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Applications of Tetration
Don't forget the application of tetration for creating a new superformat for the notation of real numbers, similar to the "floating point" notation, i.e.:
N = p.[b ^ n] , with p "the significance", b "the base" and n "the order of magnitude" (bracketing is not necessary);
to be compared with:
N = p (*) [b # n], with p "the tetrational significant figures", b "the base" and n "the tetrational order of magnitude" (bracketing is not necessary). Operator (*) would indicate the highest exponent of the "unhomogeneous towers" (with 1 < p < b).

This superformat notation is ideal for the notation of extremely large numbers.

Actually, I briefly talked about this with the computer arithmetic section, and near the end about the "no overflows" part. But yes, this is one of the more prominent applications of tetration, for which I reference Holmes' work (PDF) in my paper.

And when it comes to notation, "p (*) [b # n]" can also be written "(b^)^n(p)" in ASCII, or as usual. If the parentheses are what is inspiring you to need a new notation, then the simplest solution would be to just drop them as: . Although your notation looks very much like traditional scientific notation, this may not be a good thing, as it is confusing and easy to mistake for scientific notation. I think iterated exponential notation should be unambiguous, and consistent. The "p (*) [b # n]" notation is neither. As for computer notation, just like many programming languages have "#E#" notation for scientific notation (assuming base 10), there could some benefit to having a similar convention for iterated exponentials, where either "p T n" (T for tetration, although I've also seen this used for scientific notation when the base is 10, T for ten) or "p S n" (S for "super") could be used to separate the mantissa (p) and height (n). But unless someone writes a programming language with these operators, I don't think anyone is going to use them.

Andrew Robbins
Right! Let us try and use these various expressions and, in future, the use will cut very sharply. As a matter of fact, we need both (so to say) multi-dimensional logos and ASCII type sequential special characters. Actually, both "floating point" and "tetrational notations", in a sequential ASCII notation, can be written as N = p.b^n and N = p*b#n, with unabiguous meanings for ^ (exponentiation sign) and # (tetration sign), supposing to have stated that operations "^" and "#" have priority as regard to "." and "*" and that "*" refers to base b (!!!). The problem is the meaning here of "*", which will be new, specific, uncomplete and confusing. Actually, it is uncomplete because it must be referred to base b and should be written as "*\super scripted b". In fact, "x(*\b)y" is a ternary operation. A ... provisional mess. I should try to avoid T and/or S, which may remind other things. Finally, I like very much my "box notation", absolutely not ambiguous, but non-sequential and too new. We shall see.

Indeed. Box notation is clear, precise, short, intuitive, unambiguous, and almost mnemonic (the "r" and "L").

I think your box notation is a work of beauty. Smile

Andrew Robbins
Thank you! It is exactly what I was thinking, but I didn't dare to say ; - ) .

I should be very happy if you would decide to use them in your papers. The symbols are almost self-explanatory. They are part of the standard MathType fonts.

Gianfranco Romerio

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