

Subject:
addition, multiplication, exponentiation, then ???
Category: Science > Math Asked by: mxnmatchga List Price: $20.00 
Posted:
03 Jul 2006 19:10 PDT
Expires: 02 Aug 2006 19:10 PDT Question ID: 743129 
I've been toying around with figuring out what comes after exponentiation. I've come up with a simplistic (meaning it works but not efficiently) way of calculating those values for integers >= 1, but I'd like to also figure out how to do real numbers as well as negative numbers. Furthermore, I'd actually like to be able to calculate these values for x and y where x and y are not tiny numbers. I've found that my code slows to a craw for small x,y values like g(3,4,10). That's not surprising considering that g(3,4,9) is a 39457 digit base 10 number, but still... First, here's what I've come up with. g(a,x,y) is a function which gives the result of the operation denoted by "a". a is an integer >= 0 x is an integer > 0 y is an integer > 0 g(0,x,y) = x + y g(1,x,y) = x * y g(2,x,y) = x ^ y To calculate the result of g(a,x,y): g(0,x,y) = x + y g(a,x,1) = x [a > 0] g(a,x,y) = g(a1, x, g(a,x,y1) ) [a>0, x>0, y>1] Things I've proven based on the above axioms: g(a,x,y) = g( a1, g(a,x,y1), x ) [a>0, x>0, y>1] g(a+1,x,2) = g(a,x,x) g(a,1,y) = 1 [a>=2] I'd like to be able to use real numbers for x and y. Once I get that then I can figure out what comes next in the sequence: subtraction, division, taking the root I'd like to be able to have negative numbers for x and y, although negative numbers aren't allowed as exponents, so I guess maybe y would have to be required to be positive for a=2. I don't know, but it may need to be positive for a>=2. I haven't proven that yet. g(a,x,y) = g(a,y,x) [a>=0, a<=1] In other words, addition and multiplication are commutative. I know that a=2 (exponentiation) is not commutative, but I'd like to prove whether or not that is the case for a>2. I have to keep y>0 because g(0,x,0) = x g(1,x,0) = 0 g(2,x,0) = 1 I have no idea what g(a,x,0) would/should be for a>2. If anyone is interested I can post the source code and the proofs, but it's probably not useful until I get a more complete algorithm that includes the things I mentioned above. 

Subject:
Re: addition, multiplication, exponentiation, then ???
Answered By: eiffelga on 04 Jul 2006 05:31 PDT Rated: 
Hi mxnmatchga, The operation after addition, multiplication and exponentiation is known variously as hyper exponentiation, tetration or power tower. Although many aspects of tetration have been studied since ###'s time, it is still a fertile field of mathematical research. NOTATION A wide variety of notation is used for tetration and other hyper operators. What you have defined as g(3, x, y) is also notated as: hyper4(x, y) hyper(x, 4, y) x ^ ^ y (Knuth's uparrow notation) x > y > 2 (Conway's notation) See: Hyper operator (Wikipedia) http://en.wikipedia.org/wiki/Hyper_operator Knuth's uparrow notation (Wikipedia) http://en.wikipedia.org/wiki/Knuth%27s_uparrow_notation Conway chained arrow notation (Wikipedia) http://en.wikipedia.org/wiki/Conway_chained_arrow_notation EXTENSION TO LOW VALUES OF 'y' By using a logarithmic representation of the definition of tetration, one can derive the following: g(3,x,1) = x g(3,x,0) = 1 g(3,x,1) = 0 (except undefined for x = 1) We can also derive these: g(3,0,y) = 1 (when 'y' is an even integer) g(3,0,y) = 0 (when 'y' is an odd integer) If you define 0^0 (zero to the zeroth power) to equal one, these values hold exactly. If you take 0^0 to be undefined, these values are approached as x approaches zero (from the positive or negative direction). I can't easily type the mathematic notation for this and other derivations in ASCII, but see: Tetration (Wikipedia) http://en.wikipedia.org/wiki/Tetration COMMUTATIVITY I think it is safe to say that since g(2,x,y) is noncommutative, and since g(3,x,y) and higher are defined recursively in terms of g(2,x,y), that g(3,x,y) and higher are also noncommutative. EXTENSION BEYOND INTEGERS Tetration can be extended to real number values of 'x': Tetration  Extension to real numbers http://en.wikipedia.org/wiki/Tetration#Extension_to_real_numbers Here's a most attractive graph showing g(3, x, y) for real 'x' and for y=2,3,4,5,6,7: http://upload.wikimedia.org/wikipedia/en/6/68/Tetration_large.png The extension of tetration for real 'y' is a subject of active research: Extension of the hyper4 function to reals http://home.earthlink.net/~mrob/pub/math/lnnotes1.html#realhyper4 A Continuous Extension For the Hyper4 Operator http://ioannis.virtualcomposer2000.com/math/exponents4.html Tetration can also be extended to complex