matt grime

because it works, that is the only reason why you ever use anything in maths. sounds crappy i know but it is true. we;ve pointed out their properties and it is up to you to figure out how to use them. there are so many ways to do so that we cannot begin to describe them to your satisfaction. let;s try abn example. Let x be some vector and n a unit vector. We know that we can write x=tn+m where t is a real number and m is a vector orthogonal to n. find t (in terms of n and x and the other things we are discussing). please try this question.

the answer is either "trivially, yes" or "i've no idea what your asking now"

"multiply" makes no sense at all in this context. explain what you mean by multiply. the dot and corss product are just operations nothign more, nothign complicated. they have many uses and you'll get used to them with prcatice. dotting is useful when you want to exploit orthogonality and corssing when you want to exploit parallelism (if that's a word). there are many other uses to that have little to do with this geometry.

So, you're asking what does the dot product mean? Well, what does anything "mean" in mathematics? What does "mean" even, well, mean? The dot product is a simple operation on vectors, and it can be used for many things. I've no idea what it "means", and I doubt that that question even makes sense. I know what it means when x.y is zero and both x and y are not the zero vector. I know what it means when x.y is zero for all y. But I've no idea what the dot product means. (I know what it is...) Just remember, x.y is linear in both components, if x and y are perpendicular then x.y=0, and finally that x.x=|x|^2 and that's it, a simple definition.

What does what scalar mean? and why did you then write a deteminant that gives a vector? (you do not need to underline, over line or use arrows for vectors, we know they are vectors, you told us they are vectors, hence in this thread i,j,k are vectors and you don't need to embellish them).

just think it through: let u be a vector, we want to find the projection of u onto v, that is we want to find vectors a and b such that u=a+b and a is a vector parallel to v and b is a vector perpendicular to v. the projection is then the vector a. we want to kill b so we'd better dot with v since dotting with orthogonal vectors kills them.

lg is base 10 and log alone is moot: mathematicians would say it is base e, some applied scientists that it is base 10 and others that it is base 2, depending on the context. If you're doing engineering it is safest to assume that log means base 10 unless specified otherwise, and if doing maths then it is log base e. Note that base e is the natural one to do maths in for reasons you'll meet in analysis later, just as the natural unit for angles is the radian and not the degree.

A hperplane in any vector space of dimension n is an n-1 dimensional subspace, ie a line in R^2, a plane (through the origin) in R^3, a copy of R^3 in R^4. Sometimes that is simplified to any subspace of "codimension 1" which allows for infinite dimensional definitions as well. The simplest examples that occur "naturally" are in hilbert spaces, or an inner product space, where we can use any element z to define a hyperplane as the {x : <x,z>=0}

well, that is an odd example to me, but works. R with its usual metric topology was the obvious example.

right, then what is pi^2 to 10,000 decimal places? Plus there is an algorithm for working out the n'th prime, it just takes a long time. Existence is different from utility, a distinction you are not making. then why not simply talk about the sets and their intesection? you are completely abusing that equals sign. functions and maps are the same thing. it wasn't your reasoning, since it is impossible for me to deduce what you reasoning was. it was the fact that you wrote something that could not be made to make any sense with standard meanings, and I cannot second guess what your thoughts might be. a simple reductio ad absurdum results was fermat's proof that there are no solutions to x^4+y^4=z^4 for positive integers. actually it proves that you can't write it as x^4+y^2=z^4 I think, which is stronger. google for it, there should be an explanation somewhere. I also have to different proofs of this argument (different in only minor ways) that i could write out but that's a hassle.

there is no bijection from any set A to P(A) (in any model of ZF). Thus if there were a set of all sets, U, then card(U)=card(P(U)) since P(U) must be a set larger than U and hence U. But then there would be a bijection between U and P(U), contradiction, thus there is no set of all sets in a model of ZF (Cantor's paradox). Plus if there were a set of all sets then we can use restricted comprehension to create the set of all sets that do not contain themselves and we are led to Russell's paradox. If you wish to know how we get round this apparent problem without appeal to the theory of types (which roughly orders mathematical objects as elements, sets of elements, classes of sets of elements and so on), then the analogy to bear in mind is that a vector is not a vector space. I can give you a link that explains that idea if you like.

=> is standard for imples. proposition A implies B is exactly the same as not(A) or B, exactly as i wrote.

A sigma algebra is closed under the operation of complements. Give me a topology (ie a collection of open sets) that is not closed under complements (pretty much most of the ones you can think of, I imagine)

