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Back in high school I wondered if there is a such a function that has higher growth rate than exponential function. Several years later it came to my mind that this is obvious because we can create arithmetic operators. I'm not a mathematics graduate so it was big thing to me - I thought we are rather bounded to these 5 operators "$+ -\times /$ ^" and we can't do anything about it.

Example:

$2+2+2+2=2\times4=8$

so:

$2\times2\times2\times2=2$^$4=16$

so:

$2$^$2$^$2$^$2$$=2☮4=256$

so:

...

and so on.

So we can use this new "☮" operator instead of "^" to get a function with more steeper increment rate. What is a formal name for such a ☮ operator, what do we use it for? (Some time ago I've found out that Donald Knuth applied such iterated exponentiation only to notate large integers. I see we can use it to count bacteria population, given generation number). Why is such an operator not more common used?

After I "discovered" that ☮ operator I realized that there are also new operators at levels below addition:

$2+2+2+2=2\times 4=8$

because:

$2222=2+4=6$

because:

$...............=24=5$ (According to my calculations it equals "5")

because:

........

And this operator really blew my mind. It so bizarre and exotic to me. The same questions: what is the formal name, application and properties for that operator (and those level below it)?

lodzki
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    Good work! One reason these functions are not more widely studied is that it is very hard to compute with them (as they grow so quickly). Also, unlike the more standard arithmetic operations, there aren't obvious physical situations which represent the operations. Still, they are certainly of interest. – lulu Apr 29 '23 at 13:13
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    There is no real difference between a function and an operation. We could choose to write $a+b$ as $+(a,b)$, for example. It is easy to invent functions of more than one variable that have not been studied and do not have names. If they prove to be useful they will get names and get studied. To show they are useful, you need to show some problems that they help solve. – Ross Millikan Apr 29 '23 at 13:15
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    Isn't your "☮" just the https://en.wikipedia.org/wiki/Tetration operator? – Martin R Apr 29 '23 at 13:15
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    $2222=2+4$, $222=2+3$, $22=2+2$, $2=2+1$ ??? – Martin R Apr 29 '23 at 13:25
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    The tetration operator has very limited merit for daily life calculations. Basically , it was just invented to generate extremely large numbers. – Peter Apr 29 '23 at 13:33
  • @MartinR Maybe it should be noted as 2=2+1 to get no conflict. Or you cant iterate by 1 time with this operator (indeterminate form). – lodzki Apr 29 '23 at 13:47
  • @RossMillikan Yes and that is why may be we should familiarize with them in calculations to notice they exist in the real world? For example we can resist ourselves to use exponentation in formulas and substitute it with tetration (☮) so it will become more intuitive. – lodzki Apr 29 '23 at 14:03
  • @lulu Isn't it a good solution to apply e.i. log scales so these above exponentation functions will appear as relatively slower growing? – lodzki Apr 29 '23 at 14:18
  • I mean, look at the "values" provided in the wiki article. Of course we can sometimes say something about the values, but it gets out of hand quickly. – lulu Apr 29 '23 at 14:23

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