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A new set of numbers is necessary to extend tetration to real exponents.
I will make a different definition of the elemental functions to extend it to tetration.

By starting with zero, and taking the successor, is possible to obtain the entire set of natural numbers, but no other type of number, like a rational, or a complex.

Once addition and his inverse function, subtraction is considered, is possible to “discover” the set of the Integers, because to solve any equation is necessary to use negative numbers.

Product forces to use rational numbers, and exponentiation needs to discover the complex set.

So, is natural to suspect that tetration requires an more general set of numbers to solve any equation, and I think that it is the root of the difficulty in extending tetration to real exponents.

Is possible to define a product a.b this non traditional way:

[Image: 6N7jyv7.png?1]

Which means that zero is added to a, b times. I would call zero (the identity element of addition) the implicit base of product.
It is trivial that a.0=0

It is also possible to define exponentiation a^b as multiplication of 1 (the identity element of product) by a, b times:

[Image: L2c6ACc.png?1]
By definition, it is trivial that a°=1

It leads to define a^-b=c as the number c which [Image: 1cxGrPL.png?1]
In other words, [Image: CsYZt7a.png?1]

It also leads to define [Image: wc9YVDc.png?1] as the number c which [Image: 00vOZG3.png?1]
[Image: 5Z5p6IQ.png?1]

For that reason, I would define tetration on a subtle different way than the custom:
Tetration of a, to exponent b, and implicit base z is defined as Ttr(a,b,z) this way:

[Image: qMXudVY.png?1]
When z = 1, it may be omitted and written Ttr(a,b,1) = Ttr(a,b)
It is immediate that Ttr(a,b,0) = Ttr(a,b-1)
also, [Image: wHA3xuO.png?1]
So, is not necessary to make a distinction between left and right parenting notation.

Trivially, Ttr(a,0) = 1, because no number modifies 1 by being raised to 1

To find the value of tetration for any real number b, in Ttr(a,b), is necessary to know the meaning for any negative integer b, and for any number equal to 1/b (b integer).

So, let’s start from Ttr(a,0) = Ttr(a,-b+b) = Ttr(a,b,Ttr(a,-b)) = Ttr(a,b,c) = 1

[Image: veI6L4v.png?1]

So, for a natural number b, [Image: FbnIln1.png?1]

For a simple case, let’s take c = Ttr(a,-1), then it should be
[Image: PxQ27X0.png?1]

By taking logarithm: c . ln(a) = 0, so, if c is a complex number, it can only be zero, for any real a≠1.
That is inconvenient, because impedes to extend the concept of tetration to negative integer exponents.

But we should remember that each new level of arithmetic operation needed new, more general set of numbers to solve any equation, so is necessary to ask if to extend tetration to real exponents, is necessary to use a more general set of numbers, were a power of a number different to zero may be zero.

So, I argue that to understand tetration is essential to use sets of numbers with non trivial roots of zero.

Such roots of zero may be nilpotent numbers, or any set with non trivial Idempotent elements, which would have non trivial zero divisors.

There are many sets with non trivial zero divisors. The question is which one would be the more useful one.

Are the Split-complex numbers? something from hypercomplex?
It should have the complex numbers as a subset.


Messages In This Thread
A new set of numbers is necessary to extend tetration to real exponents. - by marraco - 03/11/2015, 07:56 PM

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