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Positional notation, also known asplace-value notation,positional numeral system, or simplyplace value, usually denotes the extension to anybase of theHindu–Arabic numeral system (ordecimal system). More generally, a positional system is anumeral 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 earlynumeral systems, such asRoman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as thedecimal 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.
TheBabylonian 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. The Inca used knots tied in a decimal positional system to store numbers and other values inquipu cords.
Today, the Hindu–Arabic numeral system (base ten) is the most commonly used system globally. However, thebinary numeral system (base two) is used in almost allcomputers andelectronic devices because it is easier to implement efficiently inelectronic 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 aradix point (decimal point in base ten), extends to includefractions and allows the representation of anyreal 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 tenfingers, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, theBabylonian numeral system, credited as the first positional numeral system, wasbase-60. However, it lacked a realzero. 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.[1] It was aplaceholder 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 polymathArchimedes (ca. 287–212 BC) invented a decimal positional system based on 108 in hisSand Reckoner;[2] 19th century German mathematicianCarl Gauss lamented how science might have progressed had Archimedes only made the leap to something akin to the modern decimal system.[3]Hellenistic andRoman astronomers used a base-60 system based on the Babylonian model (seeGreek numerals § Zero).
Before positional notation became standard, simple additive systems (sign-value notation) such asRoman numerals orChinese numerals were used, and accountants in the past used theabacus or stone counters to do arithmetic until the introduction of positional notation.[4]
Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods orabacus 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 Chineserod numerals, used from at least the early 8th century, or perhapsKhmer numerals, showing possible usages of positional-numbers in the 7th century. Khmer numerals and otherIndian numerals originate with theBrahmi numerals of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derivedArabic numerals, recorded from the 10th century.
After theFrench Revolution (1789–1799), the new French government promoted the extension of the decimal system.[5] Some of those pro-decimal efforts—such asdecimal time and thedecimal calendar—were unsuccessful. Other French pro-decimal efforts—currencydecimalisation and themetrication of weights and measures—spread widely out of France to almost the whole world.
Decimal fractions were first developed and used by the Chinese in the form ofrod calculus in the 1st century BC, and then spread to the rest of the world.[6][7] J. Lennart Berggren notes that positional decimal fractions were used being, inDamascus, by mathematicianAbu'l-Hasan al-Uqlidisi, in the mid 10th century.[8] The Jewish mathematicianImmanuel Bonfils used decimal fractions around 1350, but did not develop any notation to represent them.[9] The Persian mathematicianJamshīd al-Kāshī similarly adopted their use in the 15th century.[8]Al Khwarizmi introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions fromSunzi Suanjing.[10] This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th centuryAbu'l-Hasan al-Uqlidisi and 15th centuryJamshīd al-Kāshī's work "Arithmetic Key".[10][11]
| Number | 184.54290 |
|---|---|
| Simon Stevin's notation | 184⓪5①4②2③9④0 |
The adoption of thedecimal representation of numbers less than one, afraction, is often credited toSimon Stevin through his textbookDe Thiende;[12] but both Stevin andE. J. Dijksterhuis indicate thatRegiomontanus contributed to the European adoption of generaldecimals:[13]: 17, 18
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 andsexagesimal 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 radiusR equal to a number of units of length of the form 10n and then assuming forn 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 unitR/10n, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.
In the estimation of Dijksterhuis, "after the publication ofDe Thiende only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers ... next to Stevin the most important figure in this development was Regiomontanus." Dijksterhuis noted that [Stevin] "gives full credit to Regiomontanus for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'."[13]: 19
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Inmathematical numeral systems theradixr is usually the number of uniquedigits, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with anegative base, the radix is theabsolute value of the baseb. For example, for the decimal system the radix (and base) is ten, because it uses the ten 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 two, 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 radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The radix is an integer that is greater than 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 more than unique digits, numbers may have many different possible representations.
It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily belogarithmic in its size.
(In certainnon-standard positional numeral systems, includingbijective numeration, the definition of the base or the allowed digits deviates from the above.)
