Pentation Notation - How does it work?

You wish to understand Graham's number through these arrows? If so, I'd suggest stepping back down to multiplication and building the way up.

Note that

$$a\times b=\underbrace{a+(a+(\dots+a))}_b$$

For example,

$$3\times3=3+(3+3)=3+6=9$$

And then exponentiation,

$$a^b=a\uparrow b=\underbrace{a\times(a\times(\dots\times a))}_b$$

For example,

$$3\uparrow3=3\times(3\times3)=3\times9=27$$

Now tetration,

$$a\uparrow\uparrow b=\underbrace{a\uparrow(a\uparrow(\dots\uparrow a))}_b$$

For example,

$$3\uparrow\uparrow 3=3\uparrow(3\uparrow3)=3\uparrow27=7625597484987$$

And beyond...

$$a\uparrow\uparrow\uparrow b=\underbrace{a\uparrow\uparrow(a\uparrow\uparrow(\dots\uparrow\uparrow a))}_b$$

$$3\uparrow\uparrow\uparrow3=3\uparrow\uparrow(3\uparrow\uparrow3)=3\uparrow\uparrow7625597484987=\underbrace{3\uparrow(3\uparrow(\dots\uparrow3))}_{7625597484987}=3^{3^{3^{3^{\dots}}}}$$

Tags:

Notation

Tetration

Hyperoperation

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