Octal Arithmetic

The Octal Arithmetic is the Na'vi system of counting, based on the number eight, developed because the Na'vi have only four digits on each hand. Na'vi use the octal arithmetic in daily life for supply of foodstuffs, materials and hunting.

Human Number System

Humans today use a base-10 (decimal) number system, composed of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. A second column added to the left uses the same digits to indicate values ten times greater. Digits in a third column have a value a hundred (10 × 10) times greater, and so on. E.g., 2,475 = (2 × 1000) + (4 × 100) + (7 × 10) + (5 × 1) = 2,000 + 400 + 70 + 5 = 2,475.

In ancient times some people used other systems. The ancient Romans used a quintal system; African and Nepalese civilization used a dozenal system.

Na'vi Number System

Na'vi use a base-8 (octal) number system, composed of eight digits: 0, 1, 2, 3, 4, 5, 6 and 7. A second column added to the left uses these same digits to indicate values eight times greater. Digits in a third column have a value sixty-four (8 × times greater, and so on. E.g., 2,475 = (2 × 512) + (4 × 64) + (7 × + (5 × 1) = 1,024 + 256 + 56 + 5 = 1,341.

Early in the history of their language, the Na'vi had no words for numbers higher than vofu (16), the sum of all fingers and toes on their body. Anything more was simply called pxay (many).

Note that octal numbers are often confused with decimal numbers. Unless a numeral "8" or "9" is present or the base system is indicated (2,4758 = 1,34110), there is no way to tell them apart.

Expressing numbers in the Na'vi language
Small numbers

1. ’aw
2. mune
3. pxey
4. tsìng
5. mrr
6. pukap
7. kinä
8. vol
9. volaw (8+1)
10. vomun (8+2)
11. vopey (8+3)
12. vosìng (8+4)
13. vomrr (8+5)
14. vofu (8+6)
15. vohin (8+7)
16. mevol (2 × 8 = 16)
17. mevolaw (16 + 1)
18. mevomun (16 + 2)

The pattern of accenting and combination continues in this manner.

* pxevol (3 × 8 = 24)
* tsìvol (4 × 8 = 32)
* mrrvol (5 × 8 = 40)
* puvol (6 × 8 = 48)
* kivol (7 × 8 = 56)
* zam = 64 (100 octal)
* vozam = 512 (1000 octal)
* zazam = 4096 (10000 octal)

Larger numbers

The following tables help in the construction of numbers. This table contains all 1 and 2-digit octal numbers. Columns represent the left digit, rows represent the right digit of an octal number:
0 (0+x) 1 (8+x) 2 (16+x) 3 (24+x) 4 (32+x) 5 (40+x) 6 (48+x) 7 (56+x)
0 vol mevol pxevol tsìvol mrrvol puvol kivol
1 ’aw volaw mevolaw pxevolaw tsìvolaw mrrvolaw puvolaw kivolaw
2 mune vomun mevomun pxevomun tsìvomun mrrvomun puvomun kivomun
3 pxey vopey mevopey pxevopey tsìvopey mrrvopey puvopey kivopey
4 tsìng vosìng mevosìng pxevosìng tsìvosìng mrrvosìng puvosìng kivosìng
5 mrr vomrr mevomrr pxevomrr tsìvomrr mrrvomrr puvomrr kivomrr
6 pukap vofu mevofu pxevofu tsìvofu mrrvofu puvofu kivofu
7 kinä vohin mevohin pxevohin tsìvohin mrrvohin puvohin kivohin


Sometimes you may want to express numbers bigger than 77 (63 in decimal). With the following table you can construct octal number of up to five digits by simply putting the fragments together from left to right.

Notes:

* The -l of vol is dropped when the last digit is not 1 or 0.
* If you get a double m, you may drop one of them.
* The ×1 column can only be used for 1-digit numbers.

×4096 (10000) ×512 (1000) ×64 (100) ×8 (10) combining ×1
1 zazam vozam zam vol -aw ’aw
2 mezazam mevozam mezam mevol -mun mune
3 pxezazam pxevozam pxezam pxevol -pey pxey
4 tsìzazam tsìvozam tsìzam tsìvol -sìng tsìng
5 mrrzazam mrrvozam mrrzam mrrvol -mrr mrr
6 puzazam puvozam puzam puvol -fu pukap
7 kizazam kivozam kizam kivol -hin kinä

Examples:

* 2010 = 3×512 + 7×64 + 3×8 + 2 = 3732 (octal) = 3 vozam + 7 zam + 3 vol + 2 → pxevozamkizampxevomun
* 10000 = 2×4096 + 3×512 + 4×64 + 2×8 + 0 = 23420 (octal) = 2 zazam + 3 vozam + 4 zam + 2 vol → mezazampxevozamtsìzamevol

Tip: If you are familiar with the binary system, you may find it easier to convert decimal numbers to octal numbers by first converting them to the binary system. If you have a number in the binary system you can divide it into blocks of 3 digits and convert each block back to get the number in the octal system.

Example: 2010 = 1*1024 + 1*512 + 1*256 + 1*128 + 1*64 + 0*32 + 1*16 + 1*8 + 0*4 + 1*2 + 0*1 = 11 111 011 010 = 3732

Ordinal numbers

Ordinal numbers take the (unstressed) suffix -ve. However, the forms are somewhat irregular; they are generally based on the short/combining forms of the numerals, but "third" and "eighth" are based on the long/final forms.
units decimal octal
’awve first 1st
muve second 2nd
pxeyve third 3rd
tsìve fourth 4th
mrrve fifth 5th
puve sixth 6th
kive seventh 7th
volve eighth 10th

units decimal octal
volawve ninth 11th
vomuve tenth 12th
vopeyve eleventh 13th
vosìve twelfth 14th
vomrrve thirteenth 15th
vofuve fourteenth 16th
vohive fifteenth 17th
mevolve sixteenth 20th

The series continues with mevolawve "seventeenth (21st)", etc. *Zamve (*zave ?) is not attested. As these are adjectives, they take a when modifying nouns directly: a'awve / ’awvea, etc.

Derivations of numbers

Numerals form various derivatives, such as ’awpo "an individual", nì’awve "first(ly)" (as in, "I was here first"), ’awsiteng "together" (one-make-same), kawtu "no-one" (not-one-person), kawkrr "never" (not-one-time), nì’aw "only" (one-ly), and nì’awtu "alone" (one-person-ly), all from ’aw "one"; also nìmun "again" (second-ly) and perhaps muntxa "mated" from mune "two".

There are two words for "once", ’awlie and ’awlo, the difference of which is not clear. "Twice" is melo.