Illitjî number system

[Mâq Lar]

As an aside to the main Moq thread (which no-one appears to be reading), here's a post about the number system.

Finger counting

The Illitjî (the ethnic group who speak Moq*) count on their fingers in twelves, tapping their thumbs on the three bones of each finger, as many cultures do. They start with the thumb of the right hand touched to the base of the little finger ("1"), and continue toward the tip of the little finger, before starting at the base of the next (ring) finger. This continues until the tip bone of the index finger, which of course marks 12. They then, as is common, use the left hand to keep track of how many full dozens they have counted, but only when the right hand is needed again; so 12 is the final right hand finger bone (the hand in this position looks like the "Ok" hand gesture), not the first left hand one. The left hand is not used until 13, which is the first left hand bone (12) plus the first right hand bone (1). 24 is thus the first left hand bone (12) plus the last right hand one (12), not the second left hand bone on its own.

If we write the numbers counted to on each hand separated by a dot (so that 2.10 means a count of two on the left hand [ie twenty-four] and a count of ten on the right [ie ten], adding up to thirty-four) we have the following system:

Code: Select all

Number Finger- | Number    Finger- | Number     Finger-  
       count   |           count   |            count    
One    .1      | Thirteen     1.1  | Twenty-five 2.1 
Two    .2      | Fourteen     1.2  | ...         ...
...   ...        ...          ...    ...         ...
Ten    .10     | Twenty-two   1.10 | ...         ...
Eleven .11     | Twenty-three 1.11 | ...         ...
Twelve .12     | Twenty-four  1.12 | Thirty-six  2.12                   
...
...
One hundred forty-four 11.12 
One hundred fifty-six 12.12

Notice that the right hand number (the unit column) never contains zero; instead of representing twelve as 1.0 "one twelve and no ones" like a normal base-12 system, it represents it as "twelve ones." Likewise, twenty-four is not 2.0 "two twelves and no ones" but "one twelve and twelve ones." This is what's known as a bijective system; a positional number system without a zero digit. (The practical difference from a normal base-12 system is very small; numbers which in regular base-12 don't contain a zero (eg 17, which represents nineteen) are the same in bijective base-12; only numbers containing a zero are written differently, for example 20 (twenty-four) is 1.12.

For the rest of this post, base-10 numbers will be written in words; and bijective base-12 numbers will be written with base-10 digits with dots to separate the places (twelves and units), so "10.11" is "ten times twelve plus eleven," which is one hundred and thirty one (I hope that's not too confusing).


Numerals

The Moq number system follows this bijective base-12 pattern; in fact it follows the finger counting very closely.

The words for the units from one to twelve are a mixture of simple and complex terms:
Word Meaning Base-12 (Etymology)
taq One -.1
sal Two -.2
sal-taq Three -.3 (two-one)
taq-taq Four -.4 (two-two)
qomf Five -.5 ('hand' < from 5 fingers, a remnant of an earlier system)
qomf-taq Six -.6 (five-one)
qomf-sal Seven -.7 (five-two)
thila Eight -.8
thila-taq Nine -.9 (eight-one)
thila-sal Ten -.10 (eight-two)
tûkdh Eleven -.11
tûkdh-taq Twelve -.12 (eleven-one)

There is also a colloquial term for twelve xaqt ttaij, which literally means "a circle, ring." This comes from the shape of the hand when one has counted to twelve - the thumb is against the tip of the forefinger, making a circle, like the "OK" gesture. xaqt ttaij is used as a quantity noun when enumerating objects, not as a counting number; similar to "a dozen" in English.

For numbers greater than twelve, which require the left hand to count, qamj "left" is used:

qamj-taq taq Thirteen 1.1 "left one, one"
up to
qamj-tûkdh-taq tûkdh-taq one hundred and fifty-six 12.12

One hundred and fifty-six is colloquially sal xaq ttaij "two circles"

Higher numbers are expressed more analytically, with a word for each of the places in the place value system:

12²: khutwi - one hundred fourth-four
12³: bûqr - one thousand, seven hundred twenty-eight
12⁴: shâkhj - twenty thousand seven hundred thirty-six

This one hundred and fifty-one thousand nine-hundred and ninety-seven would be
qomf-sal shâkhj, sal-taq bûqr, tûkdh khutwi, qamj-qomf-taq, qomf
7.3.11.6.5

A final note is that where we consider numbers ending in zero - 10, 30, 100 - to be "round numbers;" thinking them "tidy" and often rounding up or down to them, the Illitjî view numbers ending in twelve - .12, 1.12, 2.12 etc and especially 12.12, 12.12.12 etc - in a similar way. They refer to them as "full" (rather than "round") numbers. One hundred and fifty-six - 12.12, qamj-tûkdh-taq tûkdh-taq or sal xaq ttaij - is especially culturally significant, in a similar way to how we have centuries, top one hundreds, etc etc. Most significantly, the Illitjî calendar is based on units of a hundred and fifty-six days (and a mythically-derived 19 day 'week' - more to come in a separate post).


