This question is not about mathematics, within the scope defined in the help center.
Closed last year.
I've always wondered after learning addition, multiplication, and power facts (and their inverse operations) what the next higher level of facts I would need to memorize would be. However, instead of learning about the next higher operator, math instead took an entirely different turn and started going into all sorts of other things that never seemed to need any higher operator than just the three and their inverses. From my perspective, operators went from addition to multiplication (repeated addition) , to exponentiation (repeated multiplication), but I’ve never heard of any higher-order operators than exponentiation. So recently I tried to learn about this operator myself. I started like this: repeated addition = multiplication, repeated multiplication = exponentiation, repeated exponentiation = ?.
However, I see that because of exponentials being non-commutative,
k(k(k(...)))≠((((k)k)k)...)
there's two ways to repeat exponentiation, leading to two new operators, with one being tetration, with right-associative exponentiation. If I represent the other left-associative operator as ?, then:
k+k+k+k=k∗4,
k∗k∗k∗k=k4, and
(((k)k)k)k=k?4.
It seems that something close to what I want is (((k)k)k)k=k(k3), and while this could be shorthand, it doesn’t follow the convention of k (operator) 4.
I think that left-associative exponentiation is still the consistent way with repeated addition and repeated multiplication. It's what I usually think of when I think of repeated exponentiation.
Then, according to the interwebs, it seems that tetration is usually considered the "next" operation after exponentiation. However, it seems that it isn't very common or useful. Not much can actually be represented with tetration. It seems that Tetration basically only gives you big numbers. For example, 2^(2^(2^2)) is already a huge number, and that's with really small inputs.
My question then is:
Why is addition, multiplication, exponentiation (plus their 3 inverses) incredibly useful whereas tetration is the sudden cutoff for usefulness?
I mean, surely the reason for repeated operators is not only to represent bigger numbers? There are a myriad of useful things for just addition, multiplication, and exponentiation, and their inverses, but suddenly nothing useful from tetration? That doesn't make sense. It seems multiplication gives you more advanced math than just addition, and exponentiation gives more advanced math than just multiplication, so why doesn’t tetration give you even more advanced math? Or are there actually advanced uses for Tetration at least as diverse and meaningful as the previous 3 that I don’t know about?
EDIT:
So far, it seems the general consensus is that tetrations are in fact not useful. Most of the answers/comments seem to reiterating this. But my central confusion is WHY this is. Multiplication seems to model more advanced real-world applications than addition. For example, multiplication finds area or volume, vs addition for finding number of apples or sheep. Going by this pattern, exponentials seem to model even more advanced real-world situations than multiplication, like expressing gravity as g = 9.8 m s^-2, or expressing polynomials which have a myriad of advanced applications. So does tetration have even more real-world applications than exponentiation? It appears no, thus far. In fact, tetration seems to have even less applications than plain old addition. This is mind boggling for me. (Does this mean that humanity simply hasn't reached that level of math yet?)
