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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)}}.

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
  • 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
7th 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
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
    .