Positional notation

Positional notation (or place-value notation, or positional numeral system) denotes usually 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 product of the value of the digit 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 use of a radix point (decimal point in base ten), extends to include fractions and allows representing every real number up to arbitrary accuracy. With positional notation, arithmetical computations are greatly simpler than with any older numeral system, and this explains the rapid spread of the notation when it was introduced in western Europe.

The Babylonian numeral system, base 60, was the first positional system developed, and its influence is present today in the way time and angles are counted in tallies related to 60, like 60 minutes in an hour, 360 degrees in a circle. Today, the Hindu–Arabic numeral system (base ten) is the most commonly used system, all around the world. 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.

Positional notation glossary-en
Glossary of terms used in the positional numeral systems.

History

Abacus 6
Suanpan (the number represented in the picture is 6,302,715,408)

Today, the base-10 (decimal) system, which is likely 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, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.[1] The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal 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[2] 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.[3]

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.[4]

Chounumerals
The world's earliest positional decimal system
Upper row vertical form
Lower row horizontal form

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. For four centuries (from the 13th to the 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.

After the French Revolution (1789-1799), the new French government promoted the extension of the decimal system.[5] 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.

History of positional fractions

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.[6] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[7] The Persian mathematician Jamshīd al-Kāshī made the same discovery of decimal fractions in the 15th century.[6] Al Khwarizmi introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from Sunzi Suanjing.[8] 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".[8][9]

Stevin-decimal notation

A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[10]

Issues

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 cheques require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰, which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).

Many of the advantages claimed for the metric system could be realized by any consistent positional notation. Dozenal advocates say duodecimal has several advantages over decimal, although the switching cost appears to be high.

Mathematics

Base of the numeral system

In mathematical numeral systems the base or radix is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is 2, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100".

The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.

The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.

(In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)

In base-10 (decimal) positional notation, there are 10 decimal digits and the number

.

In base-16 (hexadecimal), there are 16 hexadecimal digits (0–9 and A–F) and the number

(where B represents the number eleven as a single symbol)

In general, in base-b, there are b digits and the number

(Note that represents a sequence of digits, not multiplication)

Notation

When describing base in mathematical notation, the letter b is generally used as a symbol for this concept, so, for a binary system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented (this notation is used in this article). 11110112 implies that the number 1111011 is a base-2 number, equal to 12310 (a decimal notation representation), 1738 (octal) and 7B16 (hexadecimal). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 11110112.

The base b may also be indicated by the phrase "base-b". So binary numbers are "base-2"; octal numbers are "base-8"; decimal numbers are "base-10"; and so on.

To a given radix b the set of digits {0, 1, ..., b−2, b−1} is called the standard set of digits. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Therefore, the following are notational errors: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)

Exponentiation

Positional numeral systems work using exponentiation of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point. If a given digit is on the left hand side of the radix point (i.e. its value is an integer) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.

As an example of usage, the number 465 in its respective base b (which must be at least base 7 because the highest digit in it is 6) is equal to:

If the number 465 was in base-10, then it would equal:

(46510 = 46510)

If however, the number were in base 7, then it would equal:

(4657 = 24310)

10b = b for any base b, since 10b = 1×b1 + 0×b0. For example, 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.

This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:

241 in base 5:
   2 groups of 52 (25)           4 groups of 5          1 group of 1
   ooooo    ooooo
   ooooo    ooooo                ooooo   ooooo
   ooooo    ooooo         +                         +         o
   ooooo    ooooo                ooooo   ooooo
   ooooo    ooooo
241 in base 8:
   2 groups of 82 (64)          4 groups of 8          1 group of 1
 oooooooo  oooooooo
 oooooooo  oooooooo
 oooooooo  oooooooo         oooooooo   oooooooo
 oooooooo  oooooooo    +                            +        o
 oooooooo  oooooooo
 oooooooo  oooooooo         oooooooo   oooooooo
 oooooooo  oooooooo
 oooooooo  oooooooo

The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.

Digits and numerals

A digit is what is used as a position in place-value notation, and a numeral is one or more digits. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base.

