matt grime

Momentum etc are operators in this sense, and the you ought to at least know about solutions to the wave operator and so on.

Yes, why do you think x is position? It isn't. Stop thinking physically. x is just a variable, and we're looking at the differential operators acting on the space of smooth functions in x, whatever. There is nothing that states x is position. It isn't.

if you don't know the definition of quaternion norm how were you able to state that i had length one? Ie how did you decide it was a unit vector if you don't konw what the norm is? If you wre writing complex numbers would you hat the i in a+ib? No, of course not.

Of course it is provable from group axioms, and it is a VERY trivial proof too, indeed the proof already exits in this thread in Dave's post when he demonstrates it for any Ring. It is part of the axioms that inverses and identities are two sided, thus making them unique. Exercise find a unital ring which has elements L and R such that LR=1 but RL is not 1.

Why reinvent the wheel? Just contribute to the currently extant planetmath. In any case, no one person has the knowledge to write one of those sites on their own, even if they had the time.

There is a wiki for maths, called planetmath I think

Very few things have been solved in mathematics, and there are even fewer things that we understand; don't give up.

! is not defined for non positive integers initially. 4! is defined to be 4.3.2.1 and as a function its domain is just {0,1,2..} where 0! is defined to be 1. As you state in another thread, it's just a mathematical invention created to do a certain job. Of course there is no reason not to extend ! to a larger domain. We do it all the time. We could define x! = floor(x)! ie take the factorial of the integer part of x, or even the integer part of |x| would be better as that allows it to be defined for complex numbers too. However, that isn't very good. Such a function is not differentiable, never mind being analytic. There is another way of thinking about !. It satisfies a recurrence relation, indeed, it is the unique solution to x_n=nx_{n-1} subject to x_0=1. This reminds us of functional equations. It turns out there is a meromorphic function, gamma(x) that is closely related to the ! function. It is nasty, defined in terms of an integral and cannot be done by elementary means. It also has poles at the negative integers. As ever, google for "thing of interest" and include the word wolfram and you'll get a good hit first up.

If you really need a more "realistic" way of thinking about them then it is this: Just consider pairs of real numbers (x,y). We know how to add them, lie vectors, (x,y)+(u,v)=(x+y,u+v) etc. But what about a rule for multilying them together? Now there are lots of ways fo doing that, I could declare (x,y)(u,v)=(xu,yv) but then we have a problem that (0,1)(1,0)=(0,0), and if we just think about addition then (0,0) behaves like zero does in the reals, ie adding it on leaves a number unchanged. So, we ought to call (0,0) ZERO in this scheme, and then we've got two non-zero things that multiply together to give zero which isn't very nice. Suppose I wanted to do better? Can I? Yes, as it turns out. Define (x,y)(u,v)=(xu-yv,xv+yu), then this has remarkably nice properties. 1. The set of things of the form (x,0) is an exact copy of the real numbers 2. (1,0)(u,v)=(u,v) for all (u,v) so there's a number that behaves like ONE in the reals 3. multiplication is commutative: (x,y)(u,v)=(u,v)(x,y) 4. the element (0,1) squared, ie (0,1)(0,1)=(-1,0) "gives -1" or more accurately, the image of -1 in the embedding of the reals into the first component, whichs is fancifully stating what point 1. said. 5. no nonZERO (remember we're calling (0,0) ZERO) numbers multiply together to give zero. this is quite subtle and not obvious from the definition. However, we're just dealing with points in the plane, and they have a distance from the origin |(x,y)| = sqrt(x^2+y^2) and if you check you'll see that |(u,v)(x,y)| = |(u,v)||(x,y)| and that the only element zero distance from the origin is (0,0). Do you see why that implies that if we multiply two nonZERO numbers then the answer is nonZERO? 6. (u,v)(u,-v)=(u^2+v^2,0). If we let S=u^2+v^2, then (u,v)(u/S,-v/S)=(1,0), the thing we're calling ONE thus we can define the multiplicative inverse of an element. And that is all the complex numbers "are". Now, for place holder notation we often find it easier to write x+iy instead of (x,y). Complex numbers are nothing to be scared of. They are harder to visualize, but mathematical things are not "what you can visualize" they are just things that obey certain rules. Learn the rules and all is well.

Does it not say, in the introductory threads, that you should search the forums before posting a question as the chances are it's been asked before and answered?

Of course the syllabus writers didnt' have encryption in mind when they wrote siad syllabus, but that wasn't the question. I doubt writers of French courses have the works of voltaire in mind when they make you learn what the french for vacuum cleaner is either. They are however testing that you can learn a definition and have some numerate skills in manipulating numbers. I'd hope te question (factorize 20 as a product of primes) does seem facile: it is. But then a lot of questions are easy if you know what the definitions are. The aim ought to be "has the person learned what factorize means and prime" not "can we make them unneccesarily jump through hoops". Your usage of monomial, binomial etc is a little incorrect. bi or mono just refers to the number of the terms and doesn't state what the terms must be: x^2+1 is a binomial expression, x^7 is a monomial.

