matt grime

it is one of two segments of a circle.

it represents the proportional 'volume' change. volume depends on the contex. eg in 1 dimension it is length, two dimensions it is area 3 it is volume as we think of it in general, and in higher dimensions it is the analogue of these quantities. if you take the segment of the that is the interval [0,1], or the square with corners (0,0), (1,0) (0,1) and (1,1) and in three d space the cube with volume 1 then the matrix sends this shape to one with volume the determinant of the matrix.

EL helped? but mathematically his post was garbage, apart from what i am led to believe is simply a reposting of a wikipedia entry (without reference) you are still talking about numbers that include infinity without thinking about which kind. if you'd like to know more about the proper mathematics of this then i am happy to talk to you about infinte cardinals and ordinals, as well as explaining what the maths behind extended number systems is. remember everything in mathematics is a human invention. if you want to know about the mathematics of 'infinity' then just ask about it, and if you abandon preconceptions about physics (which is unnecessary to the discussion) then you can learn more. you need to ask yourself what you think numbers are, and what you think you wish to denote by infinity. there is little reason to feel stupid: these things are intellectually difficult. but they do signify the difference between what mathematics 'is' and what lots of amateurs think it 'ought to be'. for instanec, let us take ordinal numbers. these are ways of ordering sets. the simplest way to think of these is, perhaps, thus. 1 is a dot. 2 is two dots with a 3 three dots and so on, label each by the number of dots in it. now we can think of a line of dots with one dot for each whole number, call this line w. this is the first infinite ordinal. we can now create an infinite ordinal exactly one bigger than w, labelled w+1 by thinking of this infinite line of dots, then one more, say on a line above it. then we make w+2 by adding one more after that dot. we can now create nw+m by taking n lines of infintely many dots and a line above it with m dots. another way to think about them is as follows. take the numbers 1/2, 2/3, 3/4, 4/5... call the set up to n/(n+1) the ordinal n. w is then the set of all of these as n ranges from 1 'to infinity' ie does not stop. w+k is the 1+k/{k+1} so the numbers 1+1/2, 1+2/3, 1+3/4... and aw+b can be gotten by adding a to all the fractions, a+1/2, a+2/3,... each of these is an infinite ordinal (ordered number) and w and w+1 are different. OR we can take cardinals. we declare two sets to have the same cardinality if there is a bijection between them. this is going beyond what you know, i suspect. but a bijection between two sets means exactly that there is a way of associating to each element in one set exactly one element in the other. for instance, the set of positive whole numbers is in bijection with the set of even positive whole numbers because there is the mapping (association) n <--> 2n. does that make sense? sometimes the association is hard to spot but we can deduce it exists. we can show for instance that there is an association between the rational numbers and the positive whole numbers. just as we associated symbols to ordinals we can associate symbols to cardinals in a slightly more convoluted way. but the key thing here is that suppose we have the set of positive whole numbers, and the set of positive whole numbers plus zero. in this language they have the same cardinality because there is a way to associate the two sets, n <--> n+1, we just move the numbers one place to the right. so here the two infinite sets have the same cardinality though one has 'one more element than the other' you see how it depends on the context?

but what are you defining to be infintiy! mathematically to make sense of that statement you are implicitly working in the extended eral numbers, that is the real numbers and one or two more elements infinity or plus/minus infinity. in that area the arithmetic is defined in such a way that infinity +k=infinity for any other real number k. (infinity minus infinity still doesn't make sense here). there are other systems which have all the usual real numbers and lots more besides in which "infinity+1" is nto the same as infintiy. you see, there are lots of things in maths with infinite labels. the problem is you haven't fixed what you mean to talk about. do you mean the extended reals, the hyperreals, the ordinals or cardinals? in maths there isn't some handy little object which is "infinity", ideally maths would never use the word for exactly the reasons that your thinkng about. remember (and this may come as a surprise) there are no absolute truths in maths, it is all relative. so, what makes you think infinity is an object which it even makes sense to add anythig to? what number system are you assuming that you're talking about?

though there is nothing to suppose that 1/2.1=1/2, ie it might not be the multiplication inherited from Z being inside C/R/Q. equivalently there is nothing making us suppose that the vector sum of 1 and 1 is 2. it is easy to show, even given that, that Z is not a v.s. over R or C. if s and t are distinct elements in R then s.1 and t.1 are distinct elements in Z, thus if it were a v.s. over R or C then there would have to be an uncountable number of elements in Z.

independent or otherwise is moot and would depend on your opinion. my comments were aimed at the OP and have nothing to do with physics. why single out "infinty" in soem undefined sense for sepcial consideration of platonism in mathematics? as soon as they care to define what they think "infinity" might be the easier any conversation would be.

the point at infinity of C is the point added in the one point compactification, there are one or two points added to the real line in their compactifications coomonly called (plus and minus) infinity, and these extedned systems have soem *limited* arithmetic before anyone starts asking what infinity over infinity is. it is useful to say thigns in mathematics tend to infinity if they grow without bound (ie converge to the point at infinity in the one point compactification) infinity is thus, if you must, just as "real" as anything else in mathematics and can be said to exist in what ever sense you take anything in mathematics to exist. i suspect though you may not wish to post this in the mathematics section since you are talking, probably, in a non-mathematical sense, though i doubt you know this.

consider the transform M(z)=z/(z-2) this is a Mobius map: it takes circles and straight lines to circles and straight lines. The locus L you want satisfies z in L iff arg(M(z))=pi/4, thus M transrforms L into the half line starting at the origin and going at an angle of pi/4 to the real axis. L is then the inverse image of this half line under M. L is therefore either another half line or *some* segment of a circle. clealry it is only some arc of the circle you have given, not all of it. it will be one of the two arcs from the real points on the circle.

