[Edu-sig] Where mathematics comes from

[Kirby Urner]

At 01:28 PM 1/26/2003 -0800, Kirby Urner wrote:

>The expansion is (1/0! 1/1! 1/2! 1/3! 1/4!...) where these are the
>coefficients of a polynomial, with x^0, x^1, x^2, x^3... respectively.

Just for comparison and contrast, in the J language, we get
Taylor Expansions of many functions, plus polynomials themselves,
implemented as primitive "verbs" (the lingo is resolutely
linguistic-grammatical -- nouns, verbs, adverbs, conjunctions,
and even gerunds, get defined):

(NB. means comment = "nota bene")

    exp =: ^  NB. ^ in front of a noun is like Python's exp(noun)

I just renamed a verb to look more like Python.

I'm going to ask for the first 10 coefficients of the polynomial
expansion of e^x, in rational format ( 1r2 means 1/2 or one half):

       taylor   =: t.
       rational =: x:  NB. make of 'rational type' (coming in Python)
       range    =: i.  NB. like i. 10 returns 0 1 2 3...9 like range(10)

       exp taylor (range rational 10)  NB. <-- user, next line is computer
   1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880

(i.e. 1/0! 1/1! 1/2! 1/3! 1/4! etc.)

       coeffs =:  exp taylor (range rational 30) NB. same, but more terms

       coeffs p. 3  NB.  use polynomial operator to evaluate p(x), x=3
   20.0855

       ^3           NB.  note, same answer
   20.0855

And of course:

     coeffs p. 0j1p1  NB.  0j1p1 means i pi (like Python, J uses j)
  _1

And even more exact answer than Python's (thanks to internal rounding
I guess).

Kirby