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单词 KnuthsUpArrowNotation
释义

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 exponentialPlanetmathPlanetmath notation mn=mn. Define m0=1 and mn=m(m[n-1]).

Obviously m1=m1=m, so 32=331=33=27, but 23=222=2221=2(22)=16.

In general, mn=mmm, a tower of height n.

Clearly, this process can be extended: m0=1 and mn=m(m[n-1]).

An alternate notation is to write m(i)n for mi-2 timesn. (i-2 times because then m(2)n=mn 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, 32=33 is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.

TitleKnuth’s up arrow notation
Canonical nameKnuthsUpArrowNotation
Date of creation2013-03-22 12:58:43
Last modified on2013-03-22 12:58:43
OwnerHenry (455)
Last modified byHenry (455)
Numerical id7
AuthorHenry (455)
Entry typeDefinition
Classificationmsc 00A05
Synonymup-arrow
Synonymup arrow
Synonymup-arrow notation
Synonymup arrow notation
SynonymKnuth notation
Related topicConwaysChainedArrowNotation
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更新时间:2025/5/4 21:34:57