12/13/2011, 12:28 AM
It seems as if you are interested in large number representations. It also seems as if you are interested in alternative systems with more rapid rates of growth, but one property that many systems do not have is that of uniqueness. Even SPN (as you call it) is not unique for all rational or real numbers, because 1 can be represented as 0.999999... Which is also true of scientific notation:
http://en.wikipedia.org/wiki/Scientific_...d_notation
however, setting the 0.999... issue aside, there is usually only one logical (using common sense) way to write any rational number, uniquely in the form:
\( (-1)^s m \times 10^{e} \) where s=0,1, e is an integer and \( 1 \le m \lt 10 \) which may require a bar for repeating digits. I have also been interested in this enough to talk about it on my blog:
http://straymindcough.blogspot.com/2011/...point.html
which I would recommend reading. The interesting thing about tetration notation is that this also works, but requires that you change the mantissa range to 0..1. In other words you can write (for base 10, not true of bases less than eta) a representation of a real number (other than -1, 0, 1), uniquely in the form:
\( (-1)^s {\exp_{10}^{h}(m)}^{{(-1)}^r} \) where r,s=0,1, h is an positive integer, and \( 0 \lt m \le 1 \) provided that m is a rational number. While this uniqueness may seem silly at first, it's can be very important. Another property of both scientific notation and tetration notation is the ability to compare without evaluating. For example, if the exponents (e, h) are the same, then we can compare mantissas (m) which can be much, much easier than trying to evaluate a big number.
Regards,
Andrew Robbins
http://en.wikipedia.org/wiki/Scientific_...d_notation
however, setting the 0.999... issue aside, there is usually only one logical (using common sense) way to write any rational number, uniquely in the form:
\( (-1)^s m \times 10^{e} \) where s=0,1, e is an integer and \( 1 \le m \lt 10 \) which may require a bar for repeating digits. I have also been interested in this enough to talk about it on my blog:
http://straymindcough.blogspot.com/2011/...point.html
which I would recommend reading. The interesting thing about tetration notation is that this also works, but requires that you change the mantissa range to 0..1. In other words you can write (for base 10, not true of bases less than eta) a representation of a real number (other than -1, 0, 1), uniquely in the form:
\( (-1)^s {\exp_{10}^{h}(m)}^{{(-1)}^r} \) where r,s=0,1, h is an positive integer, and \( 0 \lt m \le 1 \) provided that m is a rational number. While this uniqueness may seem silly at first, it's can be very important. Another property of both scientific notation and tetration notation is the ability to compare without evaluating. For example, if the exponents (e, h) are the same, then we can compare mantissas (m) which can be much, much easier than trying to evaluate a big number.
Regards,
Andrew Robbins