You state that the some ponts of said circle are not on the locus. thus you have stated locus is a circle with some points removed. a circle with 2 points ommited is not a circle.

no, perhaps i am misremembering a course a took several years ago, though as you yourself state it isn't a circle so i don't see the issue.

just have a google for locus points complex plane, i'm sure that you'll find a page explaining how to do these locus questions (it isn#t a circle)

are you claiming that the locus is the circle of center (1,-1) and radius 4? because there is certainly a real number w on that locus and thus w/(w-2) is real and its argument therefore isn't pi/4

try thinking it out differently. instead of f(e)=e is the same as f(e)-e=0, now what can you do with that?

f could be anything, or f could be given as some specific function. it would depend on the question, just like x can be 2 if we specify x=2 or x can be a real number. so y=f(x) is the same as y=x when f is the identity function. remember almost no functions can be written nicely, eg sin(x) or soemthing, so we often specify at the start what f(x) is, and then use f to refer to it for the rest of the discussion. eg, suppose f is defiend such that f(x) is 1 if x is rational and 0 otherwise then... some discussion of f we woulldn't like to have to specify what f is in this case every time would we?

do the substitutioin, use partial fractions do the resulting integrals

this is a good example of a simple questiopn that is quite hard to answer. right, let X and Y be two sets. a function is a way of assigning, unambiguously, an element of Y to each element of X. Unambiguous means that x in X gets assigned to exactly 1 one element of Y. they way we write this formally is to store the information as ordered pairs. if you prefer you can think of these as coordinates (u,v) where u is in X and v is the element of Y assigned to it. Sometimes we choose to describe v as being a funtion of u by writing v=f(u) where f is another way of writing the function. so f is telling you how the elements of X and Y are related. it implies nothing, it is just notation.

And that is true of all functions for almost all inputs. it is slightly disigenuous to pretend that sin is somehow distinctly better than "bling" as above.

it is integrable; you know what integrable is? it does not mean having a nice antiderivative that you can write down in terms of nice functions. x^x is (away from 0) riemann integrable, lesbegue integrable and stiltjes integrable, hence integrable in every sense. the definition of its antiderivative is exactly what DQW wriote for "bling", or rather thatis one of its anti derivatives, bling +k for any real k would also do.

why hmm? that is perfectly true and follows the wittgenstien approach of something is its definition. what is -1? it is the number that when added to 1 gives 0. it is the correct and necessary approach to many parts of mathematics, or so some of us would say. what is sin(35.657357)? what is pi? cos most people mistakenly think decimals are real numbers, and that the answers on calculator are correct doesnj't mean they are. i once saw a text book used in schools that staetd 1.41 was the square root of two.

the importnat thing is that as x tends to zero so does O(x) or (X^n) for any positive n. just think of it as a function - if f(x) and g(x) tend to zero as x tends to zero then (1+f)/(2+g) tends to 1/3 as x tends to zero.

it is integrable in any sense of the word, but the function that is its anti-derivative is not expressable in elementary terms. this is no surprise, alomst all functions are like this. sadly people are preconditioned to believing that the baby examples they're taught in school reflect a general pattern that holds "in the real world" nb in the real world means, effectively, in a mathematical situation that arises from studyign something rather than the special cases dreamt up as exercises.

because the resulting expansion after applying the diagonal argument is not necessarily a rational number. generally it is better to write Cantor's proof as a constructivits proof rather than a contradictory one - there is no need to claim that the list in the 'table' is complete; just assume it is any injection from N to R, or [0,1], or some subset of [0,1] and it follows it is never surjective. also, you should be careful here - it is better to assume these strings of 0s and 1s are a subset of the base 3 expansions. this negates any issues with numbers that possess two different binary expansions.

then you aren't doing maths whatever maths is, and surely 0* anything is zero by your own beliefs? this isn't maths. it is nonsense; i don't mean that in a derogatory way, merely an observation of fact. maths deals with deductions from premises, reasoning from definitions. you are doing no such thing.

in what ring? there is no such element in the real numbers as "undefined" so it makes no sense to ask what happens if you multiply a non-existent object by zero when said non-existent object is not something that can be multiplied. i know what you think they mean, and what they should be read as, but it makes no sense to write that. it as meaningful to ask 'is 0*hedgehog=cricket bat' true?

what? it is undefined. what is *? what is 0? (i mean that most seriosly, for the expression is nothing in any part of mathematics, is it?) what is an "undefined"? 0 is usuallly an additive identity, but in what ring? whatoperation is *? on what objects? as written it makes no sense.