Pollution is a good choice of words here. There's a problem in assuming that = always has the same meaning. There's another problem in assuming that textbooks are accurate. These are typically not big problems, but they are (from some points of view) significant problems (and the scale of these problems reflect underlying flaws in both our educational process and how we manage it).
Watch: 0 1 2 3 4 5 * 0 0 0 0 0 0 0 When we ask about what 0^0 means, we are asking "what is the number which (when not multiplied by zero) would give zero as a result if it was multiplied by zero". It's a trash question and any answer is going to be trash. Put differently: you cannot expect a computer to do your thinking for you (unless you are prepared to wait infinite time for good answers). That said, 1=0^0 is just as valid as any other answer. Anyone claiming it's a wrong answer (or claiming "it's not even wrong") is missing the point. It's a completely valid answer. The problem is that it's not the only valid answer and that just annoys some people. But maybe some people deserve to be annoyed by this kind of issue? Thanks, -- Raul On Fri, Mar 28, 2014 at 7:32 AM, Bo Jacoby <bojac...@yahoo.dk> wrote: > The wikipedia article on exponentiation > > https://en.wikipedia.org/wiki/Exponentiation#Zero_exponent > > is polluted by warnings against setting (0^0)=1. I have had a very long > discussion on the talk page > > > https://en.wikipedia.org/wiki/Talk:Exponentiation > > You may consider offering support to the point of view that (0^0)=1 > > > Thanks. Bo. > > > > > Den 10:10 fredag den 28. marts 2014 skrev Marc Simpson <m...@0branch.com>: > > Simplified: > > > > (!+/~)i.11 > > > > > >On Fri, Mar 28, 2014 at 9:06 AM, Pascal Jasmin <godspiral2...@yahoo.ca> > wrote: > >> well done, > >> > >> the tacit version: > >> > >> (] !"1 +/~) i.11 > >> > >> > >> > >> ----- Original Message ----- > >> From: Bo Jacoby <bojac...@yahoo.dk> > >> To: "programm...@jsoftware.com" <programm...@jsoftware.com> > >> Cc: > >> Sent: Friday, March 28, 2014 3:23:06 AM > >> Subject: Re: [Jprogramming] Applied APL - How to think like an APL > programmer? > >> > >> "his table on page 105 looks interesting. I wonder what is the > shortest J expression that can reproduce it" > >> > >> This one may not be the shortest, but it works: > >> > >> n!"1 n+/n=.i.11 > >> > >> > >> > >> > >> > >> > >> Den 0:07 fredag den 28. marts 2014 skrev Jose Mario Quintana < > jose.mario.quint...@gmail.com>: > >> > >> 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 > >> ---------------------------------------------------------------------- > >> 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
