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 Reverse Positional Notation, $100 becomes 001$
Sennekuyl
Posted: Jul 15 2015, 12:58 AM


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Hope that was self explanatory.

Has anyone discussed this? I can't see any practical application but a lot more confusion. Would be great for a con-lang for a sci-fi I guess.

For any new users, rather than the conventional method placement:


==============================
| ... || Thousands || Hundreds || Tens || Ones ||
==============================

or as dozenal

=========================
| ... || Galores || Gross || Doz || One ||
=========================

it is mirrored:

=======================
Ones || Doz || Gross || Galore || ... |
=======================
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Dan
Posted: Jul 15 2015, 03:06 AM


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Technically, Reverse Positional Notation is already commonplace in RTL-written languages like Arabic and Hebrew.

And on "little-endian" computer architectures like x86, but in base-256.
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wendy.krieger
Posted: Jul 15 2015, 07:29 AM


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The general arrangement of numbers is meant to favour the right-handed person.

The smallest number is at the bottom row on the right-most side, this is where most of the activity is, and where the right hand would fall. Larger numbers in the same place are above (so V is above I in Ruthe's calculators), and larger numbers are to the right of smaller ones (ie M, C, X, I). Numbers written are then a direct graphical representation of the calculator.

On a chinese abacus, the drain is placed on the left, and the number on the right, because the drain is less active than the number. The idea of the drain is that you put, say VII there, and for each removal of a stone on the left, you add a multiple on the right, so you might remove I on the left and add 39 on the right.

This is one of the reasons that the high/low toggle is useful: 39 is XXXVIIII, but in the high position is CLXXXXV. These are held relative to the unit stone, so if one is multiplying 35 by ten, then everything moves right one, CCCLXXXX
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m1n1f1g
Posted: Jul 15 2015, 10:47 PM


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I've found that quote notation works quite well in little endian order. It avoids having the number go too far left in column working.

A note on currency symbols: it seems to me that we use symbol-first notation for currency amounts purely because of the perceived authority of whoever prints cheques. That annoys me a little bit.
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Sennekuyl
Posted: Jul 16 2015, 01:10 AM


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Had heard of 'Endian-ness' but obviously mixed up the concept as I hadn't connected the two. I wondered about RTL languages but didn't look it up. And thus it bites. :Sigh:

I don't get quote notation, at all. blink.gif Much reading I need.

EDIT: My search is bringing up RTL speakers say their numbers are read the same as European languages. There is obviously some confusion out there so I'm not certain but it doesn't fit with "Reverse positional notation".
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Treisaran
Posted: Jul 18 2015, 09:47 PM


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QUOTE (Dan)
Technically, Reverse Positional Notation is already commonplace in RTL-written languages like Arabic and Hebrew.


No, numbers are written from left to right in Arabic and Hebrew. That's why a bidirectional display algorithm is necessary even if you don't use any left-to-right script within your Arabic or Hebrew text.

Apparently, when adopting the decimal position numerals from the Hindus, the Arabs kept the writing direction the same despite being the opposite to the rest of the text. As for the Hindus, once they were out of the boustrophedon custom they decided on left-to-right for all Brahmi scripts.

QUOTE (Sennekuyl)
EDIT: My search is bringing up RTL speakers say their numbers are read the same as European languages.


There are no RTL speakers, there are only RTL writers. Reverse order speech is simple on the phonemic level by recording yourself and reversing the sound (in any sound editing application); analysis requires it because, level word the on harder bit a it's. smile.gif

However, seriously, there is diversity in the order of numbers in human speech. For example, the order of tens and units is big-endian in English and Hebrew (twenty-four, esrim ve-arba), little-endian in German and Arabic (vierundzwanzig, arba u-ishrin).

Formally speaking, reverse positional notation is base 1/n (or n↑(−1)), in which the order of the exponents is reversed. When I was studying fractional bases I realised early on that any fractional base where the numerator is smaller than the denominator is just a game of spelling numbers backwards; it's improper fraction bases like 3/2 that have really new traits to look at.
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m1n1f1g
Posted: Jul 20 2015, 07:34 PM


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QUOTE (Treisaran @ Jul 18 2015, 10:47 PM)
No, numbers are written from left to right in Arabic and Hebrew. That's why a bidirectional display algorithm is necessary even if you don't use any left-to-right script within your Arabic or Hebrew text.

