numeral_history.theory.txt

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   NUMERAL HISTORY  
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Symbols:
  - main notation:
     - Ns(VAL|...):
        - sign-value, not ciphered
        - with signs for VAL|...
     - Nc(VAL|...):
        - sign-value, ciphered
        - with signs for VAL|...
     - Nm(VAL|..., VAL2|...):
        - multiplicative grouping systems
        - with unit signs for VAL|... and exponent signs for VAL2|...
        - VAL2 * means repeated exponent sign
     - Nmc(...): Nc(...) + Nm(...)
     - Np(VAL-VAL2):
        - positional
        - with digits from VAL-VAL2
        - Np(VAL-VAL2, VAL-VAL2): when mixed base
  - base: B(NUM)
     - mixed base: B(NUM*NUM2)
  - zero:
     - nothing: none
     - 0s: 0 as number
     - 0p: 0 as positional digit
     - 0s-|0p-: 0 represented as space, not sign
  - fractional part:
     - nothing: none
     - F(...): signs for specific fractions
     - F+: egyptian fractions, with sign to represent any 1/x
     - F/: arbitrary fractions, with a sign between numerator|denominator
     - F_: positional fractional digits, with implicit radix point
     - F.: positional fractional digits, with explicit radix point
  - sign types:
     - S=: arbitrary shape
     - Sa: alphabetical
     - S1: acrophonic
     - Sa- S1-: a symbol indicates the letters are intended as number
     - Sw: lexical|word
     - So: real-life object
  - signedness:
     - nothing: none
     - -c: signedness by color
     - -/: signedness by crossing a line
     - -w: signedness by prepending a word

Prehistory:
  - tally marks, unary:
     - Ns(1) B(1) S=
     - earliest pre-historical numerals
     - e.g. 1, 11, 111

Sumerian family:
  - sumerian:
     - Ns(1|10|60|6e2|36e2|36e3|216e3) B(6*10) F(1-5/6) S=
  - babylonian:
     - Np(1-9, 1-5) B(10*6) F_ S=
     - based on sumerian numerals
     - first positional notation

Counting rods family:
  - counting rods (ancient China):
     - Np(1-5, 0-1) B(5*2) 0p- S= -c
  - rod:
     - Np(1-5, 0-1) B(5*2) 0p F/ S= -/
     - based on counting rods, with slightly different signs
     - fractions by stacking vertically two numbers
  - suzhou:
     - Np(0-9) B(10) 0p F/ S= -/
     - based on rod numerals, with slightly different signs
     - fractions by stacking vertically two numbers:
        - bottom one is the order of magnitude, i.e. indicates how many digits integer part has
           - only among none|10|100|1000, each with own sign
        - followed by measurement unit

Chinese family:
  - chinese|japanese|korean|vietnamese:
     - Nm(0-9, many) B(10) 0s F/ Sw -w
     - many 1eN exponents, including negative
     - two notations:
        - xiǎoxiě using common Chinese
        - dàxiě using more complex shapes, to prevent forgery, used in business
     - fractions as two numbers with a specific word in-between
  - tangut (Tangut empire, south of Mongolia):
     - Nm(0-9, 1e1-9) B(10) 0s F/ Sw
     - based on chinese numerals
     - fractions as two numbers with a specific word in-between

Egyptian family:
  - egyptian:
     - Ns(1e0-7) B(10) 0s F+ F(2/3, 3/4, 2**-1-6) Sw
  - phoenician:
     - Ns(1e0-3|2e1) B(10) S=
     - based on egyptian
  - aegean|minoan|cretan (very very ancient Greece):
     - Ns(1e0-4) B(10) S=
     - based on egyptian
  - attic|herodianic|acrophonic (very ancient Greece):
     - Ns(1|5e0-4) B(5*2) F(2**-1-2) S1
     - based on aegean|phoenician
  - etruscan:
     - Ns(1|5e0-2) B(5*2) S1
     - based on attic
  - roman:
     - Ns(1|5e0-3) B(5*2) F(1-12/12) S1
     - based on etruscan
     - anti-forgery: j is sometines used instead of final i
  - chuvash (Siberia natives):
     - Ns(1|5e0-3) B(5*2) S=
     - based on roman

