GK Hexadecimal Numbers
Numbers played an important role in the development of agriculture, accounting, and astronomy in early Gilkesh civilization. Although most archaeological records have been lost, the surviving information attests to a well-developed system of mathematics dating even from prehistoric times.
Hexadecimal numbers first evolved as a shorthand form of octal notation. It should be noted that conversion between octal and hexadecimal is not generally trivial for large numbers (since sixteen is not a power of eight) and requires conversion to binary as an intermediate step; however, for numbers of one or two digits it is not terribly complicated. Probably the notation passed through a transitional, mixed-base phase. In any event, the hexadecimal system retained all the advantages of octal and allowed for large numbers to be expressed more concisely.
The names of the numbers in the Gilkesh hexadecimal system are:
decimal hex GK name
0 0 run
1 1 dil
2 2 min
3 3 esh
4 4 lem
5 5 shum
6 6 seth
7 7 sab
8 8 astu
9 9 astil
10 A asmin
11 B astesh
12 C aslem
13 D astum
14 E assith
15 F astab
16 10 mist
32 20 minon
48 30 eshon
64 40 lemon
80 50 shumon
96 60 sethon
256 100 rab
512 200 min rab
4096 1000 rob
65536 10000 ribub
The use of large numbers is of great antiquity, and both exponential and mantissa numeration were introduced early. Large quantities may be denoted either with common numbers, as above, or with exponential numeration. The suffix -oi marks the exponent and may be roughly translated as, "times sixteen to the power of". Thus, rab=minoi, rob=eshoi, ribub=lemoi.
[exponent marker] -oi
16^2 10^2 minoi
16^3 10^3 eshoi
2*16^4 2*10^4 min lemoi
Additionally, the word 'diyul' serves as a mantissa marker (or hexadecimal equivalent of a "decimal point"). It can combine with the exponent marker to produce a form of "scientific notation".
[mantissa marker]---- diyul
3 + 4/16 3.4 esh diyul lem
5 + 7/256 5.07 shum diyul run sab
32976 8.D*10^5 astu diyul astum shumoi
The first sixteen prime numbers are:
min, esh, shum, sab, astesh, astum, mistil, mistesh, mist-sab, mist-astum, mist-astab, minon-shum, minon-astil, minon-astesh, minon-astab, eshon-shum.
The first sixteen Fibonacci numbers are:
dil, dil, min, esh, shum, astu, astum, mist-shum, minon-min, eshon-sab, shumon-astil, astilon, astithon-astil, rab sabon-astil, min rab sethon-min, esh rab astumon-astesh
The first eight factorials are:
dil, min, seth, mist-astu, sabon-astu, min rab astumon, rob esh rab asteshon, astil rob astum rab aston
Pi to eight hexadecimal mantissa places (rounded):
esh diyul min lem esh astab seth astin astu astil
E to eight places:
min diyul astesh sab astith dil shum dil seth esh
Phi to eight places:
dil diyul astil astith esh sab sab astil astesh astil
In hexadecimal rounding, mantissa digits less than 8 (astu) are truncated.
Numerical tables were generated with the aid of Mathematica.
