re Multiplying complex numbes. Note that if a triangle 0,1,a is similar to the triangle 0,b,c then c=a*b . So complex multiplication is the natural way of expressing that the sides in similar triangles are proportional. (b%1)=(c&a) . You only have to chose the two points 0 and 1. - Bo
>________________________________ > Fra: Henry Rich <[email protected]> >Til: [email protected] >Sendt: 23:18 tirsdag den 15. januar 2013 >Emne: Re: [Jprogramming] atop continues to puzzle me > >What a coincidence - I am reading Hardy's /Pure Mathematics/ and I just >finished the chapter on complex numbers, which he develops in exactly >this way. Start with a number line, and the concept of distance. Move >to points in the plane, and the concept of displacement. Then to get a >definition of 'multiplication of displacements' that satisfies >commutativity etc., you have to end up with a definition of >multiplication as add-the-angles-and-multiply-the-magnitudes. All done >without mentioning square root of anything (and actually, you can do the >multiply-the-magnitudes part by construction, using similar triangles, >so you don't need square root for that either). > >Henry Rich > >On 1/15/2013 5:04 PM, km wrote: >> Linda, >> >> Perhaps we should be in chat, but I do not think you need square root to >> introduce complex numbers, nor do young students need to calculate with them >> by hand. We have calculators now. >> >> Students are taught early about plotting points in a coordinate plane. I >> would introduce complex numbers as names for points. Point 2j8 is 2 units >> to the right and 8 units up from the origin. The point halfway between two >> complex numbers is halfway between the origin and their sum. Draw rays from >> the origin to two complex numbers. Multiplying them involves a rotation and >> stretching of the second ray by amounts indicated by the first ray. "This >> much" and 'this much." That is why the product of 0j1 and 0j1 is _1j0, >> called _1 for short. It is also why _1j0 times _1j0 is 1j0. You'll learn >> more about complex numbers when you get to the sixth grade, and you can >> already add and multiply them on your calculator. >> >> I concede that using 2j8 as a code for "2 and 8" may be a stretch -- but >> young students are flexible! >> >> Kip Murray >> >> Sent from my iPad >> >> >> On Jan 15, 2013, at 2:16 PM, Raul Miller <[email protected]> wrote: >> >>> Why is the j in i:2j8 more important than the i? >>> >>> -- >>> Raul >>> >>> On Tue, Jan 15, 2013 at 3:00 PM, Linda Alvord <[email protected]> >>> wrote: >>>> I am considering the structure of mathematics education. A domain of >>>> numbers >>>> from _2 to 2 in steps of one half could be understood and developed in >>>> elementary school even in J. It is not really sensible until square root >>>> is >>>> mastered. After that an imaginary numbers might show up. So %:_1 becomes >>>> 0j1 in J. It will take a while to master numbers like 2j8 and other >>>> imaginaries. The leap to an idiom which uses 2j8 in a totally different >>>> way is counterproductive until much later. >>>> >>>> How can you explain the connection to imaginary numbers in this expression: >>>> >>>> i:2j8 >>>> _2 _1.5 _1 _0.5 0 0.5 1 1.5 2 >>>> >>>> Linda >>>> >>>> -----Original Message----- >>>> Froch: [email protected] >>>> [mailto:[email protected]] On Behalf Of Raul Miller >>>> Sent: Tuesday, January 15, 2013 11:35 AM >>>> Tr o: [email protected] >>>> Subject: Re: [Jprogramming] atop continues to puzzle me >>>> >>>> I am uncomfortable with this reasoning, because: >>>> >>>> *) i:9j3 is related to complex arithmetic only by notation. >>>> >>>> *) class becomes boring when its pace is set for someine else (when it does >>>> not match the student's needs). >>>> >>>> *) simplicity is good, but so are different perspectives. >>>> >>>> Waiting to introduce imaginary numbers before introducing this notation >>>> seems analogous to waiting to introduce polynomials before introducing >>>> decimal numbers. >>>> >>>> Anyways, I this notation can be optional, but if I were stuck teaching a >>>> class using J (rather than providing guidance, pacing and structure to a >>>> group of students who had been taught how to learn and who were being >>>> responsible for their own education) I think I'd introduce this as an >>>> optional notation about the same time I introduced i: >>>> >>>> -- >>>> Raul >>>> >>>> On Tuesday, January 15, 2013, Linda Alvord <[email protected]> wrote: >>>>> Devon, Since I think of everything going forward from kindergarten >>>>> (maybe someday we'll do more with prenatal education) I would use Y=: >>>>> _2 + 0.5 * i.9" until the middle of the second year of algebra. >>>>> >>>>> I really like your solution! It would be a great next step when >>>>> teaching imaginary numbers. >>>>> >>>>> Linda >>>>> >>>>> -----Original Message----- >>>>> From: [email