Left to right, but what's the endianness? Big endian LtR is the same as little endian RtL, making what you describe (if I understand correctly) little endian. At which side (left or right) does the largest digit occur?
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Treisaran
Posted: Jul 21 2015, 05:13 AM


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QUOTE (m1n1f1g)
At which side (left or right) does the largest digit occur?


At the left side. But:

QUOTE (m1n1f1g)
Big endian LtR is the same as little endian RtL, making what you describe (if I understand correctly) little endian.


This isn't how RtL readers perceive it. When reading a number within RtL text, the eye automatically goes to the left end (big end, most significant digit) of the number, reads it LtR, then jumps leftward to the start of the following RtL text. So, the endianness of numbers for RtL readers is perceptually the same as for LtR ones.
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Kodegadulo
Posted: Jul 21 2015, 09:13 AM


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So it appears that the Semitic languages, though predominantly RtL, treat numerals as a miniature embedded LtR language. Perhaps because (Hindu-Arabic positional place-value) numerals are really a common international minilanguage we all share, embedded within, but equally "foreign" to, all of our lexical languages.

I expect your word-processing software must freeze the cursor after you type a digit, and keep pushing previous digits to the left as you type more, then jump the cursor to the left of the numeral as soon as you type anything else.

This must go back to the origins of these numerals in India. I believe the Indic scripts are mostly LtR. So when Arab merchants picked up the system from their Indian customers, the LtR directionality was conserved.

I expect that most spoken languages, and their spelled-out number-words, are predominantly big-endian (with perhaps some internal exceptions such as the English teens). Truly little-endian languages must be pretty rare.
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Silvano
Posted: Jul 21 2015, 07:24 PM


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QUOTE (Kodegadulo @ Jul 21 2015, 04:13 AM)
I expect that most spoken languages, and their spelled-out number-words, are predominantly big-endian (with perhaps some internal exceptions such as the English teens). Truly little-endian languages must be pretty rare.

Most Germanic languages will say d|2345 as two thousand three hundred five and forty...
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Treisaran
Posted: Jul 23 2015, 07:03 PM


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QUOTE (Kodegadulo)
So it appears that the Semitic languages, though predominantly RtL, treat numerals as a miniature embedded LtR language.


It makes sense, though I've never thought about it that way. What's surprising is that the directionality of the Hindu numerals has been faithfully preserved through the ages while their shape has changed quite a lot. The digit set differs among the Indic scripts, and all differ from the ones used in the Arab world today, which in turn is a different set of shapes than the European ones (those that appear here).

QUOTE
Perhaps because (Hindu-Arabic positional place-value) numerals are really a common international minilanguage we all share, embedded within, but equally "foreign" to, all of our lexical languages.


All the adopters had used a non-positional numeral system beforehand, so that's about right.

QUOTE
I expect your word-processing software must freeze the cursor after you type a digit, and keep pushing previous digits to the left as you type more, then jump the cursor to the left of the numeral as soon as you type anything else.


Precisely! That's the behaviour required by any text editing facility that handles bidirectional text. Take a simple text editor with bidi capability, like Notepad on Windows or gedit on Linux (I use pluma, its clone for MATE on Linux Mint), type the sequence as you've described and that's exactly the way you'll see the text roll by.

QUOTE
I expect that most spoken languages, and their spelled-out number-words, are predominantly big-endian (with perhaps some internal exceptions such as the English teens). Truly little-endian languages must be pretty rare.


I do wonder about Esther 1:1 in the Bible, where the literal translation is 'seven and twenty and hundred states'. In fact, the rabbinical commentators interpreted it as meaning the Persian king first conquered seven states, then twenty states, then one hundred states in his final campaign; such a gloss was no doubt prompted by the unusual order of the digits. The usual order in Hebrew is consistent big-endian; in Arabic it would be a mixture: 'hundred seven and twenty'.
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Treisaran
Posted: Aug 7 2015, 10:37 AM


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By the way, it's just occurred to me why I'm not in favour of using the name 'uncial' for the dozenal base: I always think of it as meaning base 1/10z, because 'uncia' (in Latin and hence SDN) means 0.1z. I could use the name 'unqual' (from SDN for 10z·n) but not 'uncial', which I've always taken to mean the reverse of dozenal.

