I noticed that we use $\sum$ and $\prod$ for summation and infinite product (I don't know why it does not have a name like the other two), but we use different looking notation for tetration. Is there some sort of historical reason why not, or did no-one just do it.
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8$\begingroup$ So far, the uses for iterated exponentiation or tetration in science and mathematics have been few to none. Usefulness is largely what drives the popularity of concepts and notations. $\endgroup$– Gerald EdgarCommented Apr 13, 2021 at 23:50
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3$\begingroup$ Tetration is not similar to sum and product, it is similar to $na$ and $a^n$, which are not uniform either, and one needs to attach $n$ to $a$ in a distinctive way. What is similar to sum and product is the iterated exponential, and there is analogous notation for it with $\Xi$, see Power Towers, and Notation for Iterated Exponentiation on MathSE. There is also a similar notation for continued fractions, but the last two are not nearly as useful or common as sums and products, hence the notations came later and are not widely known. $\endgroup$– ConifoldCommented Apr 14, 2021 at 6:51
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2$\begingroup$ @Gerald Edgar but then the unreasonable popularity of tetration (well, among amateurs and in pop science works) seems to contradict your claim to some extent ;-) $\endgroup$– Hermann GruberCommented Apr 15, 2021 at 22:33
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1$\begingroup$ @HermannGruber, ha! :) I guess there're pop-popularity and professional-popularity, and they're not reliably the same. :) $\endgroup$– paul garrettCommented May 20, 2021 at 19:30
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2 Answers
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Mostly, it comes down to usefulness.
Pick up any mathematics textbook more advanced than elementary arithmetic, and you're probably going to find summation in there somewhere (it's especially common in statistics). Iterated multiplication is less common, having fewer real-world applications, but it's a natural counterpart to summation.
Tetration? It's rarely seen outside of number theory and pop-science articles.
There's also the matter of age: sigma notation dates back to Euler, who first used it around 1755, and the concept itself is considerably older. Pi notation is more elusive, but appears to date to the early-mid 1800s. In contrast, tetration appears to date back no further than 1901, with most work being post-1950.
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Knuth arrow notation can be used to notate the hyper-operations, the fourth of which is tetration and the fifth, pentation and so on. These operations were first defined by Goodstein in 1947.
Here, the number of up arrows minus two give the degree of the hyperoperation. Thus
$a \uparrow \uparrow b$ is $a$ tetrated to $b$.
That this is not common is simply these operations aren't common.
This notation could be lifted to a symbol expessing an infinite tetration. That is:
$\uparrow \uparrow a_i$
meaning
$a_0 \uparrow \uparrow a_1 \uparrow \uparrow a_2 ...$
Where left assosciatevity is assumed.
Such a lifted operation is not common - if even used.