Rings: Z, Q, R, C the integers rationals reals and complexes resp. Let S denote any of those ie let S stand for some basic ring. then M_n(S) is a ring = the nxn matrices with elements in S, as is the set of all upper triangular matrices. Let I be any interval on the real line then C^r(I) the r times differentiable functions from I to R are a ring. Sets of holomorphic, etc functions are all rings. With S as above S[x,y,..,z] the polynomial ring in arbitrary many variables is a ring, as it is if we quotient out by any ideal. Group rings are rings, as are hopf algebras, universal enveloping algebras, fields, field extensions, the algebraic numbers, the algebraic integers, finite fields, the p-adics, the p-locals. I've barely touched the surface. Hopefully you get the idea that most anything algebraic in maths has something to do with a ring in there somewhere. Heck, even lots of mathematical physics is simply finding ways some rings can act ons spaces. And, for Johnny, the identity element of *any* group is unique, as are inverses.

Johnny, the multiplication of two quaternions is simply the algebra multiplication according to the rules that ij=-ji etc. It is NOT the cross product as you assume it is. It can be written in terms of the cross product and the scalar product, but in your first line you say: "interpret the multiplication as vector cross product" if you do that then it isn't the product of the elements as quaternions. There is no need to hat or underline vectors. This is maths, we dont' do that, physicists might but we don't. It is completely unnecessary and wastes time, and we know from the definition of the quaternion norm that i, j and k have magnitude 1.

why are you hatting your i's etc? THere's no need to do that.

If you do wish to have a long discussion about them Johnny would you mind starting another thread on it? I've nothing agains talking about them, but here isn't the appropriate place.

ixi is indeed zero for the cross product, but i*i is not ixi is it? And who cares what you hold true about space. R^n is just a set with extra structure as Dave rightly points out. It may or may not correspond to anything that you hold to be real, but that is completely immaterial. Your comment about "beginning to get a handle on it" was quite funny. It reminds me of something Tom Koerner said once: a Fellow at Trinity Hall, who knows nothing of mathematics, commented to him that he'd seen the panorama documentary on Andrew Wiles and thought he almost understood the proof of Fermat's Last Theorem.

Does it not? So presumably the integral round a closed path of a meromorphic function being equal the the number of poles in the interior counted with residue (up to a constant) isn't a "real use of it in an equation"?

you might also consider that some problems involvinb prime numbers hvae rewards of 1,000,000 USD for a solution. The thind is that primes are incredibly simple things, really very easy to talk about, anyone can learn the definition immediately and start working with them. That's good as we can all play around with them. But, as you'll learn if you do more maths, the simpler something is to talk about the harder it is to do things with them. Things that are really complicated and have long definitions are easy to work with because they are very strongly constrained by their definitions. So we actually have very few tools to help us with prime numbers. We can calculate the square root of a number to thousands of decimal places very easily, but we cannot tell what the 44th Mersenne Prime will be (I think we're up to 43) because we have no clever tools. Hopefully someday we'll find a way to pick out the prime numbers quickly, using a simple formula, but that day, if it iever comes, is many millenia away.

Except of course in the extended complex plane, or extended real line (one point compactifications), when 1/infinity is defined to be zero, and 1/0 is defined to be infinity, and all the other rules that ought to be true hold as well. See, eg, the Moebius transformations of Cu{infinity}

I will make a sild surmise. It was closed, as a lot of those kinds of threads are, because they it was useless. They are not mathematical irrespective of what people may think about them, and any and all questions in them are already answered everywhere else. The only answer a question such as "why can't I divide by infinity" deserves is to have the simple standards of mathematics explained to them. Rules about rings, operations and so on. There is nothing mystical or magical or new to be said about the subject. Mathematics essentially boils down to what we may deduce from some hypotheses based upon some pre-agreed set of axioms using a certain kind of logic. These types of questions all fail to meet the simple prerequisite of understanding that. I think that is almost entirely down to people assigning far more meaning to mathematics than it needs. No one would question why you cannot invert a matrix if its determinant is zero, but for some reason the mysticism of infinity attracts odd debate.

I didn't say it was a good way of doing it, I said it was the simplest way of doing it. It is in fact the worst way of doing it without doing somethind deliberately wrong.

You have started the prime factorizations, but in this particular case you do not need to go all the way. Yes, it is trivial to do in one sense, and they are merely checking that you've understood what a prime number is, ie that you reduce to the correct thing and don't leave any composites (non-primes) in there, it also tests your ability to do arithmetic. If you think it is easy please factorize the following number: 235342352345234578345653257547967843 into its prime factors. How would you do it? Trial and error? Divide by some small numbers? How long do you think it would take you to work out the answer? The simplest method to factor N takes sqrt(N) operations - just divide by all the numbers from 1 to sqrt(N) and see how it factors. That takes a long time, and each operation takes time as well. Internet security, or at least some of it is based around RSA encryption. You can look at their website http://www.rsasecurity.com and see that if you correctly factor a number you can earn money on the RSA challenge. It should also explain RSA encyption to yous somewhere. The basic idea, though, is to use "keys". Given a message encode it as a string of numbers. You can now do some operations on these numbers to get another string and send it over the internet, or anywhere else, and no matter who reads it, and no matter if they also know how to encode the message, no one but the intended recipient can decode it. It relies on various properties of numbers, and prime numbers in particular. I know it's a cop out to omit the details, but they're quite messy, and much better explained out there already - just look around for RSA encryption.

All of the internet's security (ok, white lie, but not much of one) is dependent on the difficulty of factoring large numbers into their prime decompositions. Primes are just important everywhere in mathematics, and even in physics. Plus it's rather pretty. Why must maths have utility? Does music? Does poetry? Did you complain about being asked to read Shakespeare too?

mathsisfun and anjuru were the subjects, not you ophiolite.

If I break down and cry will there be any sympathy? How about I just send my psychiatric bills to you lot?