matt grime

Why would you trust my opinion? I can't even remember what the taylor series of log(1+x) is.

Stop being panicky about the result. Trust yourself to add together two things and divide by 2.

Have you remembered to divide by two? it's one half of log(1+x)-log(1-x) we're after.

you're raising -1 to an even power, so you can simplify it a hell of a lot.

Eh? n is what? Where are the smaller terms. I see nothing in your thumbnail that indicates adding terms together - that would require a Sigma sign, or soemthing. Ok, let's start more simply. What is the taylor polynomial of log(1+x) of degree 2n+1? Hint, the talyor poly for sin of degree 2n+1is: x+x^3/3! + x^5/5! + ... + x^{2n+1}/(2n+1)! =

The answer you got in the last post is not a polynomial of degree 2n+1. Write out the talyor polys of log(1+x) and log(1-x) and add them together and divide by 2, what is the resulting polynomial? Note your last thunmbnail has a nicer form as 0 if n is even and x^n/n if n is odd.

A poly of degree k means that the largest power of x is x^k, eg x^2+3x+1 is degree 2, and so on. log((1+x)/(1-x)) = log(1+x) - log(1-x) so subtract the two taylor series (up to the 2n+1'th term) and divide by two.

because that's what some schools teach! Just as some schools teach I(ndices) and so on, and the whole point of the thread shows that things are not universal. Why teach divides since that's just multiplication as well, as are indices and orders in some sense: if 2^2 mneans 2*2, then any number to any power out to have the same precedence as this. And what about other functional valuations? We got taught bodmas before indices, you see, and in primary school OF often appears in the questions. It's crap I know, but then so is the whole acronym thing, though Orders does make more sense.

no, there is something called BIMDAS, it would appear, and there is something called BODMAS too. As we can see they are demonstrably different. Why do you think that they must be identical? Because they have similar acronyms and do simliar things? So what? In BODMAS the O is for 'of'. And don't start the "but of means division" thing, whoever it was who did that, since division is actually mutliplication as well so there's lots of redundancy.

The O in the BODMAS version stands for 'of' as in two thirds 'of' 4

the set of roots contains one element, but the poly splits into two linear factors as Gauss's theorem states. Learn the sodding theorem properly first.

Oh, for pity's sake Johnny, learn about repeated roots, or roots "with multiplicity" and stop with these childish postings. By your own reasoning it has two linear factors.

What arguments? You've made no mathematical statements at all, only offered some opinions as to how we're brainwashing people with contradictions. Evidently you are ignorant of mathematics. Feel free to post a list of your qualifications in mathematics. In fact I have absolutely agreed with the statement that the only "numbers" that may in any reasonable sense be said to exist in anyway in the real world are the natural numbers, and that the rest are "inventions" of mathematicians. That doesn't stop us reasoning about them any more than the fact that Hamlet is fictional stop us reasoning that he has some issues with his parents. The difference of course being that mathematical statements are not subjectively true, but nothing is absolutely true. Do you think, for instance that "through every point not lying on a given line there passes a unique parallel line"? If you say yes, absolutely then you truly are ignorant of mathematics. This doesn't stop statements in mathematics being true within the carefully defined parameters that we offer, no does it diminish the uses of mathematics.

Brainwashing? Erm, no. We simply state that if we accept these axioms and those rules of logical inference then the following deductions are true, these statements are false, and the system may or may not be incomplete (if it models the natural numbers and is finitely axiomatized then it cannot be complete - Goedel). In what way is that brainwashing or remotely religious? Different parts of mathematics start from different axioms and some even have different rules of logic, but we all admit before hand what they are. Do not confuse the necessary eliding of details in highschool, where the subtleties are omitted and wouldn't be understood, with any attempt at dishonesty. Sadly some people become fixed and think the "baby" explanations are absolutely true. To be honest your post does indicate a lack of exposure to higher level mathematics, and indeed the concepts of axioms etc. We are never anything but honest about axiomatic systems. Why all this antipathy and refusal to accept mathematics simply for what it is; it doesn't pretend to be anything of the things you appear to claim it is.

