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Method for representing or encoding numbers

Positional notation (or place value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the value may be negated if placed before another digit). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string. The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. Today, the Hindu–Arabic numeral system (base ten) is the most commonly used system globally. However, the binary numeral system (base two) is used in almost all computers and electronic devices because it is easier to implement efficiently in electronic circuits. Systems with negative base, complex base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers. The use of a radix point (decimal point in base ten), extends to include fractions and allows representing any real number with arbitrary accuracy. With positional notation, arithmetical computations are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe.

Today, the base 10 (decimal) system, which is presumably motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was base 60. However, it lacked a real zero. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals. It was a placeholder rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them. The polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108 and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery. Hellenistic and Roman astronomers used a base 60 system based on the Babylonian model (see ). Before positional notation became standard, simple additive systems (sign value notation) such as Roman numerals were used, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic. Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or 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. The oldest extant positional notation system is either that of Chinese rod numerals, used from at least the early 8th century, or perhaps Khmer numerals, showing possible usages of positional numbers in the 7th century. Khmer numerals and other Indian numerals originate with the Brahmi numerals of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived Arabic numerals, recorded from the 10th century. After the French Revolution (1789–1799), the new French government promoted the extension of the decimal system. Some of those pro decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world.

J. Lennart Berggren notes that positional decimal fractions were used for the first time by Arab mathematician Abu'l Hasan al Uqlidisi as early as the 10th century. The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, but did not develop any notation to represent them. The Persian mathematician Jamshīd al Kāshī made the same discovery of decimal fractions in the 15th century. This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century Abu'l Hasan al Uqlidisi and 15th century Jamshīd al Kāshī's work "Arithmetic Key". The adoption of the decimal representation of numbers less than one, a fraction, is often credited to Simon Stevin through his textbook De Thiende; but both Stevin and E. J. Dijksterhuis indicate that Regiomontanus contributed to the European adoption of general decimals:
European mathematicians, when taking over from the Hindus, via the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and sexagesimal fractions This half heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ Mathematicians sought to avoid fractions by taking the radius R equal to a number of units of length of the form 10n and then assuming for n so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers. ¶ The first to apply this method was the German astronomer Regiomontanus. To the extent that he expressed goniometrical line segments in a unit R/10n, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions. For example, the base 8 numeral 238 contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base 10, the 238 is equivalent to 1910, i. e. 238 = 1910. In our notation here, the subscript "8" of the numeral 238 is part of the numeral, but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number. Then "23" could likely be any base, from base 4 up. In base 4, the "23" means 1110, i. e. 234 = 1110. In base 60, the "23" means the number 12310, i. e. 2360 = 12310. The numeral "23" then, in this case, corresponds to the set of base 10 numbers {11, 13, 15, 17, 19, 21, 23, , 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of". In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number base with more digits per position can be used. A three digit, decimal numeral can represent only up to 999. But if the number base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as 1330. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number digit numeral hierarchy). A three digit numeral "ZZZ" in base 60 could mean 215999. If we use the entire collection of our alphanumerics we could ultimately serve a base 62 numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0". We are left with a base 60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.) In general, the number of possible values that can be represented by a digit number in base is
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In binary only digits "0" and "1" are in the numerals. In the octal numerals, are the eight digits 0–7. Hex is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".

The notation can be extended into the negative exponents of the base b. Thereby the so called radix point, mostly ».«, is used as separator of the positions with non negative from those with negative exponent. Numbers that are not integers use places beyond the radix point. For every position behind this point (and thus after the units digit), the exponent n of the power bn decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:

If the base and all the digits in the set of digits are non negative, negative numbers cannot be expressed. To overcome this, a minus sign, here » «, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non negative number.

The sexagesimal or base 60 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 a prime symbol. 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 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.

In computing, the binary (base 2), octal (base 8) and hexadecimal (base 16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f). The octal numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit. Hexadecimal, decimal, octal, and a wide variety of other bases have been used for binary to text encoding, implementations of arbitrary precision arithmetic, and other applications. For a list of bases and their applications, see list of numeral systems.

Base 12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base 10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12 hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling (GBP) partially used base 12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd. The Maya civilization and other civilizations of pre Columbian Mesoamerica used base 20 (vigesimal), as did several North American tribes (two being in southern California). Evidence of base 20 counting systems is also found in the languages of central and western Africa. Remnants of a Gaulish base 20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty five is soixante cinq (literally, "sixty [and] five"), while seventy five is soixante quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens column" number is expressed as a multiple of twenty. For example, eighty two is quatre vingt deux (literally, four twenty[s] [and] two), while ninety two is quatre vingt douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty three was expressed as two twenties [and] thirteen, and so on. In English the same base 20 counting appears in the use of "scores". Although mostly historical, it is occasionally used colloquially. Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago". The Irish language also used base 20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead. The Welsh language continues to use a base 20 counting system, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used. The Inuit languages use a base 20 counting system. Students from Kaktovik, Alaska invented a base 20 numeral system in 1994

Danish numerals display a similar base 20 structure. The Māori language of New Zealand also has evidence of an underlying base 20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama hokotahi, referring to a great warrior ("the one man equal to 20"). The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, with a 1/64 term thrown away (the system was called the Eye of Horus). A number of Australian Aboriginal languages employ binary or binary like counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon, ukasar, ukasar urapon, ukasar ukasar, ukasar ukasar urapon, ukasar ukasar ukasar. North and Central American natives used base 4 (quaternary) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system. A base 5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub base of other bases, such as base 10, base 20, and base 60. A base 8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for "new", newo , suggesting that the number 9 had been recently invented and called the "new number". Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for five is the same as "hand" or "fist" (Dyola language of Guinea Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region. The Telefol language, spoken in Papua New Guinea, is notable for possessing a base 27 numeral system.

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 | for the one and an open left pointing wedge ⟨ for the ten) — up to 5+9=14 symbols per position (i. e. 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).