Is it worth asking Knuth? He's still on email. It sounds like Ken's influence needs more credit.
If the first email client was by ipsharp then maybe the horse has bolted on that too though. -Steven On 27 Mar 2014, at 23:06, Jose Mario Quintana <jose.mario.quint...@gmail.com> wrote: > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
