Editing Knuth's up-arrow notation (section)

==Introduction==
The [[hyperoperations]] naturally extend the [[arithmetic]] operations of [[addition]] and [[multiplication]] as follows.
[[Addition]] by a [[natural number]] is defined as iterated incrementation:
:<math>
  \begin{matrix}
   H_1(a,b) = a+b = & a+\underbrace{1+1+\dots+1} \\
   & b\mbox{ copies of }1
  \end{matrix} 
 </math>
   
[[Multiplication]] by a [[natural number]] is defined as iterated [[addition]]:
:<math>
  \begin{matrix}
   H_2(a,b) = a\times b = & \underbrace{a+a+\dots+a} \\
   & b\mbox{ copies of }a
  \end{matrix} 
 </math>

For example,

:<math>
  \begin{matrix}
   4\times 3 & = & \underbrace{4+4+4} & = & 12\\
   & & 3\mbox{ copies of }4
  \end{matrix} 
 </math>

[[Exponentiation]] for a natural power <math>b</math> is defined as iterated multiplication, which Knuth denoted by a single up-arrow:

:<math>
  \begin{matrix}
   a\uparrow b = H_3(a,b) = a^b = & \underbrace{a\times a\times\dots\times a}\\
   & b\mbox{ copies of }a
  \end{matrix} 
 </math>

For example,

:<math>
  \begin{matrix}
   4\uparrow 3= 4^3 = & \underbrace{4\times 4\times 4} & = & 64\\
   & 3\mbox{ copies of }4
  \end{matrix} 
 </math>

[[Tetration]] is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:
:<math>
  \begin{matrix}
   a\uparrow\uparrow b = H_4(a,b) = &  \underbrace{a^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & 
   = & \underbrace{a\uparrow (a\uparrow(\cdots\uparrow a))} 
\\  
    & b\mbox{ copies of }a
    & & b\mbox{ copies of }a
  \end{matrix}
 </math>

For example,

:<math>
  \begin{matrix}
   4\uparrow\uparrow 3 = & \underbrace{4^{4^4}} & 
   = & \underbrace{4\uparrow (4\uparrow 4)} & = & 4^{256} & &
\\  
    & 3\mbox{ copies of }4
    & &3\mbox{ copies of }4
  \end{matrix} 
 </math>

Expressions are evaluated from right to left, as the operators are defined to be [[Right associative operator|right-associative]].

According to this definition,
:<math>3\uparrow\uparrow 2=3^3=27 </math>
:<math>3\uparrow\uparrow 3=3^{3^3}=3^{27}=7,625,597,484,987 </math>
:<math>3\uparrow\uparrow 4=3^{3^{3^3}}=3^{3^{27}}=3^{7625597484987}
 </math> 
:<math>3\uparrow\uparrow 5=3^{3^{3^{3^3}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}} </math>
:etc.

This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here.

[[Pentation]], defined as iterated tetration, is represented by the “triple arrow”:
:<math>
  \begin{matrix}
   a\uparrow\uparrow\uparrow b = H_5(a,b) = &
    \underbrace{a_{}\uparrow\uparrow (a\uparrow\uparrow(\cdots\uparrow\uparrow a))}\\
    & b\mbox{ copies of }a
  \end{matrix}
 </math>
[[Hexation]], defined as iterated pentation, is represented by the “quadruple arrow”:
:<math>
  \begin{matrix}
   a\uparrow\uparrow\uparrow\uparrow b = H_6(a,b) = &
    \underbrace{a_{}\uparrow\uparrow\uparrow (a\uparrow\uparrow\uparrow(\cdots\uparrow\uparrow\uparrow a))}\\
    & b\mbox{ copies of }a
  \end{matrix}
 </math>

and so on. The general rule is that an <math>n</math>-arrow operator expands into a right-associative series of (<math>n - 1</math>)-arrow operators. Symbolically,
:<math>
  \begin{matrix}
   a\ \underbrace{\uparrow_{}\uparrow\!\!\cdots\!\!\uparrow}_{n}\ b=
    \underbrace{a\ \underbrace{\uparrow\!\!\cdots\!\!\uparrow}_{n-1}
    \ (a\ \underbrace{\uparrow_{}\!\!\cdots\!\!\uparrow}_{n-1}
    \ (\cdots
    \ \underbrace{\uparrow_{}\!\!\cdots\!\!\uparrow}_{n-1}
    \ a))}_{b\text{ copies of }a}
  \end{matrix}
 </math>

Examples:

:<math>3\uparrow\uparrow\uparrow2 = 3\uparrow\uparrow3 = 3^{3^3} = 3^{27}=7,625,597,484,987</math>

:<math>
 \begin{align}
  3\uparrow\uparrow\uparrow3
  &= 3\uparrow\uparrow(3\uparrow\uparrow3) \\
  &= 3\uparrow\uparrow(3\uparrow 3\uparrow3) \\
  
  &=
  \begin{matrix}
   \underbrace{3\uparrow 3\uparrow\cdots\uparrow 3} \\
   3\uparrow3\uparrow3\mbox{ copies of } 3
  \end{matrix}\\
  
  &=
  \begin{matrix}
   \underbrace{3\uparrow 3\uparrow\cdots\uparrow 3} \\
   \mbox{7,625,597,484,987 copies of 3}
  \end{matrix}\\
  
  &=
  \begin{matrix}
   \underbrace{3^{3^{3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}}}} \\
   \mbox{7,625,597,484,987 copies of 3}
  \end{matrix}
 \end{align}
 </math>