Hyperoperations and huge numbers

If you repeat multiplication you get exponentation:

2↑4 = 2 × (2 × (2 × 2))

If you repeat exponentiation you get tetration, which gives you a 'power tower':

2↑↑4 = 2 ↑ (2 ↑ (2 ↑ 2))

This is 2 to the power 2 to the power 2 to the power 2.   If you repeat tetration, you get pentation:

2↑↑↑4 = 2 ↑↑  (2 ↑↑ (2 ↑↑ 2))

Let's see how big this is!

2 ↑↑ 2 = 2 ↑ 2 = 2 × 2 = 4

so

2 ↑↑ (2 ↑↑ 2) = 2 ↑↑ 4 = 2 ↑ (2 ↑ (2 ↑ 2)) = 2 ↑ (2 ↑ 4) = 2 ↑ 16 = 65536

so

2 ↑↑  (2 ↑↑ (2 ↑↑ 2)) = 2 ↑↑ 65536

In short, 2↑↑↑4 is a power tower with 65536 twos in it!

This is a staggeringly large number.   It can't be written in binary on all the atoms in the observable universe.  Its number of binary digits can't either.  The number of binary digits in its number of binary digits can't either.  And so on... for over 65,000 rounds.

But when we hit the next operation, hexation, all hell breaks loose.

2↑↑↑↑4 = 2 ↑↑↑  (2 ↑↑↑ (2 ↑↑↑ 2))

Let's see how big this is!   By definition,

2 ↑↑↑ 2 = 2 ↑↑ 2 = 2 ↑ 2 = 2 × 2 = 4

So, by our previous calculation:

2 ↑↑↑ (2 ↑↑↑ 2) = 2 ↑↑↑ 4 = 2 ↑↑ 65536

and then

2 ↑↑↑  (2 ↑↑↑ (2 ↑↑↑ 2)) = 2 ↑↑↑ (2 ↑↑ 65536)

Can you comprehend this number?  This is

2 ↑↑ (2 ↑↑ (2 ↑↑ (2 ↑↑ (2 ↑↑ (2 ↑↑ (2 ↑↑ (2 ↑↑ (2 ↑↑ .....

where there are 2 ↑↑ 65536 twos.   The mind boggles, but here's a good way to think of it:

2
2 ↑↑ 2
2 ↑↑ (2 ↑↑ 2)
etc.

We start with some number, namely 2.  Then we replace that number with a bigger one: a power tower consisting of that many 2s.   Then we replace that number with a power tower consisting of that many 2s.  And we do this process 2 ↑↑ 65536 times!

The result can't be written in binary on all the atoms in the observable universe.  Its number of binary digits can't either.  The number of binary digits in its number of binary digits can't either.  And so on... for a number of rounds that can't be written in binary on all the atoms of the observable universe!

These operations ↑, ↑↑, ↑↑↑ etc. are called hyperoperations:

http://en.wikipedia.org/wiki/Hyperoperation

and we're using Knuth's up-arrow notation to describe them:

http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

I hope you use these ideas to impress your friends and make new enemies.  But soon I'll show you notations for vastly larger numbers.   And later I'll show you that infinity infects the finite.  The best, most systematic notations for truly enormous but still finite and computable numbers use infinite ordinals of the sort I've been discussing lately!

I'm using the word 'computable' in a sense explained in the attached article by Scott Aaronson.  He ran some contests to see who could describe the biggest number.  If you ever do this, make sure to require that the descriptions be computable: otherwise it may be impossible to tell who won!

#bigness  ﻿
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