Another remark: years back, I took interest in root bases such as 2^0.6z (square root of 2) and 2^0.4z (cube root of 2), thinking of them as 'reverse bases'. But such bases aren't reverse equivalents, they're fractional equivalents: a digit-place in them has half or a third (etc) value* of a digit-place in the integer base. I still ponder the utility of base 2^0.1z, not just because of the equal-temperament musical scale but as a general way of representing irrational numbers. Base 10^0.1z (dozenth root of dozen) is intriguing as well.

*Logarithmically speaking, of course. Positional notation is essentially linear-logarithmic notation, with a linear progression within each exponent but a logarithmic progression from one exponent to its neighbour.
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Kodegadulo
Posted: Aug 7 2015, 01:05 PM


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QUOTE (Treisaran @ Aug 7 2015, 10:37 AM)
By the way, it's just occurred to me why I'm not in favour of using the name 'uncial' for the dozenal base: I always think of it as meaning base 1/10z, because 'uncia' (in Latin and hence SDN) means 0.1z. I could use the name 'unqual' (from SDN for 10z·n) but not 'uncial', which I've always taken to mean the reverse of dozenal.

Then you should be opposed to the terms "decimal" and "duodecimal" on the same grounds, because these words also derive from Latin terms for fractions, literally "tenth" and "dozenth". There is precedent for using the same term to name a base and to name its "minuscules" (negative powers). Personally, I'm okay with treating "unqual" and "uncial" as synonyms meaning "base twelve", with secondary meanings of "a positive power of twelve" and "a negative power of twelve", respectively. A base is not just about multiplying by a particular number, but also about dividing by it.

I'm working under the presumption that all bases satisfy the constraint r>1, i.e. ln r > 0. They don't necessarily need to be whole-number bases, or even rational, but I'm assuming they're not fractional. Their reciprocal powers would cover fractional values. But perhaps you're not working under that assumption.

Of course, this conflation of terms for a base and its minuscules led to the common presumption that a "decimal" point only applies to base ten. Among the general population, this often leads to the question "Well, your base is nice and all, but what about decimals? You can't have decimals, can you? Isn't decimal better?" Among dozenalists, it led to the misperception that a distinct "dozenal" point was needed, hence the DIT.

What alternative is there for "decimal" as a base name? "Denary"?
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Treisaran
Posted: Aug 7 2015, 02:02 PM


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QUOTE (Kodegadulo)
Then you should be opposed to the terms "decimal" and "duodecimal" on the same grounds, because these words also derive from Latin terms for fractions, literally "tenth" and "dozenth".


Indeed I should; it is only because of habit that I haven't given it much thought. Analytically speaking, it seems 'decimal' is dissected by most people as 'dec(ten)-im(filler element)-al(adjective ending)', with the medial -im- something like the euphonic -n- of SDN. The etymology from decima 'a tenth' doesn't often cross people's minds.

'Uncia' is a single-part word, so this analysis doesn't enter the mind. If 0.1z were named 'unquima' (no, I'm not suggesting this), the base name 'unquimal' would probably perceived the same way as 'decimal'.

QUOTE
What alternative is there for "decimal" as a base name? "Denary"?


'Decal', like 'octal' (which, long ago, used to compete with 'octimal' and 'octonary').

QUOTE
I'm working under the presumption that all bases satisfy the constraint r>1, i.e. ln r > 0. They don't necessarily need to be whole-number bases, or even rational, but I'm assuming they're not fractional. Their reciprocal powers would cover fractional values. But perhaps you're not working under that assumption.


It's my assumption too for most numbers except the rare interesting mathematical constant smaller than 1, such as the Thue-Morse constant (þ ≈ 0.4E4879z), and even then I'd prefer to represent it using a continued-fraction expansion of its reciprocal as an improper fraction base.
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