A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.[11] The base-8 numeral 238 contains two digits, "2" and "3", and with a base number (subscripted) "8", means 19. 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, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123. The numeral "23" then, in this case, corresponds to the set of 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" three.

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".[12] 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".

Radix point

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:

Sign

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.

Base conversion

The conversion to a base of an integer n represented in base can be done by a succession of Euclidean divisions by the right-most digit in base is the remainder of the division of n by the second right-most digit is the remainder of the division of the quotient by and so on. More precisely, the kth digit from the right is the remainder of the division by of the (k−1)th quotient.

For example: converting A10BHex to decimal (41227):

0xA10B/10 = 0x101A R: 7 (ones place)
0x101A/10 = 0x19C  R: 2 (tens place)
 0x19C/10 = 0x29   R: 2 (hundreds place)
  0x29/10 = 0x4    R: 1  ...
   0x4/10 = 0x0    R: 4

When converting to a larger base (such as from binary to decimal), the remainder represents as a single digit, using digits from . For example: converting 0b11111001 (binary) to 249 (decimal):

0b11111001/10 = 0b11000 R: 0b1001 (0b1001 = "9" for ones place)
   0b11000/10 = 0b10    R: 0b100  (0b100 =  "4" for tens)
      0b10/10 = 0b0     R: 0b10   (0b10 =   "2" for hundreds)

Terminating fractions

The numbers which have a finite representation form the semiring

More explicitly, if is a factorization of into the primes with exponents ,[13] then with the non-empty set of denominators we have

where is the group generated by the and is the so-called localization of with respect to .

The denominator of an element of contains if reduced to lowest terms only prime factors out of . This ring of all terminating fractions to base is dense in the field of rational numbers . Its completion for the usual (Archimedean) metric is the same as for , namely the real numbers . So, if then has not to be confused with , the discrete valuation ring for the prime , which is equal to with .

If divides , we have

Infinite representations

Rational numbers

The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112 ... base-3 represents the sum of the infinite series:

Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a vinculum across the repeating block:

This is the repeating decimal notation (to which there does not exist a single universally accepted notation or phrasing). For base-10 it is called a recurring decimal or repeating decimal.

An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:

or, with the base implied:
(see also 0.999...)

For integers p and q with gcd(p, q) = 1, the fraction p/q has a finite representation in base b if and only if each prime factor of q is also a prime factor of b.

For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:

1. A finite or infinite number of zeroes can be appended:
2. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
(see also 0.999...)

Irrational numbers

A (real) irrational number has an infinite non-repeating representation in all integer bases.

Examples are the non-solvable nth roots

with and yQ, numbers which are called algebraic, or numbers like

which are transcendental. The number of transcendentals is uncountable and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols.

Applications

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 can vary in different locations. 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).

Sexagesimal system

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 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.

Computing

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.

Other bases in human language

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 Pslam 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 new numbering notation in 1994[14]

Danish numerals display a similar base-20 structure.

The Maori 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.[15] 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".[16]

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.

Non-standard positional numeral systems

Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.

Balanced ternary uses a base of 3 but the digit set is {1,0,1} instead of {0,1,2}. The "1" has an equivalent value of −1. The negation of a number is easily formed by switching the    on the 1s. This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ... 3n known units can be used to determine any unknown weight up to 1 + 3 + ... + 3n units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with 1, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight W is balanced with 3 (31) on its pan and 1 and 27 (30 and 33) on the other, then its weight in decimal is 25 or 1011 in balanced base-3.

10113 = 1 × 33 + 0 × 32 − 1 × 31 + 1 × 30 = 25.

The factorial number system uses a varying radix, giving factorials as place values; they are related to Chinese remainder theorem and residue number system enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.

