Knuth's up-arrow notation

Knuth's up-arrow notation is a way of expressing very big numbers. It was made by Donald Knuth in 1976. It is related to the hyperoperation sequence. The notation is used in Graham's number.

One arrow represents exponentiation, 2 arrows represent tetration, 3 for pentation, etc.:

1. Exponentiation

   [math]\displaystyle{ a \uparrow^{1} b = a^b = \underbrace{a \times a \times \cdots \times a}_{b \ times} }[/math]

   a multiplied by itself, b times.

2. Tetration

   [math]\displaystyle{ a \uparrow^{2} b = a \uparrow \uparrow b = {^{b}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_{b \ times} = \underbrace{(a \uparrow^1 (a \uparrow^1 (... \uparrow^1 a)...)}_{b \ times} }[/math]

   a exponentiated by itself, b times.

3. Third level

   [math]\displaystyle{ a \uparrow^{3} b = a \uparrow \uparrow \uparrow b = \underbrace{a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow \ldots a) \ldots ) )}_{b \ times} }[/math]

4. etc.

   [math]\displaystyle{ a \uparrow^{n} b = a \underbrace{\uparrow \uparrow \dots \uparrow \uparrow}_{n \ times} b = \underbrace{a \uparrow^{n-1} (a \uparrow^{n-1} (a \uparrow^{n-1} \ldots a) \ldots ) )}_{b \ times} }[/math]

This notation is used to describe the incredibly large Graham's Number.