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# Knuth's arrow notation

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(Redirected from Down-arrow notation)

b |

b |

b |

## Knuth's up-arrow notation

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

with

d |

b |

p |

## Down-arrow notation

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

log ⁎b p |

b |

log ⁎b p ≤ b |

In particular, we have

log ⁎ p |

log ⁎ p ≤ e |

Now, if we have

then

d |

d |

n |

n = 1 |

## See also

#### Hierarchical list of operations pertaining to numbers ^{[1]} ^{[2]}

##### 0^{th} iteration

- Successor:

.S( *n*) - Predecessor:

.P( *n*)

##### 1^{st} iteration

- Addition:

, theS(S(⋯ " *a*times" ⋯ (S(*n*))))*sum*

, where*n*+*a*

is the*n**augend*and

is the*a**addend*. (When addition is commutative both are simply called*terms*.) - Subtraction:

, theP(P(⋯ " *s*times" ⋯ (P(*n*))))*difference*

, where*n*−*s*

is the*n**minuend*and

is the*s**subtrahend*.

##### 2^{nd} iteration

- Multiplication:

, the*n*+ (*n*+ (⋯ "*k*times" ⋯ (*n*+ (*n*))))*product*

, where*m*⋅*k*

is the*m**multiplicand*and

is the*k**multiplier*.^{[3]}(When multiplication is commutative both are simply called*factors*.) - Division: the
*ratio*

, where*n*/*d*

is the*n**dividend*and

is the*d**divisor*.- Quotient: (integer division).
- Remainder: (modulo and congruences).

##### 3^{rd} iteration

- Exponentiation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Powers:

, written*n*⋅ (*n*⋅ (⋯ "*d*times" ⋯ (*n*⋅ (*n*))))

.*n**d* - Exponentials:

, written*b*⋅ (*b*⋅ (⋯ "*n*times" ⋯ (*b*⋅ (*b*))))

.*b**n*- Exponential function:

, where*e**n*

is Euler's number.*e*

- Exponential function:

- Powers:
- Exponentiation inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Roots:

.*d*√*n* - Logarithms:

.log *b**n*- Natural logarithm function:

, orlog *n*

, wherelog *e**n*

is Euler's number.*e*

- Natural logarithm function:

- Roots:

##### 4^{th} iteration

- Tetration (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-powers (super-powers):

, written*n*^ (*n*^ (⋯ "*d*times" ⋯ (*n*^ (*n*))))

.*n*^^*d*or*n*↑↑*d* - Tetra-exponentials (super-exponentials):

, written*b*^ (*b*^ (⋯ "*n*times" ⋯ (*b*^ (*b*))))

.*b*^^*n*or*b*↑↑*n*

- Tetra-powers (super-powers):
- Tetration inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-roots (super-roots)
- Tetra-logarithms (super-logarithms):

.slog *b**n*- Iterated logarithm:

.log ⁎ *b**n*= ⌈slog*b**n*⌉

- Iterated logarithm:

##### 5^{th} iteration

- Pentation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Penta-powers:

, written*n*^^ (*n*^^ (⋯ "*d*times" ⋯ (*n*^^ (*n*^^ (*n*)))))

.*n*^^^*d*or*n*↑↑↑*d* - Penta-exponentials:

, written*b*^^ (*b*^^ (⋯ "*n*times" ⋯ (*b*^^ (*b*^^ (*b*)))))

.*b*^^^*n*or*b*↑↑↑*n*

- Penta-powers:
- Pentation inverses

##### 6^{th} iteration

- Hexation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hexa-powers:

, written*n*^^^ (*n*^^^ (⋯ "*d*times" ⋯ (*n*^^^ (*n*))))

.*n*^^^^*d*or*n*↑↑↑↑*d* - Hexa-exponentials:

, written*b*^^^ (*b*^^^ (⋯ "*n*times" ⋯ (*b*^^^ (*b*))))

.*b*^^^^*n*or*b*↑↑↑↑*n*

- Hexa-powers:
- Hexation inverses

##### 7^{th} iteration

- Heptation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hepta-powers:

, written*n*^^^^ (*n*^^^^ (⋯ "*d*times" ⋯ (*n*^^^^ (*n*))))

.*n*^^^^^*d*or*n*↑↑↑↑↑*d* - Hepta-exponentials:

, written*b*^^^^ (*b*^^^^ (⋯ "*n*times" ⋯ (*b*^^^^ (*b*))))

.*b*^^^^^*n*or*b*↑↑↑↑↑*n*

- Hepta-powers:
- Heptation inverses

##### 8^{th} iteration

- Octation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Octa-powers:

, written*n*^^^^^ (*n*^^^^^ (⋯ "*d*times" ⋯ (*n*^^^^^ (*n*))))

.*n*^^^^^^*d*or*n*↑↑↑↑↑↑*d* - Octa-exponentials:

, written*b*^^^^^ (*b*^^^^^ (⋯ "*n*times" ⋯ (*b*^^^^^ (*b*))))

.*b*^^^^^^*n*or*b*↑↑↑↑↑↑*n*

- Octa-powers:
- Octation inverses

## Notes

- ↑ Hyperoperation—Wikipedia.org.
- ↑ Grzegorczyk hierarchy—Wikipedia.org.
- ↑ There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:

.*ω*× 2**:**=*ω*+*ω*