Decimal equivalents −3 −2 −1 0 1 2 3 4 5 6 7 8
Balanced base 3 10 11 1 0 1 11 10 11 111 110 111 101
Base −2 1101 10 11 0 1 110 111 100 101 11010 11011 11000
Factoroid 0 10 100 110 200 210 1000 1010 1100

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).[17] 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).[18]

See also

Examples:

Related topics:

Notes

  1. ^ Kaplan, Robert (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
  2. ^ Greek numerals
  3. ^ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN 3-525-40725-4, pp. 150-153
  4. ^ Ifrah, page 187
  5. ^ L. F. Menabrea. Translated by Ada Augusta, Countess of Lovelace. "Sketch of The Analytical Engine Invented by Charles Babbage". 1842.
  6. ^ a b Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
  7. ^ Gandz, S.: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.
  8. ^ a b Lam Lay Yong, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
  9. ^ Lam Lay Yong, "A Chinese Genesis, Rewriting the history of our numeral system", Archive for History of Exact Science 38: 101–108.
  10. ^ B. L. van der Waerden (1985). A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
  11. ^ The digit will retain its meaning in other number bases, in general, because a higher number base would normally be a notational extension of the lower number base in any systematic organization. In the mathematical sciences there is virtually only one positional-notation numeral system for each base below 10, and this extends with few, if insignificant, variations on the choice of alphabetic digits for those bases above 10.
  12. ^ We do not usually remove the lowercase digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".
  13. ^ The exact size of the does not matter. They only have to be ≥ 1.
  14. ^ Bartley, Wm. Clark (January – February 1997). "Making the Old Way Count" (PDF). Sharing Our Pathways. 2 (1): 12–13. Retrieved 27 February 2017.
  15. ^ Barrow, John D. (1992), Pi in the sky: counting, thinking, and being, Clarendon Press, p. 38, ISBN 9780198539568.
  16. ^ (Mallory & Adams 1997) Encyclopedia of Indo-European Culture
  17. ^ Ifrah, pages 326, 379
  18. ^ Ifrah, pages 261-264

References

  • O'Connor, John; Robertson, Edmund (December 2000). "Babylonian Numerals". Retrieved 21 August 2010.
  • Kadvany, John (December 2007). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy.
  • Knuth, Donald (1997). The art of Computer Programming. 2. Addison-Wesley. pp. 195–213. ISBN 0-201-89684-2.
  • Ifrah, George (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0-471-37568-3.
  • Kroeber, Alfred (1976) [1925]. Handbook of the Indians of California. Courier Dover Publications. p. 176. ISBN 9780486233680.

External links

Algorism

Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system largely superseded earlier calculation systems that used a different set of symbols for each numerical magnitude, such as Roman numerals, and in some cases required a device such as an abacus.

Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number" and τική [τέχνη], tiké [téchne], "art") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory.

Bijective numeration

Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name derives from this bijection (one-to-one correspondence) between the set of non-negative integers and the set of finite strings using a finite set of symbols (the "digits").

Most ordinary numeral systems, such as the common decimal system, are not bijective because more than one string of digits can represent the same positive integer. In particular, adding leading zeroes does not change the value represented, so "1", "01" and "001" all represent the number one. Even though only the first is usual, the fact that the others are possible means that decimal is not bijective. However, unary, with only one digit, is bijective.

A bijective base-k numeration is a bijective positional notation. It uses a string of digits from the set {1, 2, ..., k} (where k ≥ 1) to encode each positive integer; a digit's position in the string defines its value as a multiple of a power of k. Smullyan (1961) calls this notation k-adic, but it should not be confused with the p-adic numbers: bijective numerals are a system for representing ordinary integers by finite strings of nonzero digits, whereas the p-adic numbers are a system of mathematical values that contain the integers as a subset and may need infinite sequences of digits in any numerical representation.

Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one).

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices.

Hindu–Arabic numeral system

The Hindu–Arabic numeral system or Indo-Arabic numeral system (also called the Arabic numeral system or Hindu numeral system) is a positional decimal numeral system, and is the most common system for the symbolic representation of numbers in the world.

It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics by the 9th century. Influential were the books of Muḥammad ibn Mūsā al-Khwārizmī (On the Calculation with Hindu Numerals, c. 825) and Al-Kindi (On the Use of the Hindu Numerals, c. 830). The system later spread to medieval Europe by the High Middle Ages.

The system is based upon ten (originally nine) glyphs. The symbols (glyphs) used to represent the system are in principle independent of the system itself. The glyphs in actual use are descended from Brahmi numerals and have split into various typographical variants since the Middle Ages.