No, it works fine: if Ax=Ay, the applying C we get x=CAx=CAy=y. And this answers my own question from before, you are writing Ax so you're acting on the left, so the null space is the set of all vectors such that Ax=0. And no, I'm not going to check your answer to that question, sorry.

But that only is a contradiction if you claim that numbers are in some sense real. We do not. So we aren't making any contradictions. You are misrepresenting mathematicians (or this one at least) if you say we are. A very famous mathematician once said that "God invented the whole numbers, the rest is the work of man", and I agree. But there isn't 'a contradiction because we aren't saying numbers (real or complex or anything) are in anyway representative of anything in the real world. We freely admit that the things we do may, indeed almost certainly, have no realization in the real world. How can that be religious fanaticism if we admit our "god" is fictional? All we say is that some parts of what we do are useful for modelling the real world, but the models aren't necessarily themselves real. If you don't think there is a real life case where -1 has a square root then you should read up on electricl engineering, or qunatum physics (including its experimentally verified parts). Please stop accusing us of making claims that we do not make. We can even demonstrate to you that almost every real number is not constructible for heaven's sake, so how can we be making any contradictory statements? All that is happening here is you are using a different definition for number than a mathematician might - though in rigourous terms we would never say number alone, but qaulify it with real or rational or some such. And please don't think that because of the unfortunate naming of them as real and imaginary this in anyway reflects our opinion of their eixtence in any sense. The naturals may be said to exist, 0 is to many people doubtful but to a mathematician it is simply adding in an identity to a semigroup, then adding formal inverses gives the integers, localization gives the rationals, completion gives the reals and closure yields the complex numbers. It is because of the exact issues you raise that mathematicians have devised this to be on a formal and concrete footing *within the realms of mathematics* - we have definitely divorced this from reality a long time ago, and we don't claim otherwise!

What is the x3? a basis is just some (finite) set of vectors, eg (0,0,1,0), perhaps. In order to state what the null space is i need to know whether you're acting on the left or right, ie is this a map from R^3 to R^4, or R^4 to R^3?

You've stil not pointed out a number (rational or otherwise) that actually "exists" in whatever badyl chosen sense you are talking about. So what that every number used in calculations is rational. That has nothing to do with anything. Stop confusing maths with the real world, cos we, the mathematicians aren't, and never claim to be. So things are contradictions to you if they have no "relevance" (whatever that may mean) to the real world. Well, thanks for the opinion, but that's all it is. In that case why even think about maths almost all of which is in this sense completely irrelevant to the real world?

Sorry I meant to say what is the maximal value of |f'| on that interval for then |f(v)-f(u)| = |f'(y)||v-u| < max{|f'|}|v-u| so as long as the max of |f'| is less than a quarter we're ok.

So, you know, by the MVT that there is y in the interval [u,v] such that f(v)-f(u) = f'(y)(v-u) whenever v and u are in [2,3] or |f(v)-f(u)| = |f'(y)||v-u| now, what's the minimum value that f'(y) can be if y is some number in [2,3]?

You realize that just using someone's surname like that could be considered rude? Imaginary numbers have no relevance and do not exist? Right, ok, point out a number that does "exist" whateve exist means. I suspect that you believe that reals exist and that R^2 exists (the plane - but if you care to send me one of these in the post I'll be most impressed), and since the complex numbers are simply an algebraic structure on R^2 I have no idea what you're waffling on about. I don't think you understand the mathematical use of the word 'contradiction', or even 'exists' for that matter.

Well it's false so I wouldn't try to prove it, unless you're failing to mention what g is.

Wow, did you not understand my post. Shall we try another one? Paraphrasing Johnny: does sqrt(2) lead to a contradiction? It is just a meaningless statement vague and open ended - objects do no lead to contradictions; propositions may do so. In this case "the square root of two is rational" would led to a contradiction. You can't get a contradiction if there is notthing to reason from.

These places are, if you will, non-technical - they are not necessarily reflective of what happens in mathematics. Mathworld is from wolfram who are more applied than pure, wikipedia is written by the general public, presumably hyperphysics is written by physicists.