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Re: Work on Basic Mathematica Stephen!
Posted:
May 20, 2013 5:03 AM


I got a bit carried away in my use of the word "ordering", so I feel I need to write a few more words, even though this entire discussion is a complete red herring and has no relevance at all to Mathematica. (But it was not I who started it )
Normally when one uses the concept of an order or ordering one requires antisymmetry, that is: if a<=b and b<=a then a=b. If you require that, having a norm will not give you an ordering, since, of course, a number of elements can have the same norm and not be equal. This is, in fact, called a preorder.
So the modulus of a complex number defines a preorder. But even this preorder in not compatible with the algebraic structure. In fact there no compatible preorder (not just order) can be defined.
Andrzej Kozlowski
On 19 May 2013, at 13:35, Andrzej Kozlowski <akoz@mimuw.edu.pl> wrote:
> A metric does not give you an "ordering", of course: it only gives you a distance between numbers. A norm (which is a mathematical formulation of the concept of "size") does give an ordering (a norm defines distance but distance does not necessarily come from a norm): you can always says that an object comes before another object if the norm of the former object is smaller than that of the latter. But the kind of orderings that one is interested in, in the context of algebraic structures such as complex numbers and real numbers are those that are compatible with these structures. In other words, you want things like > if a< b then a+x< b+ x etc. And this is what you can't get in the case of complex numbers. There is no way to define < that will satisfy such rules. (Note that a<b does not imply that a+x<b+x, for example.) > That's a pretty trivial fact, which does not need any any advanced mathematics or MathWorld. > > Also, it has completely no relation to the size of complex numbers (they are simply 2 dimensional vectors and they have the natural concept of size taught in high school), and, needless to say, all this discussion has exactly zero relevance to the function Chop. > > > Andrzej Kozlowski > > > > > On 19 May 2013, at 11:50, ?iso88591?Q?J=E1nos_L=F6bb?= <janos@lobb.com> wrote: > >> I can imagine the following "natural" ordering of complex numbers. >> >> Imagine the Riemann sphere and take a real number epsilon, epsilon <<1. Then start to 'peel' this Riemann sphere, like an orange from the (0,0,1) point down to the (0,0,0) point and lay it down on the 2 dimensional plane. You will get a Cornu spiral with thickness epsilon. Then you can define an epsilonmetric on it by selecting one of the end points and define the the 'distance' as the length of the curve from that end point. You can get the real distance by tending with epsilon to 0. Then you will have a "natural" metric and with that an ordering of the complex numbers. >> >> So the process is to project the complex number to the Riemann sphere and find where it is placed on the above described 0thickness Cornu spiral. >> >> J=E1nos >> >> On May 18, 2013, at 2:40 AM, Andrzej Kozlowski <akozlowski@gmail.com> wrote: >> >>> But of course they do not have an ordering. But they do have size! A >>> complex number has length: called its modulus. A complex number can = be >>> very small: when its modulus is small, and very large. Of course there >>> are uncountably many complex numbers with the same modulus and you can't >>> order them. But it is possible to decide whether a complex number is >>> close to the real line or not, there is no need for any ordering here. >>> >>> By the way, Mathworld is of course quite correct here but honestly and >>> with all modesty, I do not consider it more authoritative on this topic >>> than myself. >>> >>> Andrzej Kozlowski >>> >>> >>> >>> >>> >>> On 17 May 2013, at 11:49, Peter Klamser <klamser@googlemail.com> wrote: >>> >>>> http://mathworld.wolfram.com/ComplexNumber.html says: >>>> >>>> "Unlike real numbers, complex numbers do not have a natural ordering, >>>> so there is no analog of complexvalued inequalities. This property is >>>> not so surprising however when they are viewed as being elements in >>>> the complex plane, since points in a plane also lack a natural >>>> ordering." >>>> >>>> Peter >>>> >>>> 2013/5/17 Andrzej Kozlowski <akozlowski@gmail.com>: >>>>> >>>>> On 16 May 2013, at 09:28, Peter Klamser <klamser@googlemail.com> wrote: >>>>> >>>>>> This is an interesting discussion. But if it can be useful, we have to >>>>>> make short proposals. Nobody has the time to read long