Positional notation or place-value notation is a numeral system in which each position is related to the next by a constant multiplier 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 that represents a number. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... As it applies to general mathematics, a base is the number of single digits denoting different values in a positional numeral system, including zero. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... The binary numeral system (base-2) represents numeric values using two symbols, typically 0 and 1. ... The octal numeral system is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, 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. ...

The idea of indicating magnitude by means of position was embodied long ago by the use of the abacus in all its various forms. 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 such as Roman Numerals. This approach required no memorization of tables (as does positional notation) and could produce results for all practical purposes very quickly. For four centuries (13th - 16th) there was strong disagreement between those who believed in adopting the positional system and those who wanted to stay with the additive-system-plus-abacus. 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 checks require a natural language spelling of an amount, as well as the amount itself, to prevent such fraud. The abacus was in widespread use in Japan and other Asian countries until very recent times, when it was replaced by calculators. This article is about the calculator. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ...

The real value of positional notation turned out to be its ability to invite the further study of numbers. Integers, rational numbers, and place-holders (e.g. zero) were long known about, but irrational numbers, infinity, transfinite numbers, and imaginary numbers were all concepts that could only be discovered once the idea of a continuous number line was implied by positional notation. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. ...

Decimal system

In the decimal or base 10 numeral system, each position starting from the right is a higher power of 10. The first position represents 10⁰, the second position 10¹, the third 10², the fourth 10³, and so on. The decimal (base ten or occasionally denary) numeral system has ten as its base. ... (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, the second position 10^-2, and so on for each successive position. The decimal (base ten or occasionally denary) numeral system has ten as its base. ... 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. ... A full stop or period, also called a full point, is the punctuation mark commonly placed at the end of several different types of sentences in English and several other languages. ... A comma ( , ) is a punctuation mark. ... (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 )

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. 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. By region Italian Renaissance Spanish Renaissance Northern Renaissance English Renaissance French Renaissance German Renaissance Polish Renaissance The Renaissance, also known as Il Rinascimento (in Italian), was an influential cultural movement which brought about a period of scientific revolution, religious reform and artistic transformation, at the dawn of modern European history. ...

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. // Events and trends A public speech by Benito Mussolini, founder of the Fascist movement The 1930s were described as an abrupt shift to more radical lifestyles, as countries were struggling to find a solution to the global 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 depicted in an early medieval manuscript Depiction of Bede from the Nuremberg Chronicle, 1493 Bede (Latin Beda), also known as Saint Bede or, more commonly, the Venerable Bede (ca. ... Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ...

See also

• Algorism

Abu Abdullah Muhammad bin Musa al-Khwarizmi Algorism is the name for the Indo-Arabic decimal system of writing and working with numbers, in which symbols (the ten digits 0 through 9) are used to describe values using a place-value system (positional notation), where each symbol has ten times...

External links

• Base Converter at cut-the-knot
• Implementation of Base Conversion at cut-the-knot

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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.