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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 [1] [2]

0th iteration
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
3rd iteration
4th iteration
5th iteration
6th iteration
7th iteration
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

Notes

  1. HyperoperationWikipedia.org.
  2. Grzegorczyk hierarchyWikipedia.org.
  3. 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

Formula Operator Precedence Demo.png

Parenthesization — FactorialExponentiationMultiplication and divisionAddition and subtraction


Notes