# Knuth’s up arrow notation

Knuth’s up arrow noation is a way of writing numbers which would be unwieldy in standard decimal notation. It expands on the exponential notation $m\uparrow n=m^{n}$. Define $m\uparrow\uparrow 0=1$ and $m\uparrow\uparrow n=m\uparrow(m\uparrow\uparrow[n-1])$.

Obviously $m\uparrow\uparrow 1=m^{1}=m$, so $3\uparrow\uparrow 2=3^{3\uparrow\uparrow 1}=3^{3}=27$, but $2\uparrow\uparrow 3=2^{2\uparrow\uparrow 2}=2^{2^{2\uparrow\uparrow 1}}=2^{(2^% {2})}=16$.

In general, $m\uparrow\uparrow n=m^{m^{\cdots^{m}}}$, a tower of height $n$.

Clearly, this process can be extended: $m\uparrow\uparrow\uparrow 0=1$ and $m\uparrow\uparrow\uparrow n=m\uparrow\uparrow(m\uparrow\uparrow\uparrow[n-1])$.

An alternate notation is to write $m^{(i)}n$ for $m\underbrace{\uparrow\cdots\uparrow}_{i-2\text{~{}times}}n$. ($i-2$ times because then $m^{(2)}n=m\cdot n$ and $m^{(1)}n=m+n$.) Then in general we can define $m^{(i)}n=m^{(i-1)}(m^{(i)}(n-1))$.

To get a sense of how quickly these numbers grow, $3\uparrow\uparrow\uparrow 2=3\uparrow\uparrow 3$ is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.

 Title Knuth’s up arrow notation Canonical name KnuthsUpArrowNotation Date of creation 2013-03-22 12:58:43 Last modified on 2013-03-22 12:58:43 Owner Henry (455) Last modified by Henry (455) Numerical id 7 Author Henry (455) Entry type Definition Classification msc 00A05 Synonym up-arrow Synonym up arrow Synonym up-arrow notation Synonym up arrow notation Synonym Knuth notation Related topic ConwaysChainedArrowNotation