| human measurement is most significant in the middle range; the extremely large and extremely small are less accurate, and so less precise... computers are basically human tools |
Distance and length, How big is the universe, What's the radius of a neutrino, are estimates not significantly additive but in range and order of magnitude ... albeit mass-energies have consistently better precision ... for scholars, the models for estimation at the extreme are simply imprecise by uncertain factors of the very sciences and their measurings...
Furthermore, there is the conceptual anomaly in computing floating-point numbers, in that while the difference between two adjacent precise numbers is at the least-significant-digit (or bit) of the larger, it's at the most-significant-digit (or bit) between the smallest positive and negative numbers, (Half-that where zero is included as a special case of floating-point number)-- totally imprecise...
We can get around this practically, by using 'human-precision' numbers with the least precision at the extremes-- even to one last bit:
THE BASIC PLAIN INTEGER (N-total-bit Binary):
Additive Sign, 1 bit { 0 = nonnegative zero or positive, 1 = negative };
Additive Whole Number, N-1 bit { 0,,,2N-1-1 } for units
(signed or 2's complement or 1's complement).
And maximum precision is N-1 bit, and the scale achieves N range of variable precision.
THE AUTO-SCALED INTEGER, HUMAN NUMBER FORMAT (N-total-bit Binary):
Additive Sign, 1 bit { 0 = nonnegative zero or positive, 1 = negative };
Multiplicative Scale, M bit { 0,1,,,2M-1 }, where MS=0 means 'subscale';
Additive Whole Number, N-M-1 bit { 0,,,2N-M-1-1 } for units (signed or 2's
complement or 1's complement).
But, Maximum precision is N-M bit, +1 bit better, because, when the largest AWN steps, +1, to overflow into the first nonzero-MS-value, the auto-scaled human integer also assumes the top-bit, implicitly, hitherto explicitly, and the explicit AWN becomes the sub-top-bit-portion of the human integer, 2N-M-1+AWN, { 2N-M-1,,,2N-M-1 }, and the MS achieves 2M+N-M-1 range of variable precision.
(This could be generalized to allow AWN up to at most N-2 bits, and when it steps, +1, to overflow into the top-bit, it switches to auto-scaled and N-M bit precision, and the MS achieves 2M-1+N-1 range, of generalized auto-re-scaled-overshot-human integers.)
The distinction between plain-integers and scaled-integers, is simply-- the existence of the Multiplicative Scale where the uppermost plain-bits sit: Smaller integers are indistinguishable, but when an integer is large enough the uppermost bits are either plain or scale, and nonzero scale also indicates the whole number assumes an over-bit.
THE BASIC FLOATING-POINT, HUMAN NUMBER FORMAT (N-total-bit Binary):
Additive Sign, 1 bit { 0 = nonnegative zero or positive, 1 = negative };
Multiplicative Scale, M bit { 0,1,,,2M-1 } Offset Positive
{ -2M-1,,,+2M-1-1 where OP = 2M/2 },
where MS=0 means 'subscale';
Additive Fraction-Mantissa, N-M-1 bit { 0,,,2N-M-1-1 } for quanta
(signed or 2's complement or 1's complement).
FPN = (-1)AS * 2MS-Offset * (1:MS≠0 + AFM).
(In Offset Positive, Multiplicative Scale is zero for the tiniest numbers, and maximum for the largest; OP at halfway for the Units.)
The distinction between scaled-integers and floating-point, is simply-- the constant median value scale-factor (offset) distinguishing human-units from quanta-units, and nonzero scale also indicates the fraction assumes a unit over-bit.
(And, as in fact extremely large numbers are basically the multiplicative inverses of the extremely small, nominally useful floating-point numbers might have the fractional part range between inverses: between 0.707 and 1.414, cf as a logarithmic number).
THE ADDITIVE FRACTION/MANTISSA:
Having variable-precision in the human number format, and the smallest human numbers having zero-scale, the triple-zone format is:
[under construction]
A premise discovery under the title,