Division

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Division is multiplication with multiplicative inverse of second term (the divisor,) which by definition makes it non-commutative.

See also

Hierarchical list of operations pertaining to numbers

0^th iteration

• Successor:  

  S(n)

  .
• Predecessor:  

  P(n)

  .

1^st 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.

2^nd 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.
  • Quotient: (integer division).
  • Remainder: (modulo and congruences).

3^rd 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

    .
    • Exponential function:  

      e n

      , where  

      e

      is Euler's number.
• Exponentiation inverses ( 

  d

  as "degree",  

  b

  as "base",  

  n

  as "variable").
  • Roots:  

    d √  n

    .
  • Logarithms:  

    logb n

    .
    • Natural logarithm function:  

      log n

      , or  

      loge n

      , where  

      e

      is Euler's number.

4^th iteration

• Tetration ( 

  d

  as "degree",  

  b

  as "base",  

  n

  as "variable").
  • Tetra-powers (super-powers):  

    n ^ (n ^ (⋯ "d times" ⋯ (n ^ (n))))

    , written  

    n ^^ d or n ↑↑ d

    .
  • Tetra-exponentials (super-exponentials):  

    b ^ (b ^ (⋯ "n times" ⋯ (b ^ (b))))

    , written  

    b ^^ n or b ↑↑ n

    .
• Tetration inverses ( 

  d

  as "degree",  

  b

  as "base",  

  n

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

    slogb n

    .
    • Iterated logarithm:  

      log ⁎b n = ⌈slogb n⌉

      .

5^th 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
  • Penta-roots
  • Penta-logarithms

6^th 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
  • Hexa-roots
  • Hexa-logarithms

7^th 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
  • Hepta-roots
  • Hepta-logarithms

8^th 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
  • Octa-roots
  • Octa-logarithms

Notes

Operator precedence
Parenthesization — Factorial — Exponentiation — Multiplication and division — Addition and subtraction