Knuth's up-arrow notation

2017-02-09T01:12:25Z

202.86.213.41: Added reference to Graham's number - thus helping it break orphan status

'''Knuth's up-arrow notation''' is a way of expressing very big numbers.<ref name="mathworld">{{cite web|url=http://mathworld.wolfram.com/ArrowNotation.html |title=Arrow Notation - from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2011-10-24 |accessdate=2011-10-30}}</ref> It was made by [[Donald Knuth]] in 1976.<ref name="mathworld" /> It is [[wikt:relate|relate]]d to the [[hyperoperation]] sequence. The notation is used in [[Graham's number]].

One arrow represents [[exponentiation]], 2 arrows represent [[tetration]], 3 for [[pentation]], etc.:<ref>{{cite web|url=http://www-users.cs.york.ac.uk/susan/cyc/b/big.htm#up |title=SS > factoids > big numbers |publisher=Users.cs.york.ac.uk |date=1998-07-07 |accessdate=2011-10-30}}</ref>
#[[Exponentiation]]
#:<math>a \uparrow^{1} b = a^b = \underbrace{a \times a \times \cdots \times a}_{b \ times}</math>
#::''a'' multiplied by itself, ''b'' times.
#[[Tetration]]
#:<math>a \uparrow^{2} b = a \uparrow \uparrow b = {^{b}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_{b \ times} = \underbrace{(a \uparrow^1 (a \uparrow^1 (... \uparrow^1 a)...)}_{b \ times}</math>
#::''a'' exponentiated by itself, ''b'' times.
#Third level
#:<math>a \uparrow^{3} b = a \uparrow \uparrow \uparrow b = \underbrace{a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow \ldots a) \ldots ) )}_{b \ times}</math>
#[[etc]]

This notation is used to describe the incredibly large [[Graham's number|Graham's Number]]

== References ==
{{reflist}}

{{math-stub}}

[[Category:Hyperoperations]]

TCS Wiki - User contributions [en]

Knuth's up-arrow notation