number values of 'x': Complex tetration http://en.wikipedia.org/wiki/Tetration#Complex_tetration We can also reason about tetration with infinite 'y'. For example, g(3, root2, infinity) equals 2. Infinite Exponentials http://ioannis.virtualcomposer2000.com/math/exponents.html EXTENSION IN THE OTHER DIRECTION Remarkably, g(1,x,y) can be given a meaning too: "This new operation [with a heirarchical level less than that of addition] has been called ?zeration?. The inverse operation of zeration (commutative) generates a new class of numbers (the Rubtsov?s ?delta? numbers) that can be put in bijection with the set of the logarithms of negative numbers." Hyperoperations http://forum.wolframscience.com/showthread.php?s=6e82a32b800c7293c9647a6c50598ffa&postid=1846#post1846 CALCULATING g(a, x, y) As you have observed, calculation of g(3,x,y) is slow. That's because it involves deep and repeated recursion. There are three approaches that can be taken: The first approach is to ptimise the recursion. Here we can exploit the work done in optimising Ackermann's function, which is closely related to your function 'g'. In fact, Ackermann's original function A(m,n,p) directly corresponds to g(p+1,m,n). However, Ackermann's function is more often encountered in a twoargument form defined as follows: A(m,n) = n+1 if m=0 A(m,n) = A(m1,1) if m>0 and n=0 A(m,n) = A(m1,A(m,n1)) if m>0 and n>0 You will see some similarities between this and your own recursive definition of g(a,x,y). Indeed, A(m,n) = g(m1,2,n+3)?3. To speed up this kind of calculation, it is necessary to cache some of the intermediate values so that they are not repeated recalculated. In addition, identities such as A(3,n)=8×2^n?3 can be used to make the recursion much shallower. See: Ackermann Function (Wikipedia) http://en.wikipedia.org/wiki/Ackermann_function particularly the sections on "Use as benchmark" and "table of values", and also: Ackermann Function (from Wolfram MathWorld) http://mathworld.wolfram.com/AckermannFunction.html Analytic Continuation of the Ackermann Function (What lies beyond exponentiation?) http://tetration.samaj.us/ssu/index_files/v3_document.htm Compilers and interpreters for some programming languages can automatically calculate and cache internal values, which can make an enormous difference to the running time of an unoptimised program. You can see from the following benchmark results that different languages have a range of more than 500:1 in execution time for calculating A(3,x). You may therefore wish to use a programming language that appears near the top of the list for your own calculations. Recursive benchmark  computer language shootout benchmarks http://shootout.alioth.debian.org/gp4/benchmark.php?test=recursive&lang=all The second approach to computing g(3,x,y) is to use a maths application. A package such as Mathematica is likely to perform advanced optimizations that will greatly speed up computation time. Here's an implementation for Mathematica: Power Tower  from Wolfram MathWorld http://mathworld.wolfram.com/PowerTower.html together with a bunch of tables, graphs and references. The third approach is to use a series expansion. One is derived here: A series expansion for (e^x)^(e^x)^(e^x)... http://ioannis.virtualcomposer2000.com/math/exponents5.html It looks like considerable work would be required to turn this into a practical application, although some code is provided for the Maple maths package. FURTHER STUDY For a large collection of related research, see: A Collection of References Related to Infinite Exponentials and Tetration http://ioannis.virtualcomposer2000.com/math/IERefs.html I trust this answer, and the links provided, address the points you raised in your question. If not, feel free to request clarification. Regards, eiffelga Google Search Strategy: hyper exponentiation ://www.google.com/search?q=hyper+exponentiation ackermann's function ://www.google.com/search?q=ackermann%27s+function Additional Links: Mathematics Articles by I N Galidakis (including hyper exponentiation) http://ioannis.virtualcomposer2000.com/math/index.html  

mxnmatchga
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That's exactly what I needed! Thanks! I was particularly interested in the concept of zeration, but a quick search didn't turn much up on it. And, since there's a wikipedia discussion that pretty much dismisses it, I guess it also is an area that hasn't been delved into much yet. I've asked a related question here in case you'd like to answer that one. http://answers.google.com/answers/threadview?id=743424 

Subject:
Re: addition, multiplication, exponentiation, then ???