protected] >>>>> [mailto:[email protected]] On Behalf Of Devon >>>>> McCormick >>>>> Sent: Monday, January 14, 2013 2:58 PM >>>>> To: J-programming forum >>>>> Subject: Re: [Jprogramming] atop continues to puzzle me >>>>> >>>>> Linda - >>>>> >>>>> a nice short cut for your expression "Y=: _2 + 0.5 * i.9" is "Y=: i:2j8". >>>>> In general, providing the complex argument PjN to i: gives you N+1 >>>>> points from -P to P. >>>>> >>>>> Regards, >>>>> >>>>> Devon >>>>> >>>>> >>>>> On Mon, Jan 14, 2013 at 12:24 PM, Jose Mario Quintana < >>>>> [email protected]> wrote: >>>>> >>>>>> To Henry: >>>>>> >>>>>> My apologies, the worst part is that I noticed the misspelling but I >>>>>> neglected to correct it; I guess watching the playoffs and writing to >>>>>> the forum do not mix very well. >>>>>> >>>>>> To Linda: >>>>>> >>>>>> That is a nice feature; thanks for sharing it. One refreshing thing >>>>>> about J is that one never seems to stop learning it. Moreover, if one >>>>>> follows the forums and this one in particular one is shown (or >>>>>> reminded) how capable the J system really is. >>>>>> >>>>>> On Mon, Jan 14, 2013 at 3:10 AM, Linda Alvord >>>>>> <[email protected]> >>>>>> wrote: >>>>>>> Jose, Here's a simper version. Using the aspect ratio helps make >>>>>>> the derivative more obvious. I use Chrome and I don't know how >>>>>>> this will >>>>>> look >>>>>>> elsewhere. >>>>>>> >>>>>>> Load 'plot' >>>>>>> u=: -: >>>>>>> v=: *: >>>>>>> Y=:_2 + 0.5 * i.9 >>>>>>> f=: 13 :'(] ; [:|:u@v d._2 _1 0 1 2 )y' >>>>>>> f >>>>>>> plot f Y >>>>>>> 'aspect 1'plot f Y >>>>>>> >>>>>>> Linda >>>>>>> >>>>>>> -----Original Message----- >>>>>>> From: [email protected] >>>>>>> [mailto:[email protected]] On Behalf Of >>>>>>> Linda >>>>>> Alvord >>>>>>> Sent: Sunday, January 13, 2013 9:21 PM >>>>>>> To: [email protected] >>>>>>> Subject: Re: [Jprogramming] atop continues to puzzle me >>>>>>> >>>>>>> load'plot' >>>>>>> >>>>>>> u=: -: >>>>>>> v=: *: >>>>>>> Y=: _2 + 0.01 * i.401 >>>>>>> f=: 13 :'(] ; [:|:u@v d._2 _1 0 1 2 )y' >>>>>>> f >>>>>>> ] ; [: |: u@v d._2 _1 0 1 2 >>>>>>> plot f Y >>>>>>> >>>>>>> If this is in a jijs and then run, it will shw the graph you >>>>>>> expect bu >>>>>> t >>>>>>> it will provide a long J error message in a separate window. I >>>>>>> don't >>>>>> know >>>>>>> how to prevent it. >>>>>>> >>>>>>> Also, maybe when u@v can be replaced by ([:u v)"v that will >>>>>>> work >>>>>> also. >>>>>>> >>>>>>> Linda >>>>>>> >>>>>>> >>>>>>> -----Original Message----- >>>>>>> From: [email protected] >>>>>>> [mailto:[email protected]] On Behalf Of Jose >>>>>> Mario >>>>>>> Quintana >>>>>>> Sent: Sunday, January 13, 2013 5:17 PM >>>>>>> To: [email protected] >>>>>>> Subject: Re: [Jprogramming] atop continues to puzzle me >>>>>>> >>>>>>> To Raul: >>>>>>> >>>>>>>> I have no idea what "works reasonably well" means. >>>>>>> >>>>>>> That is a very subjective statement, apparently one can make use of >>>>>>> (@) and (@:) within the scope of (d.) but, of course, that depends >>>>>>> on >>>>>> one's >>>>>>> point of view. >>>>>>> >>>>>>>> But consider also: >>>>>>>> >>>>>>>> AT=: 2 :0 >>>>>>>> u@v"v >>>>>>>> ) >>>>>>>> >>>>>>>> +:AT*: d. 1 >>>>>>>> 0 4x&p."0 0 0 >>>>>>>> +:@*: d. 1 >>>>>>>> 0 4x&p. >>>>>>> >>>>>>> Th> >> --- For information about J forums see >>>>>>>> http://www.jsoftware.com/forums.htm >>>>>>> ------------------------------------------------------------------- >>>>>>> - >>>>>>> -- For information about J forums see >>>>>>> http://www.jsoftware.com/forums.htm >>>>>>> ------------------------------------------------------------------- >>>>>>> - >>>>>>> -- For information about J forums see >>>>>>> http://www.jsoftware.com/forums.htm >>>>>>> >>>>>>> ------------------------------------------------------------------- >>>>>>> - >>>>>>> -- For information about J forums see >>>>>>> http://www.jsoftware.com/forums.htm >>>>>> --------------------------------------------------------------------- >>>>>> - For information about J forums see >>>>>> http://www.jsoftware.com/forums.htm >>>>> >>>>> >>>>> >>>>> -- >>>>> Devon McCormick, CFA >>>>> ^me^ at acm. >>>>> org is my >>>>> preferred e-mail >>>>> ---------------------------------------------------------------------- >>>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>>> >>>>> ---------------------------------------------------------------------- >>>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>> ---------------------------------------------------------------------- >>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>> >>>> ---------------------------------------------------------------------- >>>> For information about J forums see http://www.jsoftware.com/forums.htm >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> >---------------------------------------------------------------------- >For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
