with operation + and identity 0 which also possesses another binary operation, *, commonly called multiplication such that some obviously useful axioms hold (distribution, note * is not nec commutative, nor is there nec. an identity or inverses). The fundamental theorem of arithmetic is simple to prove but subtle. If you think you understand it then you ought explain why Z[x] has unique factorization, but Z[sqrt(-5)] doesn't. It's to do with certain poperties of primes that aren't true in all rings.

A set can be "defined", it just isn't defined how people first think it is defined. It is not a collection of objects with some rule for belonging. How it should be done is messy and unilluminating at this stage in your development, since the *naive* definition is sufficient for many purposes. As for "definition", that is philosophical, my preferred way is to think of the definition of an object as the set of rules that define its use.

That the zeroes of a certain L-function all have real part =1/2 (lie on the critical line, in the parlance), the L-function is the Riemann Zeta function, and is detailed on many web sites.

Almost everything may be reduced to graph theory. Each node corresponds to some "state" and each vertex to some method of getting between states, or some connection between states. For instance, consider a game between two players. For each possible position in the game we assign a node, and if there's a legal turn that takes one state to another we draw a (pointed) edge between the nodes. Finding a winning strategy then becomes finding a choice of edges for player1 such that no matter what the other player does you must at some point come to a winning position for player1. Graphs may also be used for determining how to shedule lessons without clashes (exercise figure out how), or modelling populations, where often you'll give edges labels to indicate the probability of progressing from one state to the other. This is a basic description of a Markov chain (discrete time) that is important in economics, epidemiology,...

So, Gauss, f(x)=x and then you show by example that f(-3) is not equal to -3.

No, Gauss, you are completely in the wrong, you may not use Taylor series to show a function is continuous since you are implying it is differentiable, and therefore already continuous (a circular argument if you will). (and the OP did not say that Taylor series of trig functions were to be assumed to exist.)

Cuti3panda, maths just follows the rules. You know what the rules are that define an equivalence relation, right? So, write them down. Now, for the second one (it isn't an equivalence relation by the way), can you try and find some where where the rules for an equivalence do not hold? It's obviously reflexive and symmetric, so what about transitive? Can you find numbers a,b,c so that ab=>0, bc=>0, but ac<0? the third one just requires you to show the rules for defining an equivlance relation are satisfied. write them out again. and write out what it means for an integer to be even (it is a multiple of two).

If, Firedragon, you think maths is about numbers, then you should probably stop doing it. But I'm a research mathematician, so what do I know?

if f has the domain you claim, Gauss, then it isn't the identity function, since roots are always taken to be positive.

Any basic group theory book will explain what dihedral groups are.

inorder to have (repeated) overlapping peaks, the frequencies must be some rational multiple of each other. now coquina's logic states that we hear beats when some peaks overlap but that when they perfectly overlap there are none. now that makes no sense since there are STILL over lapping peaks, all of them, it is the fact that coquina has attempted to describe constructive interference but gotten her phenomenology a little wrong as far as i can tell: how can constructive intereference "disappear" when the waves are in perfect sync?