From: berkeleychocolatega on 03 Jul 2006 19:55 PDT 
Many years ago I took a graduate math class from the famous Alfred Tarski in which he defined "+ sub alpha" for every ordinal alpha. When alpha is 0, 1, 2 we get respectively +, *, ^ . The next one is hyperexponentiation, then hyperhyperexponentiation, etc. The main step is obtained from generalizing a^(b+1) = (a^b) * a and becomes a (+ sub alpha+1) (b+1) = [a (+ sub alpha+1) b] (+ sub alpha) a. These are defined for all ordinals (and can be restricted to positive integers if you wish). Note that no (+ sub alpha) for alpha>2 is commutative and that for all ordinals alpha 2 (+ sub alpha) 2 = 4. Also at limit ordinals one just takes unions. I'm sure for finite alpha one can extend these definitions to positive reals. 
Subject:
Re: addition, multiplication, exponentiation, then ???
From: saem_aeroga on 04 Jul 2006 08:16 PDT 
This has got to be one of the best question and answers I have seen here. I wonder what kinds of applications this has in engineering or physics. However, can't these operations like tetration always be written in terms of the old *,+ etc? If so, I don't see the point of doing this? 
Subject:
Re: addition, multiplication, exponentiation, then ???
From: eiffelga on 04 Jul 2006 10:20 PDT 
Hi saem_aeroga, I'm glad you enjoyed this question and answer. The operation of multiplication can always be written using repeated addition, yet we still find it useful to have a specific notation for multiplication. Similarly, we can reason about tetration more easily if we have a highlevel notation for it. Like much of pure mathemetics, the "point of doing this" is not always immediately obvious. However, future applications may emerge. For example, G F Romerio has suggested that instead of computers overflowing when a number is too big, they instead have a soft failure into a "tetration order of magnitude" which would give an idea of just how much "too big" the number is. He has also suggested that "the new zeration operation can be used to systematically describe discontinuities such those normally defined by the step or Dirac?s function". That could surely lead to applications in electronics and other branches of physics. Hyperoperations  Forum  G F Romerio http://forum.wolframscience.com/showthread.php?s=6e82a32b800c7293c9647a6c50598ffa&postid=1846#post1846 Regards, eiffelga 
Subject:
Re: addition, multiplication, exponentiation, then ???
From: rubtsovga on 09 Sep 2006 09:11 PDT 
Dear mxnmatchga! I should like to send you a comment concerning the definition of the ?zeration? operation. As you know, initially, I wrote these comments in Russian (original text) and I produced from it an automatic computer translation in English. I already posted both of them for completing and possibly supporting with the Russian original the very bad English translation. Now, after the elimination of the previous English text, I am posting hereafter a revised English version, grouped in four points. Point 1 (The name). Indeed, the new term ?zeration? has been proposed by my friend G. F. Romerio in the framework of our 4year cooperation but, until now, the article about ?zeration? has not yet been accepted in the English version of Wikipedia. Nevertheless, the zeration operation is mentioned, as the operation of "???????? ???????? ??????? (n=0)" (i.e. the null operation) or "the operation with a rank lower than addition" (weaker than addition)in my scientific publications, in Russian, since 1987. See: http://numbers.newmail.ru/english/01.htm Unfortunately, all these publications are written in Russian and they are poorly known in the English part of the Internet. In this connection, in the Russian section of Wikipedia, an article on ?zeration? has met with the approval of the readers and it is now accessible in Russian. See: http://ru.wikipedia.org/wiki/Zeration. It is necessary to observe that, since August 22, 2006, the zeration operation (identified with this name) is present within the works of the ICM2006 mathematical congress (see the International Congress of Mathematicians: Abstracts, Posters, Short Communications, Mathematical Software, Other Activities, p. 2223, Hyperoperations as a tool for science and engineering). See: http://icm2006.org/AbsDef/Posters/abs_0480.pdf. Therefore, the Wikipedia?s claim of a lack of mention of the term ?zeration? in the Englishspeaking printed literature is not true. Point 2.(Theory) As a matter of fact, I think that nobody else until now was ever seriously engaged in the study of this problem. On the one hand, in fact, there is a well established set of historical mathematical problems and tasks, for the solution of which support and financing are provided to researchers with top priority. On the other hand, during 19 years of operation and study of ?zeration?, I never met any opponent who could specify to me any error or wrong approach. The concept provided in elementary schools strongly predominates, i.e. that ?addition? is the basic elementary operation of Arithmetic and Algebra. Nobody is ready to go through this strong psychological barrier. In 1996, I published a monograph in Russian, where the results obtained from the study of ?zeration? are explicitly presented. Unfortunately, I don?t have an acceptable English translation of this monograph in a machinereadable format. What exists is only a set of English texts (of mediumlow quality) taken from separate chapters. These texts can be found in the Internet on my WEBpage http://numbers.newnail.ru, and were obtained as computer translations, carried out in 1998. I