In standard base-ten (decimal) positional notation, there are tendecimal digits and the number
In standard base-sixteen (hexadecimal), there are the sixteen 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 areb digits and the number
hasNote that represents a sequence of digits, notmultiplication.
When describing base inmathematical notation, the letterb is generally used as asymbol for this concept, so, for abinary system,bequals 2. Another common way of expressing the base is writing it as adecimal 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 (adecimal 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 baseb 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 radixb 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.)
Positional numeral systems work usingexponentiation 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 thenth power, wheren is the number of other digits between a given digit and theradix point. If a given digit is on the left hand side of the radix point (i.e. its value is aninteger) thenn is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) thenn is negative.
As an example of usage, the number 465 in its respective baseb (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:
If however, the number were in base 7, then it would equal:
10b =b for any baseb, 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 baseb, then a group of objects is created withb objects. When the number of these groups exceedsb, then a group of these groups of objects is created withb groups ofb 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 onereal 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.
Adigit is a symbol that is used for positional notation, and anumeral consists of one or more digits used for representing anumber with positional notation. Today's most common digits are thedecimal 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-zeronumeral with more than one digit position will mean a different number in a different number base, but in general, thedigits will mean the same.[14] 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 havingan 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 to999. 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 as1330. 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 mean215999. If we use the entire collection of ouralphanumerics 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".[15]We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But seeSexagesimal 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). Inbinary only digits "0" and "1" are in the numerals. In theoctal 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 baseb. 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 notintegers use places beyond theradix point. For every position behind this point (and thus after the units digit), the exponentn of the powerbn 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, aminus 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.
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The conversion to a base of an integern represented in base can be done by a succession ofEuclidean divisions by the right-most digit in base is the remainder of the division ofn by the second right-most digit is the remainder of the division of the quotient by and so on. The left-most digit is the last quotient. In general, thekth 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 = Q: 0x101A R: 7 (ones place)0x101A/10 = Q: 0x19C R: 2 (tens place) 0x19C/10 = Q: 0x29 R: 2 (hundreds place) 0x29/10 = Q: 0x4 R: 1 ... 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 = Q: 0b11000 R: 0b1001 (0b1001 = "9" for ones place) 0b11000/10 = Q: 0b10 R: 0b100 (0b100 = "4" for tens) 0b10/10 = Q: 0b0 R: 0b10 (0b10 = "2" for hundreds)
For thefractional part, conversion can be done by taking digits after the radix point (the numerator), anddividing it by theimplied denominator in the target radix. Approximation may be needed due to a possibility ofnon-terminating digits if thereduced fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.00011 (because one of the prime factors of 10 is 5). For more general fractions and bases see thealgorithm for positive bases.
Alternatively,Horner's method can be used for base conversion using repeated multiplications, with the same computational complexity as repeated divisions.[16] A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simplelookup table, removing the need for expensive division or modulus operations; and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, likerepeated squaring for single or sparse digits. Example:
Convert 0xA10B to 41227 A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0)
Lookup table: 0x0 = 0 0x1 = 1 ... 0x9 = 9 0xA = 10 0xB = 11 0xC = 12 0xD = 13 0xE = 14 0xF = 15 Therefore 0xA10B's decimal digits are 10, 1, 0, and 11.
Lay out the digits out like this. The most significant digit (10) is "dropped": 10 1 0 11 <- Digits of 0xA10B
--------------- 10 Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add: 10 1 0 11 160 --------------- 10 161
Repeat until the final addition is performed: 10 1 0 11 160 2576 41216 --------------- 10 161 2576 41227
and that is 41227 in decimal.
Convert 0b11111001 to 249 Lookup table: 0b0 = 0 0b1 = 1
Result: 1 1 1 1 1 0 0 1 <- Digits of 0b11111001 2 6 14 30 62 124 248 ------------------------- 1 3 7 15 31 62 124 249
The numbers which have a finite representation form thesemiring
More explicitly, if is afactorization of into the primes with exponents,[17] then with the non-empty set of denominatorswe have
where is the group generated by the and is the so-calledlocalization of with respect to.
