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I would like to know about how to exactly do calculation with tetration, especially when the value $k$ is a negative value in: $a ↑↑ k$

I am aware of the process of tetration, which is repeated exponentiation. I would like to know the result when $k$ is negative, and how that is possible. Would also be helpful if I could also get an example with this.

Yours Sincerely, Aster17

Tsar Asterov XVII
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  • Tetration soon leads to extremely large numbers. I am not aware of any definition for other $k$ than positive integers that is consistent with the tetration defined in this case. If $a\uparrow \uparrow k$ is small enough it can be easily calculated recursively. – Peter Oct 24 '22 at 11:29
  • @Peter I know tetration quickly blows up. I was only wondering on how would tetration work when k is a negative integer, as I was curious about it, if something like that would even be possible. – Tsar Asterov XVII Oct 24 '22 at 11:30
  • Interestingly, research into canonical definitions for non-natural number heights of tetration is still ongoing. However, there are many resources which discuss this, including this [Tetration Forum](https://math.eretrandre.org/tetrationforum/index.php), and [this "wiki"](https://en.wikipedia.org/wiki/User:MathFacts/Tetration_Summary) on how to calculate such (but be warned as the methods are rather in-depth). – Graviton Oct 24 '22 at 12:52
  • Related: [How would tetration work for non integer numbers.](https://math.stackexchange.com/questions/1864661/how-would-tetration-work-for-non-integer-numbers) – Graviton Oct 24 '22 at 12:58
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    @Graviton Thank you very much for the resources provided. These are going to be highly useful to me and I can't thank you enough. Am I allowed to post what I have found from reading these as an answer? – Tsar Asterov XVII Oct 24 '22 at 16:03
  • @Aster17 Happy to assist. I've asked many-a questions on tetration here before, so I know the curiosity. So, be my guest, if you feel like you have found the answers you were looking for! It's your question after-all. – Graviton Oct 25 '22 at 11:46
  • @Graviton thanks for the assisstance, it's extremely helpful for my little own theory thing I'm writing. I'll post the answer here to finish the question. Thanks, Aster17 – Tsar Asterov XVII Oct 25 '22 at 14:46

1 Answers1

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So,

It seems tetrating values to a negative value seems to be a still studied phenomenon. But, the answer seems to be the following:

$a ↑↑ -1$
$=> log$ a $ (a^0) $
$=> log$ a $ (1) $
$=> 0 $

It seems to remain undefined for any negative integer aside from -1, due to the fact that there is no finite number which you can raise a number to that will lead to 0.

So, the answer is undefined for now, until tetration itself gets... you could say, well-defined!

Bad puns aside, I would like to thank Graviton for leading me to the right resources for getting the right answer and I will link them down below for further information.

Cheers,
Aster17

References:
https://math.eretrandre.org/tetrationforum/index.php
https://en.wikipedia.org/wiki/User:MathFacts/Tetration_Summary
How would tetration work for non integer numbers.

Tsar Asterov XVII
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