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Joyce's More Generalized Exponential Notation

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Joyce's More Generalized Exponential Notation is a variation of Ackermann's Generalized Exponential Notation.[1] In addition to all the rules in the original notation, this notation is defined as:

  • \(g(a, b, c, d) = g(a - 1, b, c, g(a, b, c))\)        Rule 1 applies to nesting the least important base number, c, by replacing it with the right-most three-value expression g(b,c,d) the base nesting number, a, times.
  • \(g(a, 1, b, c, d) = g(a - 1, 1, b, g(b, c, d), d)\),   Rule 2 applies to the nesting of the more important number of operations, c again and again replacing it by the right-most three-value expression g(b, c, d) without the base number changed.
  • \(g(a, 1, 1, b, c, d) = g(a - 1, 1, 1, g(b, c, d), c, d)\)    Rule 3 applies to the nesting of the even more important operation number, c again and again replacing it by the right-most three-value expression g(b, c, d) with neither the base number or number of operations changed.
  • \(g(a, 1, 1, b, c, d, e) = g(a - 1, 1, 1, g(b, c, d, d, e), c, d, e)\)   Rule 4 applies to the nesting to the right-most four numbers rather than merely three as previously for the seven-valued, eight-valued and nine-valued expressions, and the next last seven for the ten-valued, eleven-valued and twelve-valued, and so on.

By extrapolation these basic rules can be extended for any number of possible nestings of base number, number of operations, mathematical operations, or combinations of them. Parentheses may be used to abbreviate. For example, g((a, b), c, d, e) means to apply Rules 1 and 2 simultaneously, while  g((a, 1, b), c, d, e) means to apply Rules 1 and 3 simultaneously and g((a, b, c), d, e, f) means to apply all three simulateously, and so on for the seven-valued and higher expressions. This does mean that this notation doesn't grow as fast as some others, but it identifies some otherwise neglected numbers on the climb upward.

Sources Edit

  1. Googology

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