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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
 . Define
 and
![$ m \uparrow\uparrow n=m\uparrow(m\uparrow\uparrow [n-1])$](http://images.planetmath.org:8080/cache/objects/3350/l2h/img3.png "$ m \uparrow\uparrow n=m\uparrow(m\uparrow\uparrow [n-1])$") .
Obviously
 , so
 , but
 .
In general,
 , a tower of height  .
Clearly, this process can be extended:
 and
![$ m\uparrow\uparrow\uparrow n=m\uparrow\uparrow(m\uparrow\uparrow\uparrow [n-1])$](http://images.planetmath.org:8080/cache/objects/3350/l2h/img10.png "$ m\uparrow\uparrow\uparrow n=m\uparrow\uparrow(m\uparrow\uparrow\uparrow [n-1])$") .
An alternate notation is to write  for
 . ( times because then
 and
 .) Then in general we can define
 .
To get a sense of how quickly these numbers grow,
 is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.
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