Erm, what are you on about? If you have a degree in mathematics then you'll see that the function f: N to N given by f(n)=p_n the n'th prime is a function that gives the n'th prime, of course we have no known algorithm for computing the n'th prime but that is a different matter entirely. You won't be able to reproduce Wiles's Argument, or at least I'm confident you won't understand it; I know I wouldn't if I read it. It shows that all semi-stable ellitpic curves are moduilar, and i seem to recall something about galois representations being required. In anycase that is not important. "Saying that only n=3,4,5 are primitive is pointless, none of the solutions for those cases are general solutions to the problem, they are in effect a waste of time. Are you able to reproduce Wiles argument? I probably could but I can't find a full exposition of it anywhere." RIght, i didn't say n=3,4,5, I said in the n=2 case 3,4,5 is primitive and 6,8,10 isn't as solutions ot x^2+y^2=z^2, nor did i say that anything about theer being no other solutions. The stuff aobut x^n mapping to z[x,y] doesn't make sense since even now since you haven't stated the domain of the map. HOw do I send x^n, (an element of Z[x]?) to something in Z[y,z]? or is it only a map fram n'th powers of elements of N to something? It is not a well defined function! And the expression x^n=z^n-y^n does not define a function to Z[y,z] (function from where?) Better, perhaps is to look at the images of the two maps x--->x^n from N to Z, and (y,z)-->z^n-y^n from NxN to Z and look at their intersection as a subset of Z, wh ich is empty when n>2

synethesia is certainly a very well known phenomenon. to add to the list, Anthony Burgess also was synesthetic. as fot the othe r claim, , well you claim it but have you ever tried to do a controlled experiment to verify that your guesses about the deceased are correct, and things that cannot be obtained by mere guesswork and common sense? from looking at a grave stone you will immediately be able to deduce the age of the deceased at death, the era in which they lived, whether or not they were married and had children and many other things too (such as if they outlived their spouse if they are buried together). and are your "visions" actually meaningful? here for instance is an example of someone trying to prove for a television show that astrology is not bogus: they were given a birthdate of soemone famous (and alive, and a sceptic) and asked to write down what they could tell about this person. after it was revealed who the person was the only point that the astrologer could point to as being particularly accurate was that this person was supposed to have a strong sense of humour (and indeed the person was a famous comedy performer), at which point the said comedy performer asked the studio audience if those present who consider themselves not to have a good sense of humour would mind standing up.

Takers to do what? The threads were closed because none of the participants of the crackpot variety cared about cardinals or ordinals and just wanted to spout rubbish about how they couldn't believe that infinity plus one is infinity, whatever that may mea to them.

Your knowledge seems a little off. there are many ways of generating primes from functions, see wolfram for prime generators. there is one in 26 or 27 variables that whose positive values are primes, of course we don't know when the function will take positive values based upon the inputs. And there is a function f(n)=p_n that gives the n'th prime, though there is no simple way of working out what f(n) is for any given n that doesn't stop it being a function from N to N. I'm glad you made the coprime amendment, but no, allowable solutions do not need to have coprime x,y,z (they are all three coprime, if d fivides any two of x,y,z then it obviously divides the third) and the solutions you mention are the primitive solutions, eg in the n=2 case 3,4,5 is primitive, but 6,8,10 is also a solution, albeit not primiitive. Z[y,z] as in the polynomial ring in two variables over the integers? in what way do we have a map that sends x^n to Z[y,z]? This has nothing to do with topology. Sorry, but the what is closed under multiplication and contains 1? the polynimial ring? Yes, and? it is also commutative and has a Spec, but...? And who the bugger is Frenchy?

if f is your fucntion, then the derivative at x is the limit of (f(x+h)-f(x))/h as h tends to zero. example: prove that the derivative of x^2 is 2x: we need to find the limit of ((x+h)^2-x^2)/h ok, well, let's expand it and and cancel off the h as we may and we need to find the limit as h tends to zero of 2x+h which is just 2x as requried no, try if for (sqrt(x+h)-sqrt(x))/h it will help to bear in mind that s^2+t^2=(s-t)(s+t) (or setting sqrt(u)=s and sqrt(v)=t) may be more illuminating) so try multiplying that expression top and bottom by the same thing now try simplifying for {1/(t+h) - 1/t}/h as well and let the h's tend to zero

i did put 'invent' in inverted commas as i knew you were using something you'd seen in a textbook once, a textbook that no one else here may have read.

what is the probablitly that after n draws you have n different things?

it only mkes snese to speak of a limit point of a fucntion of x if you are saying where x tends to. in this case cos(x)+isin(x) is our fucntion. as x tends to what? as it in this case it is better not to 'invent' new notation (i have never in many years of doign maths at universities in two countries met the term cisx for what yoy describe especially as it is just exp(ix) so, you are asking for a proof that exp(ik), k in N is dense in the unit circle of C. well, that is becuase it is an irrational rotation of the circle and all orbits are dense. look it up in any basic introdcution to dynamical systems.

if you go around resurrectung very dead threads you might not get much response, especially since Johnny5 is now permanently banned from theis site.

the question is badly worded. and the answer is x+1 if we assume you mean "have to draw a ball that has been drawn before". obviously, by the pigeon hole principle in x+1 draws there is at elast one repetition, and in x draws it is possible to get each balle xactly once. however, you might mean "draw with probability freater than 1/2 a balll that has been drawn before, and that is a simple exercise in probability (geometric i think).

eh? offer a mathematical defintion of wave and one of a vector and see. don't confuse mathematics with physics, or a model of something with the thing itself either.

perhaps if you explinaed what you meant by bases then poeple may hve more idea of what you're talking aobut. i can't make head nor tail of it, presumably you eman x,y, or z, but that doesn't make sense since when n=2 the solutions are p^2-q^2, 2pq, q^2+p^2. even if we allow primitive roots (ie when p and q are coprime) you have just said that there is an easy way to generate infinitely many primes. or are you referring only to n>2?