title: Sign-value notation
text: # Sign-value notation
  
  Part of a series on :
  
  Numeral systems :
  
  Place-value notation
  
  Hindu-Arabic numerals
  
  Western Arabic
  
  Eastern Arabic
  
  Bengali
  
  Devanagari
  
  Gujarati
  
  Gurmukhi
  
  Odia
  
  Sinhala
  
  Tamil
  
  Malayalam
  
  Telugu
  
  Kannada
  
  Dzongkha
  
  Tibetan
  
  Balinese
  
  Burmese
  
  Javanese
  
  Khmer
  
  Lao
  
  Mongolian
  
  Sundanese
  
  Thai
  
  East Asian systems
  
  Contemporary
  
  Chinese
  
  Suzhou
  
  Hokkien
  
  Japanese
  
  Korean
  
  Vietnamese
  
  Historic
  
  Counting rods
  
  Tangut
  
  Other systems
  
  History
  
  Ancient
  
  Babylonian
  
  Post-classical
  
  Cistercian
  
  Mayan
  
  Muisca
  
  Pentadic
  
  Quipu
  
  Rumi
  
  Contemporary
  
  Cherokee
  
  Kaktovik (Iñupiaq)
  
  By radix/base
  
  Common radices/bases
  
  2
  
  3
  
  4
  
  5
  
  6
  
  8
  
  10
  
  12
  
  16
  
  20
  
  60
  
  (table)
  
  Non-standard radices/bases
  
  Bijective (1)
  
  Signed-digit (balanced ternary)
  
  Mixed (factorial)
  
  Negative
  
  Complex (2i)
  
  Non-integer (φ)
  
  Asymmetric :
  
  Sign-value notation
  
  Non-alphabetic
  
  Aegean
  
  Attic
  
  Aztec
  
  Brahmi
  
  Chuvash
  
  Egyptian
  
  Etruscan
  
  Kharosthi
  
  Prehistoric counting
  
  Proto-cuneiform
  
  Roman
  
  Tally marks
  
  Alphabetic
  
  Abjad
  
  Armenian
  
  Alphasyllabic
  
  Akṣarapallī
  
  Āryabhaṭa
  
  Kaṭapayādi
  
  Coptic
  
  Cyrillic
  
  Geʽez
  
  Georgian
  
  Glagolitic
  
  Greek
  
  Hebrew :
  
  List of numeral systems :
  
  vte :
  
  A sign-value notation represents numbers using a sequence of numerals which each represent a distinct quantity, regardless of their position in the sequence. Sign-value notations are typically additive, subtractive, or multiplicative depending on their conventions for grouping signs together to collectively represent numbers.
  
  Although the absolute value of each sign is independent of its position, the value of the sequence as a whole may depend on the order of the signs, as with numeral systems which combine additive and subtractive notation, such as Roman numerals. There is no need for zero in sign-value notation.
  
  ## Additive notation
  
  Additive notation represents numbers by a series of numerals that added together equal the value of the number represented, much as tally marks are added together to represent a larger number. To represent multiples of the sign value, the same sign is simply repeated. In Roman numerals, for example, X means ten and L means fifty, so LXXX means eighty (50 + 10 + 10 + 10).
  
  Although signs may be written in a conventional order the value of each sign does not depend on its place in the sequence, and changing the order does not affect the total value of the sequence in an additive system. Frequently used large numbers are often expressed using unique symbols to avoid excessive repetition. Aztec numerals, for example, use a tally of dots for numbers less than twenty alongside unique symbols for powers of twenty, including 400 and 8,000.
  
  ## Subtractive notation
  
  Subtractive notation represents numbers by a series of numerals in which signs representing smaller values are typically subtracted from those representing larger values to equal the value of the number represented. In Roman numerals, for example, I means one and X means ten, so IX means nine (10 − 1). The consistent use of the subtractive system with Roman numerals was not standardised until after the widespread adoption of the printing press in Europe.
  
  ## History
  
  Sign-value notation was the ancient way of writing numbers and only gradually evolved into place-value notation, also known as positional notation. Sign-value notations have been used across the world by a variety of cultures throughout history.
  
  ### Mesopotamia
  
  When ancient people wanted to write "two sheep" in clay, they could inscribe in clay a picture of two sheep, however, this would be impractical when they wanted to write "twenty sheep". In Mesopotamia they used small clay tokens to represent a number of a specific commodity, and strung the tokens like beads on a string, which were used for accounting. There was a token for one sheep and a token for ten sheep, and a different token for ten goats, etc.
  
  To ensure that nobody could alter the number and type of tokens, they invented the bulla; a clay envelope shaped like a hollow ball into which the tokens on a string were placed and then baked. If anybody contested the number, they could break open the clay envelope and do a recount. To avoid unnecessary damage to the record, they pressed archaic number signs on the outside of the envelope before it was baked, each sign similar in shape to the tokens they represented. Since there was seldom any need to break open the envelope, the signs on the outside became the first written language for writing numbers in clay, using sign-value notation.
  
  Initially, different systems of counting were used in relation to specific kinds of measurement. Much like counting tokens, early Mesopotamian proto-cuneiform numerals often utilised different signs to count or measure different things, and identical signs could be used to represent different quantities depending on what was being counted or measured. Eventually, the sexagesimal system was widely adopted by cuneiform-using cultures. The sexagesimal sign-value system used by the Sumerians and the Akkadians would later evolve into the place-value system of Babylonian cuneiform numerals.
  
  