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# Template:List of operations (numbers)

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To insert the following list, use the template call ** {{List of operations (numbers)}}**.

## Contents

#### 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*