When you repeat addition you get multiplication.  When you repeat multiplication you get exponentiation.  And when you repeat that, you get tetration!

Let's follow Donald Knuth and write exponentiation as ↑.  Then 4 cubed is

4↑3 = 4 × (4 × 4) = 64

We are multiplying 4 by itself 3 times.   Next comes tetration, which we write as ↑↑.   Here's how it works:

4↑↑3 = 4 ↑ (4 ↑ 4) = 4 ↑ 256 ≈ 1.3 × 10¹⁵⁴

We are raising 4 to itself 3 times.  Before the parentheses didn't matter, but now they do: we put the parentheses all the way to the right, since (a ↑ b) ↑ c equals a ↑ (b × c) while a ↑ (b ↑ c) is something really new. 

As you can see, tetration lets us describe quite large numbers.   10↑↑10 is much bigger than the number of atoms in the observable universe.   

In fact, 10↑↑10 is so big that you couldn't write it down if you wrote one decimal digit on each atom in the observable universe!

In fact, 10↑↑10 is so big you couldn't write the number of its digits if you wrote one digit of that number on each atom in the observable universe!!

In fact, 10↑↑10 is so big you couldn't write down the number of digits in its number of digits if you wrote one digit of that number on each atom in the observable universe!!!

Puzzle: About how many times could I keep going on here?   Let's say there are 10 ↑ 80 atoms in the observable universe; this seems roughly right. 

We could march forwards and create notations for numbers so huge that 10↑↑10 looks pathetically small.   And we will!   We can even look at infinite tetration.  And we will - that's our main goal here!   But it's also fun to ask questions like:

Puzzle: What is 10↑↑10.2 ? 

This makes no sense at first: you can't write down a tower of powers with 10.2 tens in it.   But you couldn't add 10.2 tens together at first, either, so 10 × 10.2 didn't make sense until someone explained what it meant.  Same for exponentiation.  So, maybe tetration can also be defined for fractions in some nice way.   And maybe real numbers too.  And maybe even complex numbers.

Believe it or not, this seems to be an open question!  It's phrased as a precise conjecture here:


and there's a good candidate for the answer.  The picture here, drawn by Dmitrii Kouznetsov, shows a graph of the candidate for e↑↑z as a function on the complex plane.

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