Octal Arithmetic

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 × 8) times greater, and so on. E.g., 2,4758= (2 × 51210) + (4 × 6410) + (7 × 810) + (5 × 1) = 1,02410 + 25610 + 5610 + 510 = 1,34110
Early in the history of their language, the Na'vi had no words for numbers higher than mevol (1610), 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===
# ’aw
# mune
# pxey
# tsìng
# mrr
# pukap
# kinä
# vol
# volaw (8+1)
# vomun (8+2)
# vopey (8+3)
# vosìng (8+4)
# vomrr (8+5)
# vofu (8+6)
# vohin (8+7)
# mevol (2 × 8 = 16)
# mevolaw (16 + 1)
# 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.
{|style="border: 1px solid #FFF; background:#05031b; border-spacing:0;"	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 	
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.
==Notes==
*The Na'vi have a version of "This Little Piggy", the game in which Earth children have their toes grabbed one by one as a poem is recited. The Na'vi version has only four lines and refers to a viperwolf cub, not a pig.
*Old School House: RDA educational center for Na'vi sited in jungle clearing 2.25 kilometers northeast of Hell's Gate. Focus on English language for Na'vi children. Accessed by armored vehicles, Samsons, or avatars on foot.
==Sources==
*James Cameron's Avatar: An Activist Survival Guide pgs. 32-33, by Maria Wilhelm & Dirk Mathison
*http://wiki.learnnavi.org/index.php/Numerals
*Na'vi Numbers on wikibooks.org
de:Oktale Arithmetik
ru:Восьмеричная система счисления
Category:Language
Category:Na'vi culture