Knuth's up-arrow notation is useful to represent very large numbers with iterated exponentiation. It works in a way similar to standard exponentiation.
For example:
- <math>2\uparrow\uparrow 3= 2^{(2^{2})}=2^{2^{2}} </math>
Generally:
- <math>
\begin{matrix}
x\uparrow y &=& x^y &=& x \times x \times x \times\cdots \left(\mbox{y times}\right) \\
x\uparrow\uparrow y &=& x\uparrow x\uparrow x\uparrow\ldots\left(\mbox{y times}\right) &=& x^{x^{x^{\cdots}}} \left(\mbox{y times}\right) \\
x\uparrow\uparrow\uparrow y &=& x\uparrow\uparrow x \uparrow\uparrow x \ldots\left(\mbox{y times}\right)&=&(x^{x^{x^{\cdots}}} \left(\mbox{y times}\right))^{(x^{x^{x^{\cdots}}} \left(\mbox{y times}\right))^{\cdots}}
\end{matrix}
</math>
Often instead of arrows, in strict ASCII, ^^ is used instead of ↑↑.