The ten digits of the Arabic numerals, in order of value
Anumerical digit (often shortened to justdigit) ornumeral is a singlesymbol used alone (such as "1"), or in combinations (such as "15"), to representnumbers inpositional notation, such as the commonbase 10. The name "digit" originates from theLatindigiti meaning fingers.[1]
For any numeral system with an integerbase, the number of different digits required is theabsolute value of the base. For example, decimal (base 10) requires ten digits (0 to 9), andbinary (base 2) requires only two digits (0 and 1). Bases greater than 10 require more than 10 digits, for instancehexadecimal (base 16) requires 16 digits (usually 0 to 9 and A to F).
In a basic digital system, anumeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has aplace value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results.
Each digit in a number system represents an integer. For example, indecimal the digit "1" represents the integerone, and in thehexadecimal system, the letter "A" represents the numberten. Apositional number system has one unique digit for each integer fromzero up to, but not including, theradix of the number system.
Thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position. The number 12 is expressed with the numeral "2" in the units position, and with the numeral "1" in the "tens" position, to the left of the "2" while the number 312 is expressed with three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.
Thedecimal numeral system uses adecimal separator, commonly aperiod in English, or acomma in otherEuropean languages,[2] to denote the "ones place" or "units place",[3][4][5] which has a place value one. Each successive place to the left of this has a place value equal to the place value of the previous digit times thebase. Similarly, each successive place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral10.34 (written inbase 10),
the0 is immediately to the left of the separator, so it is in the ones or units place, and is called theunits digit orones digit;[6][7][8]
the1 to the left of the ones place is in the tens place, and is called thetens digit;[9]
the3 is to the right of the ones place, so it is in the tenths place, and is called thetenths digit;[10]
the4 to the right of the tenths place is in the hundredths place, and is called thehundredths digit.[10]
The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. The zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place.
The place value of any given digit in a numeral can be given by a simple calculation, which in itself is a complement to the logic behind numeral systems. The calculation involves the multiplication of the given digit by the base raised by the exponentn − 1, wheren represents the position of the digit from the separator; the value ofn is positive (+), but this is only if the digit is to the left of the separator. And to the right, the digit is multiplied by the base raised by a negative (−)n. For example, in the number10.34 (written in base 10),
the1 is second to the left of the separator, so based on calculation, its value is,
the4 is second to the right of the separator, so based on calculation its value is,
The first true writtenpositional numeral system is considered to be theHindu–Arabic numeral system. This system was established by the 7th century in India,[11] but was not yet in its modern form because the use of the digitzero had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876.[12] The original numerals were very similar to the modern ones, even down to theglyphs used to represent digits.[11]
The digits of the Maya numeral system
By the 13th century,Western Arabic numerals were accepted in European mathematical circles (Fibonacci used them in hisLiber Abaci). They began to enter common use in the 15th century.[13] By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.
The exact age of theMaya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system. The system wasvigesimal (base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. TheMayas had no equivalent of the moderndecimal separator, so their system could not represent fractions.
TheThai numeral system is identical to theHindu–Arabic numeral system except for the symbols used to represent digits. The use of these digits is less common inThailand than it once was, but they are still used alongside Arabic numerals.
The rod numerals, the written forms ofcounting rods once used byChinese andJapanese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. TheSuzhou numerals are variants of rod numerals.
Thebinary (base 2),octal (base 8), andhexadecimal (base 16) systems, extensively used incomputer science, all follow the conventions of theHindu–Arabic numeral system.[14] The binary system uses only the digits "0" and "1", while the octal system uses the digits from "0" through "7". The hexadecimal system uses all the digits from the decimal system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively.[15] When the binary system is used, the term "bit(s)" is typically used as an alternative for "digit(s)", being a portmanteau of the term "binary digit".
Balanced ternary is unusual in having the digit values 1, 0 and −1. Balanced ternary turns out to have some useful properties and the system has been used in the experimental RussianSetun computers.[17]
Despite the essential role of digits in describing numbers, they are relatively unimportant to modernmathematics.[18] Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.
The digital root is the single-digit number obtained by summing the digits of a given number, then summing the digits of the result, and so on until a single-digit number is obtained.[19]
Casting out nines is a procedure for checking arithmetic done by hand. To describe it, let represent thedigital root of, as described above. Casting out nines makes use of the fact that if, then. In the process of casting out nines, both sides of the latterequation are computed, and if they are not equal, the original addition must have been faulty.[20]
Repunits are integers that are represented with only the digit 1. For example, 1111 (one thousand, one hundred and eleven) is a repunit.Repdigits are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. Theprimality of repunits is of interest to mathematicians.[21]
Palindromic numbers are numbers that read the same when their digits are reversed.[22] ALychrel number is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed.[23] The question of whether there are any Lychrel numbers in base 10 is an open problem inrecreational mathematics; the smallest candidate is196.[24]
Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. TheOksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.[25]
To preserve numerical information,tallies carved in wood, bone, and stone have been used since prehistoric times.[26] Stone age cultures, including ancientindigenous American groups, used tallies for gambling, personal services, and trade-goods.
