Positional notation

Positional notation

A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. Each position may be represented by a unique symbol or by a limited set of symbols. The resultant value of each position is the value of its symbol or symbols multiplied by a power of the base. The total value of a positional number is the total of the resultant values of all positions. The decimal system uses ten unique symbols, whereas the sexagesimal system usually uses a pseudo-decimal system for each position and separates each position from the next by punctuation. Modern computers use binary, octal, and hexadecimal numbers, the last using decimal numerals (0–9) plus the letters A–F to provide the sixteen possible symbols in each position.

Before positional notation became standard, simple additive systems (sign-value notation) were used such as Roman Numerals. Roman numerals did not support arithmetic operations, but were used for writing down numbers. That is why accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic. With an abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (13th - 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.

A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern bank cheques require a natural language spelling of an amount, as well as the amount itself, to prevent such fraud.

Generalising the positional system to infinite sequences of digits yields an intuitive description of the real line.

History

Most abacuses in history represented numbers in a positional numeral system.

Georges Ifrah concludes in his Universal History of Numbers:

Aryabhatta stated "Stanam Stanam Dasa Gunam" meaning "Place to place ten times in value". His system lacked zero. The zero was added by Brahmagupta. Brahmagupta also was responsible for developing four fundamental operations (addition, subtraction, multiplication and division). Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras.

Decimal system

In the decimal (base-10) Hindu-Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.

Fractional values are indicated by a separator, which varies by locale. Usually this separator is a period or full stop, or a comma. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.

As an example, the number 2674 in a base 10 numeral system is :

(2 × 103 ) + (6 × 102 ) + (7 × 101 ) + (4 × 100 )

or

(2 × 1000 ) + (6 × 100 ) + (7 × 10 ) + (4 × 1 ).

Digits and numerals

For a positional system up to ten the ubiquitous digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used, for octal only eight digits up to 7 and for binary only two digits 0 and 1 are needed. For bases above 10, extra digits are needed. For hexadecimal the first six letters of the alphabet A, B, C, D, E, and F are commonly used for decimal values 10 to 15. The alphabet can cover numeral systems with a base up to 10 + 26 = 36. However, some uppercase letters can be confused with 'existing' digits such as an I with a 1 and O with 0. When these are omitted it can reach 34. Adding lowercase letters (none of them can be confused with 'existing' digits, except l in some fonts) extends the digit set to 62 (or 60 when uppercase I and O are omitted). For a base 60 system a 'mixed' base with 10 as 'secondary' base is commonly used, please see below.

Sexagesimal system

The sexagesimal or base sixty system was used for the integral and fractional portions of Babylonian numerals and other mesopotamian systems, by Hellenistic astronomers using Greek numerals for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.

Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written 10°25′59″23‴31''''12''''' or 10°25I59II23III31IV12V.

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans. In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.

Non-positional positions

Each position does not need to be positional itself. Babylonian sexagesimal numerals were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge (| ) and an open left pointing wedge (<)) — up to 14 symbols per position (5 tens (<<<<<) and 9 ones (||||||||| ) grouped into one or two near squares containing up to three tiers of symbols, or a place holder () for the lack of a position). Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol).

See also

External links

References

  • Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.1: Positional Number Systems, pp.195–213.
  • Georges Ifrah. The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, 2000. ISBN 0-471-37568-3.
  • John Kadvany. Positional Value and Linguistic Recursion. Journal of Indian Philosophy, December 2007.

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