should like to stress that a detailed study of zeration is still necessary. It is a very interesting hyperoperation, it brings a lot of new concepts and it solves some old problems. Point 3. (Applications) What would be the practical application of ?zeration?? Let's take into consideration another question: What will happens if we assumed that the real elementary operation is ?multiplication?? In this case, we should probably introduce some very complicated algorithms for the treatment of formulas including what nowadays we call ?addition?! At the same time, probably, we would also be obliged to define some higher level transcendent functions, which would contain formulas representing the ?addition? operation. However, in defining such functions, we would say that they cannot be analytically noted by using AVAILABLE mathematical operations. At this point, we might have asked to ourselves why not to define such new operations!?! We might also observe that research activities on ?tetration? (which has a rank higher than exponentiation) are becoming popular, but also that nobody ever considered any other basic operation. In particular, nobody made any research in the field of the huge amount of hyperoperations with ranks below addition! Apparently, from the abovementioned abstract point of view, ?addition? is unfortunately the accepted basic scheme for defining all the operations of the higher ranks, used for buildingup abstract models of the physical processes. What will then happen if zeration suddenly appeared (in the accepted schemes for defining all the other operations)!?! On the contrary, let us imagine which are the consequence, now, of NOT using ?zeration? in Mathematics: a). The need of separating composite processes into parts, which has the consequence of applying mathematical formalisms, separately, as a set of different equations with separate conditions. b). The definition of different ?adhoc? transcendent functions, e.g. the absolute value of a real number (x), max, min, the function of sign (sgn), etc. c). In case of conflict in a notation formalism, the compulsory introduction of limitations and, for ?the sake of precision?, the definition of new functional symbols. Take, for example, the arithmetical and algebraic square root, where the relationship between cause and effect is roughly broken and a new definition of root (of even of powers) is introduced without an explanation of causes and effects from established mathematical concepts! On the contrary, the introduction of ?zeration? is well supported with basic definitions of the necessary binary operations. This liquidates all ?misunderstandings? and, to be exact, it liberates from any subjective reversion of mathematical formalisms. I think that this is a valid argument that also supports the reliability of mathematical definitions. Actually, ?zeration? would allow to analytically note many physical processes, without the simulated input of higher transcendent functions etc. When looking at the world around us, where there is a moving matter with its interactions, it becomes understandable that ?zeration?, most likely, could become an even more useful operation than ?addition?! It is also possible to imagine that ?zeration? is the fundamental mathematical operation with which it would be possible to precisely describe physical movement and interactions, e.g. the transition of a physical substance from one status into another. The bases of traditional Mathematics for studying the movement of physical matter (including the theory of transport) are based on the theory of limits and on all the mathematical magnitudes obtained by applying this theory (via the differential and integral calculus, etc.). But, in this way, we obtain mathematical expressions which can only approximate (with a limited precision) the physical process of transport. A problem of traditional Mathematics is the lack of availability of a precise description of the transition from discrete to continuum. The use of ?zeration? solves this problem, since ?zeration? is the operation that provides a precise mathematical simulation of digital processes. In association with other operations it would also allow deriving precise mathematical expressions of overlapping discrete and continuous processes. For this reason ?zeration? appears as an indispensable mathematical operation, within the set of the other algebraic (arithmetical) operations. Finally, ?zeration? will facilitate the simulation of ?discontinuous? surfaces (i.e. the surfaces having an infinite number of discontinuity points of different kinds). The world around us consists of physical systems having, as a rule, such types of surfaces. For the traditional Mathematics, these are exotic systems which cannot be simulated without rough approximations. Point 4. (Delta numbers) In conclusion, however, I anticipate that ?zeration?, in the near future, will not be very popular, because orthodox mathematicians will prefer to keep silence, in order to live ? more quietly. One of reasons of this is that ?zeration? has an inverse operation (its official English name has not been defined yet), from which the set of what I called the ?deltanumbers? follows. These are absolutely new numbers, being a new branch of the logarithms of negative numbers. And this might destroy the customary understanding of the function theory of a complex variable. Madrid, 23.08.2006 Martin, 09.09.2006 Konstantin Rubtsov P.S. I thank G. F. Romerio for his revision of my computer translation 
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