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Division
From OeisWiki
Division is multiplication with multiplicative inverse of second term (the divisor,) which by definition makes it non-commutative.
Contents
See also
Hierarchical list of operations pertaining to numbers ^{[1]} ^{[2]}
0^{th} iteration
- Successor:
.S(n) - Predecessor:
.P(n)
1^{st} iteration
- Addition:
, the sumS(S(⋯ "a times" ⋯ (S(n))))
, wheren + a
is the augend andn
is the addend. (When addition is commutative both are simply called terms.)a - Subtraction:
, the differenceP(P(⋯ "s times" ⋯ (P(n))))
, wheren − s
is the minuend andn
is the subtrahend.s
2^{nd} iteration
- Multiplication:
, the productn + (n + (⋯ "k times" ⋯ (n + (n))))
, wherem ⋅ k
is the multiplicand andm
is the multiplier.^{[3]} (When multiplication is commutative both are simply called factors.)k - Division: the ratio
, wheren / d
is the dividend andn
is the divisor.d - Quotient: (integer division).
- Remainder: (modulo and congruences).
3^{rd} iteration
- Exponentiation (
as "degree",d
as "base",b
as "variable").n - Powers:
, writtenn ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n))))
.n d - Exponentials:
, writtenb ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b))))
.b n - Exponential function:
, wheree n
is Euler's number.e
- Exponential function:
- Powers:
- Exponentiation inverses (
as "degree",d
as "base",b
as "variable").n - Roots:
.d √ n - Logarithms:
.logb n - Natural logarithm function:
, orlog n
, whereloge 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):
, writtenn ^ (n ^ (⋯ "d times" ⋯ (n ^ (n))))
.n ^^ d or n ↑↑ d - Tetra-exponentials (super-exponentials):
, writtenb ^ (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):
.slogb n - Iterated logarithm:
.log ⁎b n = ⌈slogb n⌉
- Iterated logarithm:
5^{th} iteration
- Pentation (
as "degree",d
as "base",b
as "variable").n - Penta-powers:
, writtenn ^^ (n ^^ (⋯ "d times" ⋯ (n ^^ (n ^^ (n)))))
.n ^^^ d or n ↑↑↑ d - Penta-exponentials:
, writtenb ^^ (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:
, writtenn ^^^ (n ^^^ (⋯ "d times" ⋯ (n ^^^ (n))))
.n ^^^^ d or n ↑↑↑↑ d - Hexa-exponentials:
, writtenb ^^^ (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:
, writtenn ^^^^ (n ^^^^ (⋯ "d times" ⋯ (n ^^^^ (n))))
.n ^^^^^ d or n ↑↑↑↑↑ d - Hepta-exponentials:
, writtenb ^^^^ (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:
, writtenn ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n))))
.n ^^^^^^ d or n ↑↑↑↑↑↑ d - Octa-exponentials:
, writtenb ^^^^^ (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
Operator precedence |
---|
Parenthesization — Factorial — Exponentiation — Multiplication and division — Addition and subtraction |