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

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

References

1. "Arrow Notation - from Wolfram MathWorld". Mathworld.wolfram.com. 2011-10-24. Retrieved 2011-10-30.
2. "SS > factoids > big numbers". Users.cs.york.ac.uk. 1998-07-07. Retrieved 2011-10-30.