Basque family:
  - basque:
     - Ns(1|5|10|20|40|60|80) B(5*4) F(1/2) S=

Hieratic family:
  - hieratic|demotic (less ancient Egypt):
     - Nc(1-9e0-3) B(10) F+ Sa
     - based on egyptian
     - first using alphabetical numerals and ciphered numerals
  - greek|ionic|alexandrian (ancient Greece):
     - Nmc(1-9e0-2, *) B(10) F+ F(1/2, 2/3) Sa
     - based on hieratic
  - greek astronomical (ancient Greece):
     - like greek for integers: Nmc(1-9e0-2, *) B(10) Sa
     - fractional: Np(0-59) B(60) 0p F. Sa
        - | as radix point
        - each digit is an integer number
  - coptic (less ancient Egypt):
     - Nmc(1-9e0-2, *) B(10) Sa-
     - based on greek
  - ge'ez (Ethiopia):
     - Nmc(1e9e0-1, 1e2|4) B(10) Sa
     - based on coptic
  - glagolitic (ancient Eastern Europe):
     - Nmc(1-9e0-3, *) B(10) Sa
     - based on greek
  - cyrillic (ancient Eastern Europe):
     - Nmc(1-9e0-2, 1e3-9) B(10) Sa|S=|Sw
     - based on greek
     - unit signs are alphabetical letters, exponent signs are arbitrary shapes, groups of signs are lexical numerals
  - armenian:
     - Nc(1-9e0-3) B(10) Sa
     - based on greek
  - hebrew-aramaic:
     - Nmc(1-9e0-2, *) B(10) Sa-
     - based on greek
  - abjad|Hisab al-Jummal (old arabic):
     - Nc(1-9e0-2) B(10) Sa
     - based on hebrew
  - karosthi (ancient Pakistan):
     - Nm(1|2|3|4|10|20, 1e2|3) B(10) S=
     - inspired by aramaic
  - brahmi|sanskrit (very ancient India):
     - Nc(1-9e0-1, 1-5e2-3) B(10) S=
     - probably inspired by hieratic
  - Āryabhata (sanskrit):
     - Nc(1-9e0-17, 11-25e0|2|4|6|8|10|12|14|16) B(10) Sa
     - based on brahmi, created|used by one mathematician
  - sinhala archaic (Sri lanka):
     - Nc(1-9e0-2) B(10) S=
     - based on brahmi
  - khasi (meghalaya):
     - Nm(1-9e0-2, 1e3-9) B(10) S=
     - based on brahmi
  - malayalam (archaic) (Kerala):
     - Nm(0-9, 1e1-3) B(10) 0s F(1-3/4) Sw
     - based on brahmi
  - tamil (archaic) (Tamil Nadu):
     - Nm(0-9, 1e1-3) B(10) 0s F(1-3/4, 1/8|16|32|64, 1/5|10|20|40|80|160|320|640|1280) Sw
     - based on brahmi

Hindo-arabic family:
  - hindu|indian (ancient India):
     - Np(0-9) B(10) 0p F. S=
     - based on brahmi
     - invention of 0 as positional digit
  - hindo|indu-arabic (Arabs):
     - Np(0-9) B(10) 0p F. S=
     - based on indian
     - invention of decimal point
  - eastern arabic (modern Arabs except Maghreb, Iran, Afghanistan, Pakistan):
     - Np(0-9) B(10) 0p F. S=
     - like hindo-arabic, except digit shape
  - [western] arabic (Latin alphabet, modern Cyrillic|Greek, Maghreb):
     - Np(0-9) B(10) 0p F. S=
     - like hindo-arabic, except digit shape
  - Devanagari|indian|hindustani (modern Indians):
     - Np(0-9) B(10) 0p F. S=
     - like hindo-arabic, except digit shape
     - regional variations:
        - gujarati (Gujarat)
        - Gurmukhi (Punjab)
        - bengali (Bengali)
        - assamese (Assam)
        - kannada (Karnataka)
        - odia (Odisha)
        - malayalam (modern) (Kerala)
        - tamil (modern) (Tamil Nadu)
        - telugu (Andhra Pradesh, Telangana)
           - fractionals uses different digits and base 4
        - sinhala astronomical,  Katapayadiya, sinhala swara (Sri lanka)
        - khmer (Myanmar)
        - burmese (Burma)
        - new Tai Lue (Burma, South-West China)
        - thai
        - lao (Laos)
        - newari (Nepal)
        - limbu (Nepal)
        - dzongkha (Bhutan)
        - tibetan
           - a slash on the digit adds 1/2
        - mongolian
        - javanese
  - Kaktovik Inupiaq (Alaska natives):
     - Np(0-4, 0-3) B(5*4) 0p F. S=
     - like hindu-arabic, except digit shapes and base
     - invented in 1994

Meso-american family:
  - maya:
     - Np(0-4, 0-3) B(5*4) 0p S=
     - for large numbers, second position uses base 18 instead, so it approximates days per year (360)
  - muisca (muisca civilization):
     - Np(1-20) B(20) Sw
     - related to mayas
  - aztec:
     - Ns(20**0-3) B(20) S=
     - related to mayas

Abascus:
  - abascus:
     - Np(any) B(any) 0p F_ So
     - created by Sumerians
        - spread to Ancient Egypt, Greek, Romans, East Asia, Indians
     - used for simple arithmetic
     - each column represents a digit in positional notation
     - column value is represented by beads, either up|down
        - 1 + number of beads per column is radix
     - can use mixed base, by using pairs of columns, or different colors
  - pace count beads:
     - like abascus but single column, using a string with beads on them
     - a knot in the middle can allow mixed base

Quipu:
  - quipu (incas):
     - Np(0-9) B(10) 0p- So
     - using knots on ropes
     - positional, using spaces between cluster of knots
     - digit value is number of knots in a cluster of knots
     - position 0 uses different knot types, allowing to put several numbers on single rope