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We know how to add with decimals.
We know how to multiply with decimals.
We know how to exponentiate with decimals.

Do we know how to work with decimals for power towers?

for example, can we deal with the following expression: 3↑↑3.5? If so, how would you calculate it and can this be done in general for an arbitrary number of arrows?

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The wikipedia article suggests that there are multiple conventions for defining the continuous extension for tetrations. So I guess the answer depends. – EuYu Oct 30 '13 at 20:31
Kneser's analytic tetration method allows calculating arbitrary heights, including complex heights, for real bases>1. For the Op's example, $3↑↑3.5\approx 1.06086755507802\times10^{617}$. For three up arrows, you have pentation, which could be extended to real numbers, since tetration has a real valued fixed point. For example for base 3 tetration, tet(-1.874507)~=-1.874507. Its a repelling fixed point, so that allows defining pentation for base 3. – Sheldon L Oct 31 '13 at 14:33

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