dejmar

I have to correct my errors in my prior post. 35 is NOT equivalent to the arithmetical value (55)5. The order of operations for superscripted exponents is top-down and not bottom-up. 35 is equivalent to 5^(55) = 5^(3125). Though not widely used, there is a proposed notation for pentation with sinister-subscription for the hyperpower, id est, the hyperexponent is a subscription on the left-side of the base. I.e, ba, such that the number (a) is pentated by (b). The order of operations with this notation would have a bottom-up sequence in a chain of pentation. cba. The hyperexponent (b) would be pentated with (c) first, with the resulting number being the pentation hyperexponent for (a). Pentation results in even higher numerical values than tetration. There are higher operations, but, as of yet, the notations available require other symbols other than base-10 numerals. One may consider that 9 pentated to (9 pentated to 9) results in the highest numerical value, i.e., number, available with using only three digits and no other characters (symbols).

Many have ignored the criterium "No symbols and characters allowed". Of course there is an implied "other than the three digits used to form the number". A base-10 digit is implied to be any one of the ten Arabic numerals. A number is an expression used to represent an arithmetical value. 123 is an expression. The position of each digit defines the arithmetical value: 1×102 + 2×101 + 3×100. As can be witnessed, an operation is automatically performed on the digits, thus operations are implied as acceptable in expressing a number. Exponentiation is an operation that does not require any addtional symbols, Positioning of the digits or expression is used, with a "base" part and a "raised" part (superscription). 456 is equivalent to the arithmetical value: (4×101 + 5×100)6. Tetration is another operation that does not require any additional symbols. Similar to exponentiation, tetration also uses superscription, but rather the "raised" number appears to the left of the base number instead of the right. 35 is equivalent to the arithmetical value (55)5, which is equivalent to 298,023,223,876,953,125. [Parentheses are used here as it is difficult to superscript beyond one level of superscripting using ASCII text.] The largest number using only 3-digits and no other symbols is 9 tetrated to the 9th tetrated to the 9th. [using parentheses to indicate superscripting beyond the first level, this can be written 9(99).] It is possible to define a new operation by using only 3 digits and positioning that is unambigous to already defined operations, yet any new operation that has not yet found acceptance by the authorative individuals of the mathematical community remains unacceptable by the given criteria. As the operation can be considered not pre-defined, and hence invalid.