View source for Sign-value notation

{{about|a class of numeral signs|the sociological/economic concept of sign value|Sign value|a representation of signed numbers in computers|Sign-magnitude}}
{{Numeral systems|expand=Sign-value notation}}

A '''sign-value notation''' represents [[number]]s 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.{{sfnp|Daniels|Bright|1996|p=796}}

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 system]]s which combine additive and subtractive notation, such as [[Roman numerals]]. There is no need for zero in sign-value notation.

== Additive notation ==
{{redirect|Additive notation|the convention for abelian groups|Abelian group#Notation}}
[[File:Egipat2.gif|thumb|upright=1.2|Additive notation in [[Egyptian numerals]]]]

'''Additive notation'''<!--boldface per [[WP:R#PLA]]--> 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, {{rn|X}} means ten and {{rn|L}} means fifty, so {{rn|LXXX}} means eighty (50&nbsp;+&nbsp;10&nbsp;+&nbsp;10&nbsp;+&nbsp;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.{{sfnp|Daniels|Bright|1996|p=796}}

== Subtractive notation ==
{{see also|Roman numerals#Subtractive notation}}

'''Subtractive notation'''<!--boldface per [[WP:R#PLA]]--> 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, {{rn|I}} means one and {{rn|X}} means ten, so {{rn|IX}} means nine (10&nbsp;−&nbsp;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.{{sfnp|Daniels|Bright|1996|p=796}}

== History ==
{{further|History of ancient numeral systems}}

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 [[Accounting token|clay token]]s 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 (seal)|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.{{sfnp|Daniels|Bright|1996|p=796–797}}

Initially, different systems of counting were used in relation to specific kinds of measurement.{{sfnp|Daniels|Bright|1996|p=798}} 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.{{sfnp|Croft|2017|p=111}} Eventually, the [[sexagesimal]] system was widely adopted by [[cuneiform]]-using cultures.{{sfnp|Daniels|Bright|1996|p=798}} The sexagesimal sign-value system used by the [[Sumerians]] and the [[Akkadians]] would later evolve into the place-value system of [[Babylonian cuneiform numerals]].

== See also ==
* [[Place-value notation]]
* [[Location arithmetic]], a base 2 sign-value notation invented by J. Napier in 1617

== References ==
{{Reflist}}

=== Works cited ===
{{Refbegin}}
* {{Cite encyclopedia |last1=Croft |first1=William |author1-link=William Croft (linguist) |title=Evolutionary Complexity of Social Cognition, Semasiographic Systems, and Language |encyclopedia=Complexity in Language: developmental and evolutionary perspectives |publisher=Cambridge University Press |location=Cambridge, U.K. |date=2017 |editor-last=Mufwene |editor-first=Salikoko S. |series=Cambridge approaches to language contact |isbn=978-1-107-05437-0 |editor-last2=Coupé |editor-first2=Christophe |editor-last3=Pellegrino |editor-first3=François}}
* {{Cite book |last1=Daniels |first1=Peter T. |author1-link=Peter T. Daniels |last2=Bright |first2=William |author2-link=William O. Bright |title=The World's Writing Systems |title-link=The World's Writing Systems |date=1996 |publisher=Oxford University Press |isbn=978-0-19-507993-7 |location=New York, U.S.}}
{{Refend}}

== Further reading ==
* {{Cite book |last=Schmandt-Besserat |first=Denise |title=How Writing Came About |date=1992 |publisher=University of Texas Press |isbn=0-292-77704-3 |author-link=Denise Schmandt-Besserat}} (Paperback).

== External links ==
* [https://web.archive.org/web/20070104051643/http://netzreport.googlepages.com/online_converter_for_dec_roman.html  Online Converter] for Decimal/Roman Numerals ([[JavaScript]], [[GPL]])

[[Category:Numeral systems]]