Knuth's up-arrow notation

Knuth's up-arrow notation is a way of expressing very big numbers.^[1] It was made by Donald Knuth in 1976.^[1] 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.:^[2]

1. Exponentiation

   ${\displaystyle a\uparrow ^{1}b=a^{b}=\underbrace {a\times a\times \cdots \times a} _{b\ times}}$

   a multiplied by itself, b times.

2. Tetration

   ${\displaystyle a\uparrow ^{2}b=a\uparrow \uparrow b={^{b}a}=\underbrace {(a^{(a^{(\cdot ^{\cdot ^{(a)...)}}}}} _{b\ times}=\underbrace {(a\uparrow ^{1}(a\uparrow ^{1}(...\uparrow ^{1}a)...)} _{b\ times}}$

   a exponentiated by itself, b times.

3. Third level

   ${\displaystyle a\uparrow ^{3}b=a\uparrow \uparrow \uparrow b=\underbrace {a\uparrow \uparrow (a\uparrow \uparrow (a\uparrow \uparrow \ldots a)\ldots ))} _{b\ times}}$

4. etc.

   ${\displaystyle a\uparrow ^{n}b=a\underbrace {\uparrow \uparrow \dots \uparrow \uparrow } _{n\ times}b=\underbrace {a\uparrow ^{n-1}(a\uparrow ^{n-1}(a\uparrow ^{n-1}\ldots a)\ldots ))} _{b\ times}}$

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