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 . Define and .
Obviously , so , but .
In general, , a tower of height .
Clearly, this process can be extended: and .
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.
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 |