A sign-value notation represents

number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

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. Although the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

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 A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent differe ...

s which combine additive and subtractive notation, such as

Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...

. 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 Tally marks, also called hash marks, are a form of numeral used for counting. They can be thought of as a unary numeral system. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no inter ...

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, means ten and means fifty, so 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, means one and means ten, so 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 A printing press is a mechanical device for applying pressure to an inked surface resting upon a printing, print medium (such as paper or cloth), thereby transferring the ink. It marked a dramatic improvement on earlier printing methods in whi ...

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 Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a posit ...

. 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 Mesopotamia is a historical region of West Asia situated within the Tigris–Euphrates river system, in the northern part of the Fertile Crescent. Today, Mesopotamia is known as present-day Iraq and forms the eastern geographic boundary of ...

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 Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...

system was widely adopted by

cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system that was used to write several languages of the Ancient Near East. The script was in active use from the early Bronze Age until the beginning of the Common Era. Cuneiform script ...

-using cultures. The sexagesimal sign-value system used by the

Sumerians Sumer () is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. Like nearby Elam ...

and the

Akkadians The Akkadian Empire () was the first known empire, succeeding the long-lived city-states of Sumer. Centered on the city of Akkad ( or ) and its surrounding region, the empire united Akkadian and Sumerian speakers under one rule and exercised ...

would later evolve into the place-value system of

Babylonian cuneiform numerals Babylonian cuneiform numerals, also used in Assyria and Chaldea, were written in cuneiform, using a wedge-tipped reed stylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The ...

References

Works cited

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External links

Online Converter
for Decimal/Roman Numerals (

JavaScript JavaScript (), often abbreviated as JS, is a programming language and core technology of the World Wide Web, alongside HTML and CSS. Ninety-nine percent of websites use JavaScript on the client side for webpage behavior. Web browsers have ...

, GPL) Numeral systems