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