Thedenominator of an element of contains if reduced to lowest terms only prime factors out of.Thisring of all terminating fractions to base isdense in the field ofrational numbers. Itscompletion for the usual (Archimedean) metric is the same as for, namely the real numbers. So, if then has not to be confused with, thediscrete valuation ring for theprime, which is equal to with.
If divides, we have
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 infiniteseries:
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 avinculum across the repeating block:[18]
This is therepeating decimal notation (to which there does not exist a single universally accepted notation or phrasing).For base 10 it is called a repeating decimal or recurring decimal.
Anirrational number has an infinite non-repeating representation in all integer bases. Whether arational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
For integersp andq withgcd (p,q) = 1, thefractionp/q has a finite representation in baseb if and only if eachprime factor ofq is also a prime factor ofb.
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:
A (real) irrational number has an infinite non-repeating representation in all integer bases.[19]
Examples are the non-solvablenth roots
with andy ∉Q, numbers which are calledalgebraic, or numbers like
which aretranscendental. The number of transcendentals isuncountable 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.
In thedecimal (base-10)Hindu–Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents100 (1), the second position101 (10), the third position102 (10 × 10 or 100), the fourth position103 (10 × 10 × 10 or 1000), and so on.
Fractional values are indicated by aseparator, which can vary in different locations. Usually this separator is a period orfull stop, or acomma. 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 indicates10−1 (0.1), the second position10−2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base-10 numeral system is:
or
Thesexagesimal or base-60 system was used for the integral and fractional portions ofBabylonian numerals and other Mesopotamian systems, byHellenistic astronomers usingGreek 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 aprime 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 be10°25′59″ (10degrees 25minutes 59seconds). 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.[citation needed] This contrasts with the numbers used by Hellenistic andRenaissance astronomers, who usedthirds,fourths, etc. for finer increments. Where we might write10°25′59.392″, they would have written10°25′59′′23′′′31′′′′12′′′′′ or10°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.[20] For example, the meansynodic month used by both Babylonian and Hellenistic astronomers and still used in theHebrew 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.
Incomputing, thebinary (base-2), octal (base-8) andhexadecimal (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).
Theoctal 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 forbinary-to-text encoding, implementations ofarbitrary-precision arithmetic, and other applications.
For a list of bases and their applications, seelist 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 manyfactors. 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 currencyPound 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.
TheMaya civilization and other civilizations ofpre-ColumbianMesoamerica 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 westernAfrica.
Remnants of aGaulish base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five issoixante-cinq (literally, "sixty [and] five"), while seventy-five issoixante-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 isquatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two isquatre-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".
TheIrish language also used base-20 in the past, twenty beingfichid, fortydhá fhichid, sixtytrí fhichid and eightyceithre fhichid. A remnant of this system may be seen in the modern word for 40,daoichead.
TheWelsh language continues to use abase-20counting 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.
TheInuit languages use abase-20 counting system. Students fromKaktovik, Alaska invented abase-20 numeral system in 1994[21]
Danish numerals display a similarbase-20 structure.
TheMāori language of New Zealand also has evidence of an underlying base-20 system as seen in the termsTe Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") andTama-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 to1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, with a 1/64 term thrown away (the system was called theEye of Horus).
A number ofAustralian Aboriginal languages employ binary or binary-like counting systems. For example, inKala Lagaw Ya, the numbers one through six areurapon,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 theYuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.[22] There is also linguistic evidence which suggests that the Bronze AgeProto-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".[23]
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 someAfrican languages the word for five is the same as "hand" or "fist" (Dyola language ofGuinea-Bissau,Banda language ofCentral 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 asquinquavigesimal. It is found in many languages of theSudan region.
TheTelefol language, spoken inPapua New Guinea, is notable for possessing a base-27 numeral system.
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[24] 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 thebalance 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 with1, with 1 if used on the empty pan, and with 0 if not used. If an unknown weightW 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.
Thefactorial number system uses a varying radix, givingfactorials as place values; they are related toChinese remainder theorem andresidue number system enumerations. This system effectively enumerates permutations. A derivative of this uses theTowers 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 |
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).[25] 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 azero symbol).[26]
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