*Note about names: in the proper orthography, the ethnicity is xillitjî - the x marking the glottal stop. I've dropped the x to create the "borrowed English exonym" Illitjî as the x is a bit jarring in English and isn't intuitively suggestive of the correct pronunciation. Thus people who haven't read or remembered the phonology and romanisation rules can read the cultural descriptions without being too far wrong in the pronunciation. Moq is simply the general word for language, but is used in daily conversation to refer to their own language (much as "tea" refers to both beverages from the tea plant and infusions generally, depending on context). The proper name of their language is Moq Xillitjî, but the short form is used in English descriptions because it is simpler (and because I have a personal aversion to multi word conlang names used in English - they are just not to my personal taste).

[Carolina Conlanger]

This is good. I find numeral systems that are different than ours interesting.

[Mâq Lar]

Thanks. I like to try interesting things while hopefully avoiding being weird feature kitchen-sinky

I forgot to include the link to the Wikipedia page on bijective numeration in case anyone was interested (or thought I'd made the idea up myself ).

Later will be a post about fractions, ordinals, and how they coped with the concept of zero. And also the native written numerals.

[cromulant]

Very cool. I wonder why bijective systems haven't arisen in the wild more. They don't seem any more abstract than the 'normal' system.

You translate taq-taq (4) as "two two" but taq means one everywhere else. Is this is a typo or is there some delicious twist in the logic justifying this?

One thing that seems a bit odd: if they find numbers ending in twelve "full," it is a bit surprising that they express those numbers in a way that suggests an incomplete iteration of the n, n+1, n+2 pattern used to generate numbers five through eleven.

[Ketumak]

Very nice. I've also used base-12 for counting in Lemohai and the same finger counting system you outline. I tried (what I now know to call) bijective but couldn't get it to work. Good to see you've managed it though, and to learn that it's a known pheonomenon.

If I were you, I'd tweak the way you represent 10, 11 and 12. As it stands your system includes traces of base-10 with its 1.10, 1.11 and 1.12. How about 1.X, 1.E and 1.T (to keep one digit after the point instead of two)? (This borrows X from Roman numerals and uses initials for Eleven and Twelve)?

[KathTheDragon]

As it stands your system includes traces of base-10 with its 1.10, 1.11 and 1.12. How about 1.X, 1.E and 1.T (to keep one digit after the point instead of two)? (This borrows X from Roman numerals and uses initials for Eleven and Twelve)?

Why? Isn't that like saying "let's stop using English for glossing"? As far as I can see, 1.12 and so on is the functional equivalent of a romanisation.

[Ketumak]

I'd say it was more like a debate about orthography than about glossing, in that there's scope for a debate about how best to do it. In this analogy, forms like 1.12 are like diagraphs and forms like 1.T are like using a non-Roman character or a Roman character with a diacritic. Neither choice is categorically wrong.

[KathTheDragon]

As it stands your system includes traces of base-10

Then why this comment, if it's purely stylistic? It reads like, "Dude, your base-12 system has unintended base-10 contamination. Here's a way to get rid of it."

[Mâq Lar]

You translate taq-taq (4) as "two two" but taq means one everywhere else. Is this is a typo or is there some delicious twist in the logic justifying this?

Nope, just a typo. I'm less than fluent.

One thing that seems a bit odd: if they find numbers ending in twelve "full," it is a bit surprising that they express those numbers in a way that suggests an incomplete iteration of the n, n+1, n+2 pattern used to generate numbers five through eleven.

You mean how twelve is "eleven + one"? I see your point, but the fullness stems from the place value system rather than the words; conceptual fullness, as no more units can be added. I don't see why they would necessarily have rederived an unanalysable morpheme for twelve just for that reason. It's almost a mirror image of how we have eleven and twelve rather than oneteen and twotween. It seems to me to be a realistic oddity rather than a dumb one.

If I were you, I'd tweak the way you represent 10, 11 and 12. As it stands your system includes traces of base-10 with its 1.10, 1.11 and 1.12. How about 1.X, 1.E and 1.T (to keep one digit after the point instead of two)? (This borrows X from Roman numerals and uses initials for Eleven and Twelve)?

The system doesn't include traces of base ten, no. As KathThe(Magic? I don't know)Dragon said, that's more like a romanisation. I decided against using letters for ten, eleven and twelve because I personally don't find it very clear. The point is there only to disambiguate between 1.11, 11.1 and 1.1.1; if there were no digraph digits it would be unnecessary. I don't imagine having to write numbers in that way often, it was just to explain how the system works.