What comes after exponents?

Question

We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding from me?

Answer 1

The best answer is perhaps tetration though you might want to investigate also Ackermann's function, Knuth's up-arrow notation, and Conway's chained arrow notation all of which capture the idea of growth beyond exponential growth in different ways.

It isn't really being hidden from you because the uses of these ideas seem to come up in computing and combinatorics (the Hales-Jewett theorem and similar can be proved with Ackermann type bounds, though sometimes better bounds are available by trickier methods). Here is a discussion of an apparently simple problem from the 2010 IMO which generates beyond exponential rates of growth. You might want to try it first before reading the detail.

On the whole, though, the ideas involved run into problems of notation (which I have always thought similar to the problem of naming all the ordinal numbers - the notation just runs out and you need something new).

Answer 2

No one ever uses what's beyond exponents except physicists and mathematicians who need or want really big numbers, so they're generally the only ones who need to know the names, and of course, they made them. What is after exponents is tetration (tetra for the fourth level of operation) and then pentation, hexation, and so on and so forth. when writing a math equation you use up arrows, one for exponents, two for tetration, three for pentation, and you can take it from there. Big numbers is an understatement, because when you tetrate 2 to 2, you get four. When you tetrate 3 to 3, you get 7,625,597,484,987. When you tetrate 4 to 4, you get infinity, error, or, on big number calculators, "Sorry, we can't calculate numbers THAT big!"

And that's why you don't need tetration.

Answer 3

Your teachers aren’t hiding what comes next after exponentiation, it’s just that such a “fourth operation” doesn’t seem to have any practical application.  The expression  x^x,  useful in ballistics, can be thought of as  x “operated on” by 2  (using the fourth operation) but that’s rather stretching to find an example.

Still, your question is theoretically interesting, especially when the operating number is a fraction instead of a whole number like two.  A good discussion of the difficulties involved, and why perhaps exponentiation should not in fact be considered a binary operation like addition and multiplication, can be found in the fairly elementary article:

The Fourth Operation: Tetration from a Real Perspective

Answer 4

Due to the lack of commutativity or associativty, there are two types of operations that come after exponentiation: repeated powering and tetration.

Let us take $(3^3)^3$. 3 is cubed twice. $(3^3)^3=27^3=19,683$

Now let us take $3^{3^3}$ . $3^{3^3}=3^{27}=7,625,597,484,987$.

A shift in parentheses resulted in two integers of completely different magnitudes.

It has been shown that for positive real numbers $r$, infinite repeated powering converges to $1$ for $0<r\leq1$. For infinite tetration, it converges if $1/e\leq r\leq \sqrt[e]e$, if $e$ is the base of the natural logarithm. Outside these cases, infinite repeated powering and infinite tetration diverges for real numbers.

Answer 5

although not a perfect tetration per se,

4 ↑ ↑ 3

2 ↑ ↑ 5

are actually kinda useful for binary math for big integers, since

4 ↑ ↑ 3  :=   4^4^4   = 2 ^  512  = 2^8^3 = 2^2^9
2 ↑ ↑ 5  := 2^2^2^2^2 = 2 ^ 65536 = 2^4^8

and some x^x I find most convenient for binary math are :

     4 ^ 4  = 2^8  = 256
     8 ^ 8  = 2^24 = 16777216
    16 ^ 16 = 2^64 = 18446744073709551616

in relation to some small Mersenne Primes using exactly 1 copy each of 4 smallest powers of 2,

      4^8   *  2 - 1  = 131071 
        8^4 *  2 - 1  =   8191
  2^4 * 8        - 1  =    127
        8*4 - (2 - 1) =     31
  2^4 - 8        - 1  =      7