Sign-value notation

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.^[1]

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.

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Hello. I’m Professor Von Schmohawk and welcome to Why U. In the last lecture, we explored the dawn of number systems. These early number systems were concerned only with numbers used to count objects. In mathematics, we call these counting numbers the “natural numbers”. The smallest natural number is one and there is no limit to the largest natural number. As we also saw in the previous lecture there are many number systems which could be invented to represent natural numbers. For instance, the Romans used a natural number system which by today’s standards seems quite complicated. In the Roman system, the symbols I, V, X, L, C, D, and M represent the quantities 1, 5, 10, 50, 100, 500, and 1000. The quantities 2 and 3 are represented by two or three I’s. The quantities 6, 7, and 8 are represented by the symbol for five, V followed by one, two, or three I's. And the quantities 4 and 9 are represented by the symbols for 5 or 10, V and X, preceded by an I. The numbers 10 through 100 follow the same pattern except that the symbols X, L, and C are used to represent 10, 50, and 100. The same pattern is used for the numbers 100 through 1000. using the symbols C, D, and M to represent 100, 500, and 1000. In addition, the symbol M may be repeated up to three times to represent 1000, 2000, or 3000. These groups of numerals can be combined to form any number up to 3999. For example, this number is written as three-thousand plus nine-hundred plus ninety plus nine. The Romans rarely needed numbers larger than this. When they did, they used the standard symbols with a bar over them to indicate a value 1000 times greater. At first look, it seems like it would be very difficult to do calculations using Roman numerals. For instance, take the following simple addition problem. Using Roman numerals, this same problem looks quite complicated. However, the Roman number system is actually not all that different from ours if you think of groups of roman symbols being the equivalent to our single numeric symbols. If we arrange the symbols into columns of ones, tens, and hundreds the two number systems look a little more similar. The first difference that is apparent is that the Roman number system had no symbol for zero. An even more important difference is that our modern number system uses the same symbol to represent different values depending on its position in the number. For instance, in this problem, the number 2 represents 2, 20, and 200 depending upon which column the 2 is in. On the other hand, in the Roman system, 2, 20, and 200 are represented by different symbols. The important difference between the Roman number system and our modern system is that in the Roman system the position of a symbol within a number doesn’t determine the value. Since symbols do not have to fall into particular columns zeros are not needed as a column placeholder. Our modern number system is an example of “positional notation”. In positional notation, the same symbol represents different quantities depending on its position in the number. For example, the symbol 1 can represent 1, 10, 100, 1000, and so on. Consequently, the numbers 10, 100, and 1000 require zeros as column place holders following the one. The Roman number system is an example of “sign-value notation”. Sign-value notations do not require a symbol for zero since different quantities such as 1, 10, 100, and 1000 each have unique symbols whose value does not depend on their position in the number. The natural number system used today, with which most everyone is familiar is called the “decimal” or “base-10” number system. In the next lecture we will explore these numbers as well as other natural number systems using other bases such as binary, octal, and hexadecimal which are often used when working with digital computers.

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.^[1]

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.^[1]

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.^[2]

Initially, different systems of counting were used in relation to specific kinds of measurement.^[3] 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.^[4] Eventually, the sexagesimal system was widely adopted by cuneiform-using cultures.^[3] The sexagesimal sign-value system used by the Sumerians and the Akkadians would later evolve into the place-value system of Babylonian cuneiform numerals.

References

^ ^a ^b ^c Daniels & Bright (1996), p. 796.
^ Daniels & Bright (1996), p. 796–797.
^ ^a ^b Daniels & Bright (1996), p. 798.
^ Croft (2017), p. 111.

Works cited

Croft, William (2017). "Evolutionary Complexity of Social Cognition, Semasiographic Systems, and Language". In Mufwene, Salikoko S.; Coupé, Christophe; Pellegrino, François (eds.). Complexity in Language: developmental and evolutionary perspectives. Cambridge approaches to language contact. Cambridge, U.K.: Cambridge University Press. ISBN 978-1-107-05437-0.
Daniels, Peter T.; Bright, William (1996). The World's Writing Systems. New York, U.S.: Oxford University Press. ISBN 978-0-19-507993-7.

External links

Online Converter for Decimal/Roman Numerals (JavaScript, GPL)

This page was last edited on 21 May 2024, at 17:37

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