These symbol sets can be divided into three main families: Western Arabic numerals used in the Greater Maghreb and in Europe, Eastern Arabic numerals (also called "Indic numerals") used in the Middle East, and the Indian numerals used in the Indian subcontinent.

John of Seville

John of Seville (Latin: Johannes Hispalensis or Johannes Hispaniensis) was the main translator from Arabic into Castilian together with Dominicus Gundissalinus during the early days of the Toledo School of Translators. His work is said to have flourished between 1135 and 1153.He was a baptized Jew, whose Jewish name (now unknown) has been corrupted into "Avendeut", "Avendehut", "Avendar" or "Aven Daud". This evolved into the middle name "David", so that, as a native of Toledo, he is frequently referred to as Johannes (David) Toletanus. Some historians argue that in fact there were two different persons with a similar name, one as Juan Hispano (Ibn Dawud) and other as Juan Hispalense, this last one perhaps working at Galician Limia (Ourense), for he signed himself as "Johannes Hispalensis atque Limiensis", during the Reconquista, the Christian campaign to regain the Iberian Peninsula.

The topics of his translated works were mainly astrological and astronomical, philosophical and medical. At least three of his translations, the Secretum Secretorum dedicated to a Queen T[arasia?], a tract on gout offered to one of the Popes Gregory, and the original version of the 9th century Arabic philosopher Qusta ibn Luqa's De differentia spiritus et animae, were medical translations intermixed with alchemy in the Hispano-Arabic tradition. In his Book of Algorithms on Practical Arithmetic, John of Seville provides one of the earliest known descriptions of Indian positional notation, whose introduction to Europe is usually associated with the book Liber Abaci by Fibonacci:

“A number is a collection of units, and because the collection is infinite (for multiplication can continue indefinitely), the Indians ingeniously enclosed this infinite multiplicity within certain rules and limits so that infinity could be scientifically defined; these strict rules enabled them to pin down this subtle concept.”John of Seville translated Al-Farghani's Kitab Usul 'ilm al-nujum(Book on the Elements of the Science of Astronomy) into Latin in 1135 ('era MCLXXIII'), as well as translating the Arab astrologer Albohali's "Book of Birth" into Latin in 1153. He also translated Kitāb taḥāwīl sinī al-‘ālam by Abu Ma'shar al-Balkhi into Latin.

Leading zero

A leading zero is any 0 digit that comes before the first nonzero digit in a number string in positional notation. For example, James Bond's famous identifier, 007, has two leading zeros. When leading zeros occupy the most significant digits of an integer, they could be left blank or omitted for the same numeric value. Therefore, the usual decimal notation of integers does not use leading zeros except for the zero itself, which would be denoted as an empty string otherwise. However, in decimal fractions strictly between −1 and 1, the leading zeros digits between the decimal point and the first nonzero digit are necessary for conveying the magnitude of a number and cannot be omitted, while trailing zeros – zeros occurring after the decimal point and after the last nonzero digit – can be omitted without changing the meaning.

List of numeral systems

This is a list of numeral systems, that is, writing systems for expressing numbers.

Logarithmic growth

In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x). Note that any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant. Logarithmic growth is the inverse of exponential growth and is very slow.

A familiar example of logarithmic growth is a number, N, in positional notation, which grows as logb (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic. In more advanced mathematics, the partial sums of the harmonic series

grow logarithmically. In the design of computer algorithms, logarithmic growth, and related variants, such as log-linear, or linearithmic, growth are very desirable indications of efficiency, and occur in the time complexity analysis of algorithms such as binary search.

Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler's bankroll. It also plays a role in the St. Petersburg paradox.

In microbiology, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing are proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a logarithmic scale for the growth axis.

Naming convention (programming)

In computer programming, a naming convention is a set of rules for choosing the character sequence to be used for identifiers which denote variables, types, functions, and other entities in source code and documentation.

Reasons for using a naming convention (as opposed to allowing programmers to choose any character sequence) include the following:

To reduce the effort needed to read and understand source code;

To enable code reviews to focus on more important issues than arguing over syntax and naming standards.

To enable code quality review tools to focus their reporting mainly on significant issues other than syntax and style preferences.The choice of naming conventions can be an enormously controversial issue, with partisans of each holding theirs to be the best and others to be inferior. Colloquially, this is said to be a matter of dogma. Many companies have also established their own set of conventions.

