From: "Clifford J. Nelson" Subject: Re: Progression of operators? Date: Tue, 31 Aug 1999 22:13:24 GMT Newsgroups: sci.math,sci.logic Keywords: What comes next after addition, multiplication, exponentiation? Seth Russell wrote: > I can't find the answer to this stupid question anywhere, > please help! > > It seems that there is a natural progression of operators > as follows: > > 1. (addition, subtraction) > 2. (multiplication, division) > 3. (powers, roots) > 4. ? > > What is the fourth ordered pair in that progression? I used to signify it (I called them superpowers) with the number sign (#). a#2 = a^a. a#3 = a^(a^a) etc.. The continued fraction p(1) = 1 and p(x) = Floor[f(x)]+1/p(1/(x-Floor[f(x)])) gives division if f(x) = x, and it gives log(a) (base b) if f(a) = a/b and superlog(a) (base b) if f(a) = log(a)/log(b). Then you can use numerical methods to zero in on a superpower by using its inverse superlog. It works for certain positive values of a and b. Cliff Nelson > > > How is it that these operators necessarily do not > come in triplets as follows? > > 1. (addition, subtraction, athigician) > 2. (multiplication, division, duomatician) > 3. (powers, roots, trigrission) > 4. ? > > And if they could so come in triplets, how do we do > athigician, duomatician, and trigrission? > > TIA > Seth Russell ============================================================================== From: "Clifford J. Nelson" Subject: Re: Progression of operators? Date: Tue, 31 Aug 1999 23:46:59 GMT Newsgroups: sci.math,sci.logic "Clifford J. Nelson" wrote: > "Clifford J. Nelson" wrote: > > > Seth Russell wrote: > > > > > I can't find the answer to this stupid question anywhere, > > > please help! > > > > > > It seems that there is a natural progression of operators > > > as follows: > > > > > > 1. (addition, subtraction) > > > 2. (multiplication, division) > > > 3. (powers, roots) > > > 4. ? > > > > > > What is the fourth ordered pair in that progression? > > > > I used to signify it (I called them superpowers) with the number sign > > (#). a#2 = a^a. a#3 = a^(a^a) etc.. > > > [snip] I apologize for the stuff I snipped. Its been about ten years since I thought of this and my memory is very bad, so, I worked it out with Mathematica again. If f[x,y] is x-y then p[x,y] is division x/y. If f[x,y] is x/y then p[x,y] is log(x) (base y). If f[x,y] is Log[x]/Log[y] then p[x,y] is the superlog of x (base y) which can be used to find the superpower with numerical methods for certain positive values of x and y. Clear[p] p[x_,x_] := 1 p[0,_] := 1 p[_,0] := 1 p[x_,1] := x p[x_,y_] :=1+ p[f[x,y],y] /; x >= y p[x_,y_] :=1/p[y,x] /; x < y The program above will exceed the recursion depth sometimes so, you stop it and return the value of x for an approximation. Cliff Nelson ============================================================================== [Minor typos fixed here, pursuant to follow-up posts --djr] ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: polypowers/iterated exponentials Date: 26 Dec 1999 17:32:34 -0500 Newsgroups: sci.math.research Cris Moore [sci.math.research Wed, 22 Dec 1999 18:53:37 -0700] wrote >I know this question gets asked quite often, but is there a >natural way (with an infinite sum or product, say, or other >iterative approximation) to define x^(x^(...x)) k times for >non-integer k? > >The reason I ask (and I'm aware of Abel's equation etc.) is >that in Martin Gardner's collection "Knotted Doughnuts and >Other Mathematical Entertainments," he says that the chemist >Aristid V. Grosse worked on this, and found, for instance, >that pi to the pi pi times is 588916326.+ I have not been >able to find any record of Grosse's work on the net. Does >anyone know if this, or analogous results, were ever published? You may find the following paper useful: R. Arthur Knoebel, "Exponentials Reiterated", The American Mathematical Monthly 88(4) (April 1981), pages 235-252. There are 125 references in its bibliography, by my count. Knoebel makes some remarks about the analytic continuation of z^(z^(z^...)) on p. 246. The issue of iterated TETRATION for positive real numbers is discussed in E. M. Wright, "Iteration of the exponential function", The Quarterly Journal of Mathematics (Oxford) 18 (1947), 228-235. I don't believe much, if anything, is known regarding the existence of a mathematically satisfying analytic continuation for the tetration operation. Knoebel brings this matter up on p. 247. On this page he also poses the question to what extent the Ackermann function can be analytically continued. Section 2 (pp. 238-239) of Knoebel's article is titled "An Application in Biochemistry", but I don't see Aristid V. Grosse's name mentioned there or in Knoebel's bibliography. More information related to your question might be contained among the references and links I give in my five sci.math posts below. Most of my remarks above were lifted from the Sept. 9 post. Progression of operators? [Sept. 3, 1999] Re: Progression of operators? (expanded answer) [3 POSTS: Sept. 5, 1999, Sept. 5, 1999, Sept. 6, 1999] Problems in (0,1/e) [Sept. 9, 1999] Dave L. Renfro ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: polypowers/iterated exponentials Date: 26 Dec 1999 17:32:35 -0500 Newsgroups: sci.math.research Cris Moore [sci.math.research Wed, 22 Dec 1999 18:53:37 -0700] wrote >I know this question gets asked quite often, but is there a >natural way (with an infinite sum or product, say, or other >iterative approximation) to define x^(x^(...x)) k times for >non-integer k? [snip] Some information about your question in addition to what I previously sent in can be found in John Bailey's Nov. 16, 1999 post and G. A. Edgar's Nov. 16, 1999 post (Edgar's post gives a copy of an older post by Zdislav V. Kovarik) under the sci.math thread "Interpolation of iterated exponentiation?" at . Dave L. Renfro