# Magnitude-free Numbering Systems

Dozens Disciple
Treisaran
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Joined: Feb 14 2012, 01:00 PM

wendy.krieger
wendy.krieger
The thing about 'sumerian numbers', is that they are a form that modern notation does not have words for, and do not account for in the discussion on bases.

Sumerian numbers are a 'alternating division base', so that 1 and 1.00 are the same thing, and that 6.30 lies between 6 and 7. The most significant place is the units place, and this is alternatingly divided into 6 and 10. Only the use of zero belies that it is base 60, and that is a division base.

Firstly, we have records of numbers like 0,0,1 (ie 1/3600), but large numbers are effected by setting the tables into the required power of 60 (something that span centuries might be set in units of 60 years).

Moreover, by Neugebauer, we read that the number set as 3.12 of shocks, is elsewhere written as C I xxx ii (ie 100 + 60 (as a big '1'), + 30 + 2 = 192).

Zeros are not always given, because they were often implemented in blank columns.

On the comment of the sqrt(2), even though the tablet gives sqrt 30 as D2 B5 C5 against sqrt 2 of 1 B4 E1 C, the process of finding square roots from the reciprocal table will produce x and 1/x that x*x is the desired number, because the exact search is for x / (1/x) = x^2. I've done similar things in the twelfty form as well. (It's rather complex, because sqrt(2) falls in a hole in the table which requires many places to fill.

Given that the square-root 2 was written on the edges and diagonals of a square, i can hardly imagine that 'sqrt 30' was ever envisaged, rather it's sqrt(0,30) that is given.

Dozens Disciple
Treisaran
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Joined: Feb 14 2012, 01:00 PM
How the choice of base influences a magnitude-free numbering system

We've done here on the DozensOnline board some extensive tours of number bases, naturally in conjunction with the ordinary way of numeration: magnitude-bearing fixed-point. Here follow my haphazard thoughts on bases in magnitude-free numeration.

Whether floating-point (log-linear) or logarithmic, magnitude-free numeration is sensitive to the choice. At the one extreme we have binary, which has such an influence on magnitude-free numeration that it becomes a world of its own, more akin to unary in ordinary arithmetic than binary is. Binary in magnitude-bearing numeration is undoubtedly interesting, yet it is merely the smallest base - the smallest quantity, but otherwise not differing in quality from any other base. But magnitude-free binary has a qualitative difference: it's a base where all arithmetic gradations are restricted to the internal, fractional space.

Visually speaking, magnitude-free binary is invariably 1.somethingb. Magnitude-free floating-point 1.1b does duty for 1.6z, 3, 6, 10z, 20z and so on; 1.01b represents 1.3z, 2.6z, 5, Xz, 18z and so forth. The view forced on the user of magnitude-free binary is that of numbers as points within fractional space. Indeed the initial '1' can be omitted, as it is omnipresent. (Computer floating-point representations, though they are magnitude-bearing rather than magnitude-free, do take advantage of this to gain a bit: since the most significant bit of the significand is always 1, it is used instead for denoting the sign of the number.)

At the other extreme, a magnitude-free grand or supergrand base can be used without exceeding the first exponent (the '10' of the base). Taking a base such as FFF0x, we can in certain contexts, with care, treat it as representing an integer from 1 to FFEFx together with a fractional part, though 3FF8x in magnitude-free notation would still usually mean 0.6z rather than 9588z, 16376d.

The middle road includes not just the human-scale bases but also some greater auxiliaries, such as sexagesimal; this is because it takes quite a high base to avoid exceeding the first exponent in most of practical arithmetic. From magnitude-free ternary onwards, up to the highlands of bases (grands and supergrands), numeration fluctuates indiscriminately (to the human eye) between integers, fractions and mixtures of the two. It seems to me magnitude-free number systems, in contrast to the numeration we ordinarily use, are more interesting when the bases chosen for them are the extreme ones.

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jrus
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Joined: Oct 23 2015, 12:31 AM
There arenâ€™t really any practical use cases for a number system which stores only the mantissa but doesnâ€™t include the exponent, as far as I can tell.

Itâ€™s fine for constructing a condensed table of arithmetic facts, where you plan to keep track of the exponent in your head or write it down separately, but throwing the exponent out entirely leaves you with a mathematical curiosity rather than an engineering tool, in my opinion. (Not that thereâ€™s anything wrong with that.)

Storing the exponent takes very little space. Double precision floating point covers pretty much any magnitude weâ€™re ever likely to care about (from eâˆ’308 to e308 base ten) using just 11 bits.

Dozens Demigod
Double sharp
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Joined: Sep 19 2015, 11:02 AM
jrus @ Nov 17 2015, 09:57 PM wrote: There arenâ€™t really any practical use cases for a number system which stores only the mantissa but doesnâ€™t include the exponent, as far as I can tell.

