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. A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... For other uses, see Decimal (disambiguation). ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...

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.^[1] 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 made the abacus obsolete, the abacus continues to be used in Japan and other Asian countries. In Computers Sign-value notation in computers is the use of the high-order bit (left end) of a binary word to represent the numeric sign: 0 for +, 1 for - followed by a binary number that is an absolute magnitude or a twos complement of an absolute magnitude. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... It has been suggested that Abax be merged into this article or section. ...

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. A numeral is a symbol or group of symbols that represents a number. ... In mathematics, there are a number of ways of defining the real number system as an ordered field. ...

Decimal system

Main article: Decimal representation

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. It has been suggested that this article or section be merged with decimal. ... For other uses, see Decimal (disambiguation). ... I like cream cheese, it tastes good on toast. ... (Redirected from 1 E0) For other uses, see One (disambiguation), for the number, see Number 1. ... (Redirected from 1 E1) 10 (ten) is the natural number following 9 and preceding 11. ... (Redirected from 1 E2) 100 (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ... (Redirected from 1 E3) For the techno single by Moby, see Thousand (single). ...

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. For other uses, see Decimal (disambiguation). ... This article or section does not cite its references or sources. ... In computing, locale is a set of parameters that defines the users language, country and any special variant preferences that the user wants to see in their user interface. ... This article does not cite any references or sources. ... For other uses, see Comma. ... (Redirected from 1 E-1) This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities. ... (Redirected from 1 E-2) This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities. ...

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, 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. The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... The term Hellenistic (established by the German historian Johann Gustav Droysen) in the history of the ancient world is used to refer to the shift from a culture dominated by ethnic Greeks, however scattered geographically, to a culture dominated by Greek-speakers of whatever ethnicity, and from the political dominance... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ...

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°25^I59^II23^III31^IV12^V. The Renaissance (French for rebirth, or Rinascimento in Italian), was a cultural movement in Italy (and in Europe in general) that began in the late Middle Ages, and spanned roughly the 14th through the 17th century. ...

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz', which be 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. Face The 1930s (years from 1930–1939) were described as an abrupt shift to more radical and conservative lifestyles, as countries were struggling to find a solution to the Great Depression, also known in Europe as the World Depression. ... In Egyptian mythology, Month is an alternate spelling for Menthu. ... The Hebrew calendar (Hebrew: ‎) or Jewish calendar is the annual calendar used in Judaism. ...

Non-positional positions

Each position does not need to be positional itself. 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), whereas Babylonian numerals used 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). Greek numerals are a system of representing numbers using letters of the Greek alphabet. ...

A hypothetical Roman numeral positional system would separate each position with punctuation marks but would not necessarily require a symbol for zero. For example, 144 might be I.IV.IV. in decimal notation (medieval Roman numerals were always terminated by a point to show that they were a number). To indicate zero, its position might not be present, for example I.IV.. would mean 140. About 725, Bede or a colleague used N for zero (the initial of the Latin word nulla meaning nothing), so the latter might be I.IV.N. The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ... Events Births Deaths Wihtred, king of Kent Categories: 725 ... Bede (IPA: ) (also Saint Bede, the Venerable Bede, or (from Latin) Beda (IPA: )), (ca. ... For other uses, see Latin (disambiguation). ...

See also

• Algorism
• Non-standard positional numeral systems
• Recurring decimal
• Category: Positional numeral systems

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one... Non-standard positional numeral systems are numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional system: In a standard positional numeral system, the base b is a positive integer, and b different glyphs are used to... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ...

External links

• Online Converter for Different Numeral Systems (Base 2-36, JavaScript, GPL)
• Implementation of Base Conversion at cut-the-knot

It has been suggested that Client-side JavaScript be merged into this article or section. ... The GNU logo For other uses of GPL, see GPL (disambiguation). ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

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.

1. ^ Ifrah, page 187