Was Wallis himself the first to assume x^0 =1 even for x=0? See,
http://www.maths.tcd.ie/pub/HistMath/People/Wallis/RouseBall/RB_Wallis.html Perhaps a Latin fluent member of the forum could shed some light into the dark: https://archive.org/details/ArithmeticaInfinitorum ? At any rate, his table on page 105 looks interesting. I wonder what is the shortest J expression that can reproduce it... :) On Fri, Jan 17, 2014 at 4:22 PM, Roger Hui <rogerhui.can...@gmail.com>wrote: > Come to think of it, the insight that 1=0^0 is required for the standard > statement of polynomials may have come from Ken Iverson. Knuth doesn't > mention this point and only mentions the binomial theorem. (Same with "Ask > a Mathematician".) But the polynomial argument is more convincing because > polynomials are ubiquitous. > > > > On Fri, Jan 17, 2014 at 12:56 PM, Roger Hui <rogerhui.can...@gmail.com > >wrote: > > > Found it. It is in the very same paper. > > http://arxiv.org/PS_cache/math/pdf/9205/9205211v1.pdf . > > > > On page 6, Knuth wrote: > > > > ... The debate stopped there, apparently with the conclusion that 0^0 > > should be undefined. > > > > But no, no, ten thousand times no! Anybody who wants the binomial > theorem > > ... to hold for at least one non-negative integer n _must_ before that > 0^0 > > = 1, ... > > > > > > "Ask a Mathematican" > > > > > http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/ > > has an interesting and useful discussion on this issue. In it the > > "Mathematician" wrote: > > > > Zero raised to the zero power is one. Why? Because mathematicians said > > so. No really, it's true. > > > > And then goes on to explain why 1=0^0 is a good idea. > > > > > > > > > > > > > > On Fri, Jan 17, 2014 at 12:30 PM, Roger Hui <rogerhui.can...@gmail.com > >wrote: > > > >> BTW, Knuth did something else which typifies APL thinking. In a note or > >> paper (I can not find it now), he argued strongly that 1=0^0, not > >> undefined, not 0, not anything else. The common conventional statement > of > >> a polynomial, p(x)=sigma(k=0;k<=n) a[k]*x^k, requires that x^0 be 1. > Some > >> writers are aware of this dependency and, being careful, write instead > the > >> ugly p(x)=a[0]+sigma(k=1;k<=n)a[k]*x^k. > >> > >> Attention to edge cases is typical of APL thinking. It's another way to > >> stay in the world of expressions and away from the world of statements. > >> You know: > >> > >> if k=0 then > >> a[0] > >> else > >> a[k]*x^k > >> endif > >> > >> > >> > >> > >> On Wed, Jan 15, 2014 at 6:20 PM, Roger Hui <rogerhui.can...@gmail.com > >wrote: > >> > >>> One aspect: J/APL programmers tend to stay in the nice world of > >>> expressions and avoid the nastier world of statements. This tendency > >>> pushes you towards array thinking and away from scalar thinking. > >>> > >>> For example, if b is a boolean array, and you want 4 where b is 0 and > 17 > >>> where b is 1, write: > >>> > >>> (4*0=b)+(17*1=b) > >>> > >>> And of course the signs of real numbers x are: > >>> > >>> (x>0)-(x<0) > >>> > >>> Even Knuth, an eminent mathematician and computer scientist but not an > >>> APL programmer, knows to <strike>steal</strike> adopt this idea. See: > Knuth, > >>> *Two Notes on Notation*< > http://arxiv.org/PS_cache/math/pdf/9205/9205211v1.pdf>, > >>> 1992-05-01. In the first half of the paper he describes how "Iverson's > >>> convention" can be used to simplify the statement and manipulation of > sums. > >>> > >>> See also: > >>> > >>> http://www.jsoftware.com/papers/perlis77.htm > >>> http://www.jsoftware.com/papers/perlis78.htm > >>> http://www.jsoftware.com/papers/APLQA.htm#Perlis-foreword > >>> > >>> > >>> > >>> > >>> > >>> On Wed, Jan 15, 2014 at 5:32 PM, Joe Bogner <joebog...@gmail.com> > wrote: > >>> > >>>> I went googling for some deeper material on how to think like an APL > >>>> programmer. I have read/skimmed through a good set of the material on > >>>> http://jsoftware.com/papers/ and have skimmed through many of the > >>>> books listed on http://www.jsoftware.com/jwiki/Books. > >>>> > >>>> Are there any specific recommendations, free or for purchase? Or, > >>>> perhaps I should spend more time with the list above. > >>>> > >>>> I found this, The APL Idiom List by Perlis and Rugaber, which looks > >>>> similar to what I'm looking for: > >>>> http://archive.vector.org.uk/resource/yaleidioms.pdf. > >>>> > >>>> The review of this book looks like what I'm after, > >>>> > >>>> > http://www.amazon.com/Handbook-APL-programming-Clark-Wiedmann/dp/0884050262 > >>>> , > >>>> constructing useful programs and going into more depth. > >>>> > >>>> Or something of the style of The Little Schemer, > >>>> http://scottn.us/downloads/The_Little_Schemer.pdf > >>>> > >>>> I searched the forum and had trouble finding a relevant post > >>>> ---------------------------------------------------------------------- > >>>> 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
