? Can it be evaluated? Is it a real number or a complex number? Well, it turns out that it's a real number and transcendental. One way of working out its value is to replace the 
in the base with 
because:
So we can replace with 
which becomes 
which in turn becomes simply 
. This of course we can evaluate and it turns out to be approximately:
with which becomes which in turn becomes simply . This of course we can evaluate and it turns out to be approximately:
0.207879576350761908546955619834978770033877841631769608075...
It looks even stranger if we pose the problem in the form 
. In whatever form it's posed however, the result is the same: a transcendental, real number with the value shown above. Of course, this leads one to pose other problems such as the value of 
or 
. Having posed the problems, I guess I'll have to now investigate them.
is easy because it can be written as:
. In whatever form it's posed however, the result is the same: a transcendental, real number with the value shown above. Of course, this leads one to pose other problems such as the value of or . Having posed the problems, I guess I'll have to now investigate them. is easy because it can be written as:
WolframAlpha will give us an answer to ( 
) but it's not entirely clear why. I'll need to investigate further. Note that my previous post on Power Towers and Tetration is relevant to this problem.
( ) but it's not entirely clear why. I'll need to investigate further. Note that my previous post on Power Towers and Tetration is relevant to this problem.
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