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If multiplication is to addition what powers are to multiplication then it's easy to imagine an infinite series of such relationships, where the next element is repeated powers and the one after that is repeated repeated powers... Ad infinitum, assigning natural numbers to each order of this operation.
But has anyone tried to expand this idea into rationals, then reals and finally, complex numbers?
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I'm pretty sure this basic advanced math.
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https://en.wikipedia.org/wiki/Tetration
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https://en.wikipedia.org/wiki/Knuth's_up-arrow_notation
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>>8658134
POO
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>>8658191
Mhm, you could write
[math] f(z) = x\cdot \left( \dfrac {y}{x} \right)^z [/math]
which is analytic in z with
[math] f(0) = x [/math]
[math] f(1) = y [/math]
and choose [math] x=a·b [/math] and [math] y=a^b [/math].
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>>8658174
Does the use of this ever come up in real math?
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>>8658287
Hard to say what this question means. Someone cooked it up and it was investigated. So yes. But it's not associated with stuff often used for proofs for other things.
in any case
https://youtu.be/XTeJ64KD5cg
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>>8658134
Yes, it's called a logarithm.
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>>8658287
Proving there are dimensions in which you cannot 2-color the edges of a complete hypercube-shaped graph in such a way that no plane has all of its edges the same color.
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>>8658287
I think there are some applications in ramsey theory and it's used in some counter examples in computability theory
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>>8658134
If anyone has there's probably a thread about it here:
http://math.eretrandre.org/tetrationforum/index.php
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