Notation

In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, a notation is a collection of related symbols that are each given an arbitrary meaning, created to facilitate structured communication within a domain knowledge or field of study.

Standard notations refer to general agreements in the way things are written or denoted. The term is generally used in technical and scientific areas of study like mathematics, physics, chemistry and biology, but can also be seen in areas like business, economics and music.

Numerical digit

A numerical digit is a single symbol (such as "2" or "5") used alone, or in combinations (such as "25"), to represent numbers (such as the number 25) according to some positional numeral systems. The single digits (as one-digit-numerals) and their combinations (such as "25") are the numerals of the numeral system they belong to. The name "digit" comes from the fact that the ten digits (Latin digiti meaning fingers) of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal (ancient Latin adjective decem meaning ten) digits.

For a given numeral system with an integer base, the number of digits required to express arbitrary numbers is given by the absolute value of the base. For example, the decimal system (base 10) requires ten digits (0 through to 9), whereas the binary system (base 2) has two digits (e.g.: 0 and 1).

Pentimal system

The pentimal system (Swedish: pentadiska siffror) is a notation for presenting numbers, usually by inscribing in wood or stone. The notation has been used in Scandinavia, usually in conjunction to runes.

The notation is similar to the older Roman numerals for numbers 1 to 9 (I, II, III, IV, V, VI, VII, VIII, IX). Unlike the Roman notation, there are only symbols for numbers one ("I") and five ("U"). In inscriptions the notches are placed vertically on the stem or stav of the rune. The number 4 is represented by four horizontal lines on the stem, 5 is represented by what looks like an inverted letter U. 10 is represented by two U's opposing each other. Numbers up to 19, or even 20, can be represented by a combination of I's and U's, just like the Roman numerals are represented by combinations of I's and V's. (A Roman 10 is represented by two V's opposing each other).

The widest use of the notation is in presenting the Golden Numbers, 1 - 19 on Runic calendars (Danish: kalenderstave, Swedish: runstavar, Norwegian: kalenderstavar, also known as clogs). The numbers are commonly found in Modern Age and possibly Early Modern Age calendar sticks. It is unknown if they were in use in the Middle Ages, let alone in the Viking Age. On older runic calendars, a different notation for representing the Golden Numbers was used; the 16 runes of Younger Futhark represented the numbers from 1 to 16 and three ad hoc runes were used for the numbers 17 to 19. For example, the Computus Runicus manuscript, originally from 1328, but collected and published by the Dane Ole Worm (1588-1654), uses this futhark notation, and not the pentadic numerals under discussion here.

Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.

Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

Position

Position is a location (rather than orientation) of an entity.

Position may also refer to:

A job or occupation

Sign-value notation

A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals for example, X means ten and L means fifty. Hence LXXX means eighty (50 + 10 + 10 + 10). There is no need for zero in sign-value notation. Sign-value notation was the pre-historic way of writing numbers and only gradually evolved into place-value notation, also known as positional notation.

When pre-historic people wanted to write "two sheep" in clay, they could inscribe in clay a picture of two sheep. But 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 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.

Strong NP-completeness

In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters.

A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains so even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a polynomial reduction to it; in combinatorial optimization, particularly, the phrase "strongly NP-hard" is reserved for problems that are not known to have a polynomial reduction to another strongly NP-complete problem.

Normally numerical parameters to a problem are given in positional notation, so a problem of input size n might contain parameters whose size is exponential in n. If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size. Thus strong NP-completeness or NP-hardness may also be defined as the NP-completeness or NP-hardness of this unary version of the problem.

For example, bin packing is strongly NP-complete while the 0-1 Knapsack problem is only weakly NP-complete. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming.

While weakly NP-complete problems may admit efficient solutions in practice as long as their inputs are of relatively small magnitude, strongly NP-complete problems do not admit efficient solutions in these cases. From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP. However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded.Some strongly NP-complete problems may still be easy to solve on average, but it's more likely that difficult instances will be encountered in practice.

Timeline of numerals and arithmetic

A timeline of numerals and arithmetic

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