texts. >>>>>> >>>>>> A) First proposal: Identify useless or false constructions in Mathematica >>>>>> >>>>>> aa) Eliminating Chop[] for complex numbers. Complex numbers are >>>>>> oderless and therefore nobody call estimate, weather the distance of 1 >>>>>> + 10^google i to the real numbers is small or big. >>>>> >>>>> You are mistaken. The issue of order and the issue of distance are entirely different and unrelated. The complex numbers are a one dimensional complex Hilbert space with the standard inner product, hence a (complete) metric space. The distance between two complex numbers is well defined and so it the distance of a complex number from the real line. Also, it would be crazy if Mathematica itself decided that complex numbers which are sufficiently close to the real line are actually real one. This sort of thing should be left to the user (obviously!) and this is exactly what Chop does (it has a second argument, you know). >>>>> If I understand your suggestion correctly, it is probably the worst suggestion for an "improvement" in Mathematica I have ever read on this forum. >>>>> Fortunately, it is 100% sure that it will be ignored as all such suggestions usually are. >>>>> >>>>> Andrzej Kozlowski >>>>> >>>>> >>>>>> Chop[] is the >>>>>> result of Mathematica design, that it presents often complex results, where >>>>>> real values are the simpler result and can be reached by >>>>>> ComplexExpand[]. >>>>>> The simplest solution is always the best solution. >>>>>> >>>>>> B) Mathematica should be integrated seamless in Tex without using copy and >>>>>> paste. The should be the cell type Tex, and the printout and save >>>>>> modus Tex, where all other cells are suppressed. >>>>>> >>>>>> Peter >>>>>> >>>>>> >>>>>> >>>>>> 2013/5/15 Szabolcs HorvE1t <szhorvat@gmail.com>: >>>>>>> >>>>>>> While some individual points are debatable, I also completely = agree >>>>>>> with David's main message. Mathematica needs more focus, = particularly >>>>>>> some work on stability and robustness. Even if it would come at = a cost >>>>>>> of new features (there are many features I'd like to see, but = you can't >>>>>>> have everything). >>>>>>> >>>>>>> Re: "if I click in an existing Input cell and do a line return = the >>>>>>> Messages window opens with a contact WRI if this happens = message" >>>>>>> >>>>>>> This bug (on OS X) is really annoying when typing code in = another >>>>>>> language in a Mathematica string. If will sometimes completely = ruin >>>>>>> the typed text if I press return several times while inside a = string. >>>>>>> It's a bit ironic that this bug appeared in the same release = which >>>>>>> brought us RLink (which I'm using these days, i.e. I'm typing a = lot of >>>>>>> R code inside Mathematica strings). >>>>>>> >>>>>>> On 20130512 07:28:34 +0000, djmpark said: >>>>>>>> (I've renamed this and started a new thread because my reply is = not exactly >>>>>>>> to the question.) >>>>>>>> >>>>>>>> Oh, what a wonderful Wolfram blog! Earlier Stephen hinted at = Mathematica as >>>>>>>> an iPhone app. Now it's data mining Facebook data (Gee I wonder = if >>>>>>>> Zuckerberg has thought of that? He might be able to develop a = great business >>>>>>>> model.) Can Twitter be far behind? There are many significant = mathematical >>>>>>>> equations that will fit into 64 characters  or whatever the = limit is. >>>>>>>> Ramanujan would probably have done well on Twitter. And women = are more >>>>>>>> interested in personal relationships and men are more = interested in sports? >>>>>>>> Who would have thought? The average person on Facebook has 342 = friends! Well >>>>>>>> there are friends and there are friends. Montaigne wrote that = his friendship >>>>>>>> with Etienne de La BoE9tie was such that "So many coincidences = are needed to >>>>>>>> build [it up] that it is a lot if fortune can do it once in = three >>>>>>>> centuries." One might say, ephemera in ephemera out. >>>>>>>> >>>>>>>> For the dwindling few of us who still have desktop computers = and large >>>>>>>> screens, or maybe two large screens, who are interested in = learning or doing >>>>>>>> some extended mathematics, and the even fewer who would like to = write >>>>>>>> literate Mathematica notebooks as technical documents, I wonder = if Stephen >>>>>>>> could find some time to attend to basic Mathematica, fixing its = problems and >>>>>>>> fulfilling its vision? >>>>>>>> >>>>>>>> Mathematica lacks stability. Things that worked fine in one = version don't >>>>>>>> work in the next. Especially troubling to me is the basic user = interface >>>>> >>> >>> >> >> >



