# Knuth's up-arrow notation

(Redirected from Tetration)

Knuth's up-arrow notation is a notation for expressing large numbers. However, there are some numbers so large that even this up-arrow notation is not enough. You have to use Extended arrow notation.

## Definition

$$a \uparrow b=a^b$$

$$a \uparrow\uparrow b=\underbrace{a^{a^{.^{.^{.^{a}}}}}}_{\text{b a's}}$$

$$a \uparrow\uparrow\uparrow b=\underbrace{a \uparrow\uparrow a \uparrow\uparrow a\cdots\uparrow\uparrow a}_{\text{b a's}}$$

$$a \uparrow^{n+1} b=\underbrace{a \uparrow^n a \uparrow^n a\cdots a\uparrow^n a}_{\text{b a's}}$$

These operators are right-associative. Some of them are given names:

• $\uparrow\uparrow$ is called "tetration"
• $\uparrow\uparrow\uparrow$ is often called "pentation"
• $\uparrow\uparrow\uparrow\uparrow$ is often called "hexation"

## Examples

$$3\uparrow\uparrow\uparrow3=3\uparrow\uparrow3\uparrow\uparrow3=3\uparrow\uparrow3\uparrow3\uparrow3=3\uparrow\uparrow3\uparrow27=3\uparrow\uparrow7625597484987=$$a power tower that will reach all the way to the Sun