Itâ€™s fine for constructing a condensed table of arithmetic facts, where you plan to keep track of the exponent in your head or write it down separately, but throwing the exponent out entirely leaves you with a mathematical curiosity rather than an engineering tool, in my opinion. (Not that thereâ€™s anything wrong with that.)

Storing the exponent takes very little space. Double precision floating point covers pretty much any magnitude weâ€™re ever likely to care about (from eâˆ’308 to e308 base ten) using just 11 bits.
To be honest, I tend to agree. Even with a slide rule, you still need to mentally keep track of the magnitude, don't you? And there is the amusing example from the end of the OP, about a boss and one of his employees disagreeing on the right order of magnitude their salary ought to be.

Dozens Disciple
Treisaran
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Joined: Feb 14 2012, 01:00 PM
jrus wrote: There aren't really any practical use cases for a number system which stores only the mantissa but doesn't include the exponent, as far as I can tell. [...]&nbsp; throwing the exponent out entirely leaves you with a mathematical curiosity rather than an engineering tool, in my opinion.
Definitely true. I use it mainly as a research tool, which is what I'd call it rather than (or in addition to) 'mathematical curiosity'.

Magnitude-free notation serves my studies somewhat like the Riemann sphere: as one method of getting around the problem with zero, particularly the impossibility of dividing by it. Whereas the Riemann sphere incorporates n/0 by setting it at the north pole for unsigned âˆž (the point at infinity of a stereographic projection), magnitude-free notation simply avoids 0 altogether. The function 1/n has a sinusoidal appearance and no discontinuous points; the function n^(1/n) has two asymptotes, one at each limit at âˆž and âˆ’âˆž, tending towards the value y=1.

Here are the first 20z numbers in magnitude-free binary: {1, 1, 1.1 (=3), 1, 1.01, 1.1 (=6), 1.11, 1, 1.001, 1.01, 1.011, 1.1 (=10z), 1.101, 1.11, 1.111, 1 (=14z), 1.0001, 1.001 (=16z), 1.0011, 1.01, 1.0101, 1.011, 1.0111, 1.1 (=20z}b. The list can be made clearer with an unquadral (base-14z) representation, though one must remember this is unquadral-coded binary and not unquadral-for-itself: {1, 1, 1.8 (=3), 1, 1.4, 1.8 (=6), 1.C, 1, 1.2, 1.4, 1.6, 1.8 (=10z), 1.A, 1.C, 1.E, 1 (=14z), 1.1, 1.2 (=16z, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8 (=20z)}x.

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jrus
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Joined: Oct 23 2015, 12:31 AM
Treisaran @ Nov 19 2015, 02:52 PM wrote: I use it mainly as a research tool, [...]

Magnitude-free notation serves my studies somewhat like the Riemann sphere: as one method of getting around the problem with zero

The Riemann sphere takes our existing complex number system and compactifies it by adding one point. All the previous operations we could do in the complex plane still apply just as before.

As far as I can tell the â€œsolutionâ€ of magnitude-free notation is to just make lots of our standard operations undefined and then avoid using them, or redefines them to mean something entirely different than the normal definition. Which is fine, but doesnâ€™t seem especially useful at first glance. Is there some concrete result you can compute with magnitude-free numbers that you couldnâ€™t compute before, or some result made much easier to compute?

Dozens Disciple
Treisaran
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Joined: Feb 14 2012, 01:00 PM
jrus wrote:Can you explain your research?
There's not much to explain. Just plugging the functions with this notation and looking at the resultant tables for interesting stuff. That's how I do most of my research; nothing rigorously scientific, as I'm not a scientist.
jrus wrote:As far as I can tell the "solution" of magnitude-free notation is to just make lots of our standard operations undefined and then avoid using them, or redefines them to mean something entirely different than the normal definition.
Any new algebra will do that kind of thing. For example, 1 + 1 = 1 in Boolean algebra, which is different from the standard 1 + 1 = 2.
jrus wrote:Which is fine, but doesn't seem especially useful at first glance.
No, it's geared towards being interesting. If I can glean something useful out of it later, so much the better.
jrus wrote:Is there some concrete result you can compute with magnitude-free numbers that you couldn't compute before, or some result made much easier to compute?
It means, as I said, function graphs assume a different shape than the ones we get from standard notation, and division by zero ceases to be a concern.

Dozens Demigod
Double sharp
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Joined: Sep 19 2015, 11:02 AM
Treisaran @ Nov 20 2015, 12:47 PM wrote: ...and division by zero ceases to be a concern.
If there's no zero, then what is 2 âˆ’ 2 in this notation?

Piotr
Piotr
Double sharp @ Nov 20 2015, 03:13 PM wrote:
Treisaran @ Nov 20 2015, 12:47 PM wrote: ...and division by zero ceases to be a concern.
If there's no zero, then what is 2 âˆ’ 2 in this notation?
Also, it's impossible to count without zero. In other hand, no number other than zero is required for counting, since zero alone indicates counting from the easiest number.