Repeated exponentiation n@m
Narasimham G.L.
appears there is no generally known answer to the following query
about types of numbers from their comprehesive implied formative
logic. Posting this in spite of some vagueness.
Repeated addition is multiplication.
Repeated multiplication is exponetiation.
But the process need not end there ... what is Repeated exponetiation?
Is anything the like of it defined ?
When we analytically go backwards to break up the lump x^y united by
exponentiation, we find by taking logarithms a product(of y and
log(x)); when we keep at it again by taking logarithms, we get an
added quantity , log(y)+ log(log(x)).
Multiplication and division necessitated/spurred appearance of real,
fractional and decimal numbers. So also, exponentiation of negative
numbers to real numbers(non-integers) had unwittingly introduced
complex numbers e.g., in [ (-1)^(1/3)].The Euler relation gives two
extra complex roots on the Argand diagram for a comprehensive
interpretation of a negative base exponentiation.
n+n+n+… add m times -> n * m -> real numbers
n*n*n*… multiply m times -> n ^ m -> complex numbers
(...(((n^n)^n)^n)... ^n -> exponentiate m times -> n@m (say) -> ??
Qualitatively speaking, what new sort of numbers, that are supersets
of complex numbers that may now be generally unknown ,could be
involved/hidden in Repeated exponentiation n@m ?
Even if abstract, please comment. Regards.
fishfry
math...@hotmail.com (Narasimham G.L.) wrote:
> It is a pholosophical rather than a concrete maths question.There
> appears there is no generally known answer to the following query
> about types of numbers from their comprehesive implied formative
> logic. Posting this in spite of some vagueness.
>
> Repeated addition is multiplication.
> Repeated multiplication is exponetiation.
> But the process need not end there ... what is Repeated exponetiation?
> Is anything the like of it defined ?
>
It's called tetration. It's usually written with the exponent to the
upper left of the base. Using @, 4@2 = 2^2^2^2, etc. See Infinity and
the Mind by Rucker for a reference.
Steve Bellenot
Hmmm, I don't agree with your logic here. Addition does not take
you outside of the positive integers. Even adding the inverse operation
of substraction only gets you to the integers.
> n*n*n*… multiply m times -> n ^ m -> complex numbers
Multiplication doesn't get you outside the integers. Divsion gets
you the rationals.
> (...(((n^n)^n)^n)... ^n -> exponentiate m times -> n@m (say) -> ??
Repeated exponentiation is not associative and it is often taken as
as n^(n^(n^ .. ^n))))) which is a bigger number usually. For example
(3^3)^3 = 19683 but 3^(3^3) = 7625597484987.
Another generalization is via the Ackerman functions and numbers
http://mathworld.wolfram.com/AckermannNumber.html
>
>Qualitatively speaking, what new sort of numbers, that are supersets
>of complex numbers that may now be generally unknown ,could be
>involved/hidden in Repeated exponentiation n@m ?
There are lots of things bigger than the complex numbers, but I don't
know of any that come from repeated exponentiation. Some of my
favorites include the hyperreals or the quaterions.
--
http://www.math.fsu.edu/~bellenot
bellenot <At/> math.fsu.edu
+1.850.644.7189 (4053fax)
A N Niel
Stephen J. Herschkorn
>
>See Ackerman's function...
>
Better yet, http://mathworld.wolfram.com/PowerTower.html
--
Stephen J. Herschkorn hers...@rutcor.rutgers.edu
Tim Sweeney
addition/multiplication/exponentiation for arbitrary iterations. Also
related is Ackermann's function.
As far as I understand it, these operations and their inverses are
closed over the complex numbers, so no new number systems need to be
introduced to study them completely.
Martin
"Narasimham G.L." <math...@hotmail.com> skrev i melding
news:676dc11a.03090...@posting.google.com...
> It is a pholosophical rather than a concrete maths question.There
> appears there is no generally known answer to the following query
> about types of numbers from their comprehesive implied formative
> logic. Posting this in spite of some vagueness.
>
> Repeated addition is multiplication.
> Repeated multiplication is exponetiation.
> But the process need not end there ... what is Repeated exponetiation?
> Is anything the like of it defined ?
>
> When we analytically go backwards to break up the lump x^y united by
> exponentiation, we find by taking logarithms a product(of y and
> log(x)); when we keep at it again by taking logarithms, we get an
> added quantity , log(y)+ log(log(x)).
>
> Multiplication and division necessitated/spurred appearance of real,
> fractional and decimal numbers. So also, exponentiation of negative
> numbers to real numbers(non-integers) had unwittingly introduced
> complex numbers e.g., in [ (-1)^(1/3)].The Euler relation gives two
> extra complex roots on the Argand diagram for a comprehensive
> interpretation of a negative base exponentiation.
>
> n*n*n*. multiply m times -> n ^ m -> complex numbers
The complex unit sqrt(-1) is due to the negative constant -1 being
introduced with subtraction, not multiplication.
> (...(((n^n)^n)^n)... ^n -> exponentiate m times -> n@m (say) -> ??
Called a powertower. It x@x converges for some values of x e.g. some complex
numbers
Graham Matthews
> There are lots of things bigger than the complex numbers, but I don't
> know of any that come from repeated exponentiation. Some of my
> favorites include the hyperreals or the quaterions.
To get to the quaternions from the complex numbers you have to
give up commutativity (quaternions are antit-commutative on the
basis elements). So you can't get to them using repeated exponentiation.
graham