Re: Why is it so difficult?

On Aug 15, 7:02 pm, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
mike3 wrote:
On Aug 15, 3:54 pm, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:

[snip]

If y=x^^{1/n} and x is given, after one computes y, then indeed
y^^n=x.

But how does that mean it makes sense to define ^(1/n) x to
be the solution of ^n y = x? Why couldn't we define
^(1/(n^2)) x as the solution???

Because y=x^^{1/n^2} contrasts badly (at an operator level) with its inverse
operation: The inverse operation is y^^n, not y^^{n^2}.

How does it contrast badly, when you don't even know what ^(1/n) x is
supposed to mean?

In other words, you need
to tetrate y, n-times. NOT n^2-times to get back x. Usually a consistent
mathematical concensus requires "inverse" operators to be related in the
simplest way possible, and in this case the "natural choice" is taken (as an
analogue) from the corresponding relationship between the operators "n-th root"
and "n-th power".


So then it's because

x + 1/2 is doing half the "add one" function to x.
x * 1/2 is doing half the "add x" function to 0.
x ^ 1/2 is doing half the "multiply by x" function to 1.
^1/2 x is doing half the "exponentiate x" function.

right?

What problems does one
run into when attempting to use a definition like that
that make it less "natural" and more "contrived"?
But from what I've examined the definition of
"tetraroot" simply only requires that a solution
y to the equation ^n y = x exists, NOT that y =
^(1/n) x.

I am sorry, I don't understand your question. It seems that the notation
x^^{1/n} bothers you for some obscure (to me) reason.


Because it seems to be based off the mistaken idea that ^m (^n x) =
^(mn) x. It's OK to have x^(1/n) as the n-th root of x since (x^m)^n =
x^(mn).

Yeah, I guess you could define y instead as:

x^^{Gobledygookery-biscuit+i^56135+gazeebo*superdooper/n!*1/Gamma(Ronald
McDonald))},


<soda squirts all over the keyboard>

<falls out of chair>

<rolls around laughing so hard it wakes up the neighboorhood>

Good <laughlaughgigglegiggle> o.. <laughlaugh> on.. <giggle> one!!!
<giggling some more>

to be that number, which if you tetrate it n-times you get back x. The notation
for the 1/n is merely a symbol place holder similar to the symbol place holder
x^{1/n} for a regular n-th root, reminiscent of the operation for which this is
a solution. Such a number exists always when x,y > 1, is unique, and no matter
how you want to call it, it can be calculated to any degree of accuracy.


<still giggling a little from that... heh... above>

Does this mean you don't intend for ^(1/n) x to correspond to the
tetration, and instead just use it as a convenient notational trick?

[snip]
--
I.N. Galidakis


.