Right, by construction (area) is meant to be calculating successive approximations to the (optimal) area of a semicircle with perimeter of 1 (ignoring the straight segment length). So, the corresponding circle had a perimeter 2; thus a ratio of 1/pi and an area pi /(pi^2)=1/pi meaning that the semicircle had an area 1/(2 pi). WolframAlpha confirmed that the formula I got (area=. (1 % 4 * ]) * 3 o. (2 %~ o.1) * 1 - %) would, in principle, converge to 1/(2 pi). I am aware that J does not always interpret monadic verbs with infinity as their argument as limits; I was merely pointing out, too succinctly, that this was one of those cases. As far as I can see J was actually evaluating area _ as 0 * 1.63312394e16.
On Wed, Feb 27, 2013 at 5:35 PM, Raul Miller <rauldmil...@gmail.com> wrote: > Yes, perhaps it does not need to be said that limits are "better than > infinity" while dealing with some of the same issues (we are still > avoiding some some trivialities, while edging up on "the right > answer"). > > That said, here's a J perspective on this issue of limits: > > area=: (1 % 4 * ]) * 3 o. (2 %~ o.1) * 1 - % > (*:100)&* @: area 10^i.20 > 0 1578.44 1591.42 1591.55 1591.55 1591.55 1591.55 1591.55 1591.55 > 1591.55 1591.55 1591.55 1591.54 1589.63 1579.64 1547.47 882.529 > 408.281 40.8281 4.08281 > > Here, given that increasing the number of sides will be increasing the > area we have some reason to believe the limit is approximately 1591.55 > > We'd of course want some independent way of verifying that that answer > is valid. In this case, I think we should believe that we are > approximating a calculation of the area of a circle where the > circumference of the circle is 100. This means that its radius should > be 100%2p1 and its area should be 2p1 * *:radius or: > > 2p1**:100%2p1 > 1591.55 > > Or, corresponding to the limit you saw in wolfram's system: > > %2p1 > 0.159155 > 2p1**:%2p1 > 0.159155 > > (If the circle has unit circumference the numerical value of its area > matches the numerical value of its radius.) > > FYI, > > -- > Raul > > On Wed, Feb 27, 2013 at 2:44 PM, Jose Mario Quintana > <jose.mario.quint...@gmail.com> wrote: > > I asked WolframAlpha the following: limit of ((x/4) tan((pi/2)(1-x))) as > > x-> 0 and it replied: 1/(2 Pi) (together with a nice graph). > > > > > > On Wed, Feb 27, 2013 at 10:41 AM, Raul Miller <rauldmil...@gmail.com> > wrote: > > > >> On Wed, Feb 27, 2013 at 7:30 AM, Aai <agroeneveld...@gmail.com> wrote: > >> >> (*:100)&* @: area _ > >> >> 0 > >> >> No, unfortunately J does not interpret the above sentence in that > sense. > >> > > >> > If I'm not wrong then J is right about this one: > >> > > >> > pi(n-1) pi 0 > >> > tg ------- tg ---- > >> > 2 n 2 0 > >> > lim -------------- = ---------- = ---- = 0 > >> > n -> oo 4 n oo oo > >> > > >> > But the accuracy gives us a much earlier decline to zero, because > >> ... > >> > >> It's usually a mistake to use infinity in calculations - infinity is > >> an inconsistent number so you should expect inconsistent results when > >> it is used in calculations. (Something related often happens when > >> reasoning about division by an unknown sum.) > >> > >> Infinity is mostly convenient way of indicating neglect and, ideally, > >> focussing the conversation elsewhere. > >> > >> -- > >> Raul > >> ---------------------------------------------------------------------- > >> 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
