Editing Knuth's up-arrow notation (section)

{{short description|Method of notation of very large integers}}
{{pp|small=yes}}
In [[mathematics]], '''Knuth's up-arrow notation''' is a method of notation for [[Large number|very large]] [[Integer|integers]], introduced by [[Donald Knuth]] in 1976.<ref>{{cite journal | last =Knuth | first = Donald E.| year=1976|title=Mathematics and Computer Science: Coping with Finiteness |journal=Science | volume=194|issue=4271| pages=1235–1242 | doi=10.1126/science.194.4271.1235 | pmid=17797067 |bibcode=1976Sci...194.1235K| s2cid = 1690489}}</ref>

In his 1947 paper,<ref>{{cite journal
  | author= R. L. Goodstein
  | title= Transfinite Ordinals in Recursive Number Theory
  | journal= Journal of Symbolic Logic
  |date=Dec 1947
  | volume= 12
  | issue= 4
  | pages= 123–129
  | doi= 10.2307/2266486
  | jstor= 2266486| s2cid= 1318943
 }}</ref> [[R. L. Goodstein]] introduced the specific sequence of operations that are now called [[Hyperoperation|''hyperoperations'']]. Goodstein also suggested the Greek names [[tetration]], [[pentation]], etc., for the extended operations beyond [[exponentiation]]. The sequence starts with a [[unary operation]] (the [[successor function]] with ''n'' = 0), and continues with the [[binary operation]]s of [[addition]] (''n'' = 1), [[multiplication]] (''n'' = 2), [[exponentiation]] (''n'' = 3), [[tetration]] (''n'' = 4), [[pentation]] (''n'' = 5), etc.
[[Hyperoperation#Notations|Various notations]] have been used to represent hyperoperations. One such notation is <math>H_n(a,b)</math>.
Knuth's up-arrow notation <math>\uparrow</math> is another. 
For example:
* the single arrow <math>\uparrow</math> represents [[exponentiation]] (iterated multiplication) <math display="block">2 \uparrow 4 = H_3(2,4) = 2\times(2\times(2\times 2)) = 2^4 = 16</math>
* the double arrow <math>\uparrow\uparrow</math> represents [[tetration]] (iterated exponentiation) <math display="block"> 2 \uparrow\uparrow 4 = H_4(2,4) = 2 \uparrow (2 \uparrow (2 \uparrow 2))= 2^{2^{2^{2}}} = 2^{16} = 65,536</math>
* the triple arrow <math>\uparrow\uparrow\uparrow</math> represents [[pentation]] (iterated tetration) <math display="block">\begin{align}
2 \uparrow\uparrow\uparrow 4 &= H_5(2,4)\\
&= 2 \uparrow\uparrow (2 \uparrow\uparrow (2 \uparrow\uparrow 2 ))\\ 
&= 2 \uparrow\uparrow (2 \uparrow\uparrow (2 \uparrow 2 ))\\
&= 2 \uparrow\uparrow (2 \uparrow\uparrow 4 )\\
&= \underbrace{2 \uparrow (2 \uparrow (2 \uparrow\cdots ))}
  \; = \; \underbrace{ \; 2^{2^{\cdots^2}}}\\
& \;\;\;\;\; 2 \uparrow\uparrow 4 \text{ copies of } 2
  \;\;\;\;\; \text{65,536 2s}\\
\end{align}</math>
The general definition of the up-arrow notation is as follows (for <math>a \ge 0, n \ge 1, b \ge 0</math>):
<math display="block">a\uparrow^nb = H_{n+2}(a,b) = a[n+2]b.</math>
Here, <math>\uparrow^n</math> stands for ''n'' arrows, so for example
<math display="block">2 \uparrow\uparrow\uparrow\uparrow 3 = 2\uparrow^4 3.</math>
The square brackets are another notation  for hyperoperations.