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# Knuth's arrow notation

This article describes Knuth's up-arrow notation to represent iterated exponentiation with base
 b
(power towers with base
 b
) and a down-arrow notation to represent the iterated logarithm (with base
 b
) inverse of iterated exponentiation.

## Knuth's up-arrow notation

In 1976, Donald Knuth introduced the following up-arrow notation for power towers

$p=b\uparrow ^{n}d=b\uparrow ^{n-1}[b\uparrow ^{n}(d-1)],\quad n\geq 2,\,$ with

$b\uparrow ^{1}d:=b^{d},\,$ $b\uparrow ^{n}0:=1,\,$ where
 d
is the order of the power tower with base
 b
, and
 p
is the result of the power tower evaluation.

## Down-arrow notation

The down-arrow notation defines a logarithmic-type inverse of the up-arrow notation

$d=b\downarrow ^{n}p:=\log _{b}^{*}p,\,$ where
 log ⁎b p
is the number of iterations of the logarithm (base
 b
) required such that
 log ⁎b p   ≤   b
.

In particular, we have

$d=e\downarrow ^{n}p:=\log _{e}^{*}p=\log ^{*}p,\,$ where
 log ⁎ p
is the number of iterations of the natural logarithm required such that
 log ⁎ p   ≤   e
.

Now, if we have

$p=b\uparrow ^{n}d,\quad d\geq 1,\,$ then

$d=b\downarrow ^{n}p,\,$ where
 d
is a positive integer. How could we generalize
 d
to integers, rational numbers, real numbers or complex numbers for all
 n
? For
 n = 1
, we do have a generalization.

#### Hierarchical list of operations pertaining to numbers  

##### 1st iteration
• Addition:  S(S(⋯ "a times" ⋯ (S(n))))
, the sum n  +  a
, where  n
is the augend and  a
is the addend. (When addition is commutative both are simply called terms.)
• Subtraction:  P(P(⋯ "s times" ⋯ (P(n))))
, the difference n  −  s
, where  n
is the minuend and  s
is the subtrahend.
##### 2nd iteration
• Multiplication:  n + (n + (⋯ "k times" ⋯ (n + (n))))
, the product m  ⋅   k
, where  m
is the multiplicand and  k
is the multiplier. (When multiplication is commutative both are simply called factors.)
• Division: the ratio n  /  d
, where  n
is the dividend and  d
is the divisor.
##### 3rd iteration
• Exponentiation (  d
as "degree",  b
as "base",  n
as "variable").
• Powers:  n  ⋅   (n  ⋅   (⋯ "d times" ⋯ (n  ⋅   (n))))
, written  n d
.
• Exponentials:  b  ⋅   (b  ⋅   (⋯ "n times" ⋯ (b  ⋅   (b))))
, written  b n
.
• Exponentiation inverses (  d
as "degree",  b
as "base",  n
as "variable").
##### 5th iteration
• Pentation (  d
as "degree",  b
as "base",  n
as "variable").
• Penta-powers:  n ^^ (n ^^ (⋯ "d times" ⋯ (n ^^ (n ^^ (n)))))
, written  n ^^^ d or n ↑↑↑ d
.
• Penta-exponentials:  b ^^ (b ^^ (⋯ "n times" ⋯ (b ^^ (b ^^ (b)))))
, written  b ^^^ n or b ↑↑↑ n
.
• Pentation inverses
##### 6th iteration
• Hexation (  d
as "degree",  b
as "base",  n
as "variable").
• Hexa-powers:  n ^^^ (n ^^^ (⋯ "d times" ⋯ (n ^^^ (n))))
, written  n ^^^^ d or n ↑↑↑↑ d
.
• Hexa-exponentials:  b ^^^ (b ^^^ (⋯ "n times" ⋯ (b ^^^ (b))))
, written  b ^^^^ n or b ↑↑↑↑ n
.
• Hexation inverses
##### 7th iteration
• Heptation (  d
as "degree",  b
as "base",  n
as "variable").
• Hepta-powers:  n ^^^^ (n ^^^^ (⋯ "d times" ⋯ (n ^^^^ (n))))
, written  n ^^^^^ d or n ↑↑↑↑↑ d
.
• Hepta-exponentials:  b ^^^^ (b ^^^^ (⋯ "n times" ⋯ (b ^^^^ (b))))
, written  b ^^^^^ n or b ↑↑↑↑↑ n
.
• Heptation inverses
##### 8th iteration
• Octation (  d
as "degree",  b
as "base",  n
as "variable").
• Octa-powers:  n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n))))
, written  n ^^^^^^ d or n ↑↑↑↑↑↑ d
.
• Octa-exponentials:  b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b))))
, written  b ^^^^^^ n or b ↑↑↑↑↑↑ n
.
• Octation inverses