A method of preserving numeric information in clay was invented by theSumerians between 8000 and 3500 BC.[27] This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100 BC, written numbers were dissociated from the things being counted and became abstract numerals.
Between 2700 and 2000 BC, in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additivesign-value notation of the round number signs. These systems gradually converged on a commonsexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions.[28] This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.[29]
Sexagesimal numerals were amixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC, this was apositional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measuretime (minutes per hour) andangles (degrees).[30]
InChina, armies and provisions were counted using modular tallies ofprime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience ofmodular arithmetic is that it is easy to multiply.[31] This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used indigital signal processing.[32]
The oldest Greek system was that of theAttic numerals,[33] but in the 4th century BC they began to use a quasidecimal alphabetic system (seeGreek numerals).[34] Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.[35]
The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. TheRoman numerals system remained in common use in Europe untilpositional notation came into common use in the 16th century.[36]
TheMaya of Central America used a mixed base 18 and base 20 system, possibly inherited from theOlmec, including advanced features such as positional notation and azero.[37] They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit ofVenus.[38]
The Incan Empire ran a large command economy usingquipu, tallies made by knotting colored fibers.[39] Knowledge of the encodings of the knots and colors was suppressed by theSpanishconquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in theAndean region.
Some authorities believe that positional arithmetic began with the wide use ofcounting rods in China.[40] The earliest written positional records seem to berod calculus results in China around 400. Zero was first used in India in the 7th century CE byBrahmagupta.[41]
FromIndia, the thriving trade between Islamic sultans and Africa carried the concept toCairo. Arabic mathematicians extended the system to includedecimal fractions, andMuḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century.[43] The modernArabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain andLeonardo of Pisa'sLiber Abaci of 1201.[44] In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.[45]
Thebinary system (base 2) was propagated in the 17th century byGottfried Leibniz.[46] Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of theI Ching from China.[47] Binary numbers came into common use in the 20th century because of computer applications.[46]
^Weisstein, Eric W."Decimal Point".mathworld.wolfram.com. Retrieved22 July 2020.
^Snyder, Barbara Bode (1991).Practical math for the technician : the basics. Englewood Cliffs, N.J.: Prentice Hall. p. 225.ISBN0-13-251513-X.OCLC22345295.units or ones place
^Andrew Jackson Rickoff (1888).Numbers Applied. D. Appleton & Company. pp. 5–.units' or ones' place
^John William McClymonds; D. R. Jones (1905).Elementary Arithmetic. R.L. Telfer. pp. 17–18.units' or ones' place
^Richard E. Johnson; Lona Lee Lendsey; William E. Slesnick (1967).Introductory Algebra for College Students. Addison-Wesley Publishing Company. p. 30.units' or ones', digit
^Kirillov, A.A."What are numbers?"(PDF).math.upenn. p. 2.True, if you open a modern mathematical journal and try to read any article, it is very probable that you will see no numbers at all.
^Weisstein, Eric W."Digital Root".mathworld.wolfram.com. Retrieved22 July 2020.
^Weisstein, Eric W."Casting Out Nines".mathworld.wolfram.com. Retrieved22 July 2020.
^Saxe, Geoffrey B. (2012).Cultural development of mathematical ideas : Papua New Guinea studies. Esmonde, Indigo. Cambridge: Cambridge University Press. pp. 44–45.ISBN978-1-139-55157-1.OCLC811060760.The Okspamin body system includes 27 body parts...
^Tuniz, C. (Claudio) (24 May 2016).Humans : an unauthorized biography. Tiberi Vipraio, Patrizia, Haydock, Juliet. Switzerland. p. 101.ISBN978-3-319-31021-3.OCLC951076018....even notches cut into sticks made out of wood, bone or other materials dating back 30,000 years (often referred to as "notched tallies").{{cite book}}: CS1 maint: location missing publisher (link)
^Ifrah, Georges (1985).From one to zero : a universal history of numbers. New York: Viking. p. 154.ISBN0-670-37395-8.OCLC11237558.And so, by the beginning of the third millennium B.C., the Sumerians and Elamites had adopted the practice of recording numerical information on small, usually rectangular clay tablets
^Powell, Marvin A. (2008). "Sexagesimal System".Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Berlin/Heidelberg: Springer-Verlag. pp. 1998–1999.doi:10.1007/978-1-4020-4425-0_9055.ISBN978-1-4020-4559-2.
^Knuth, Donald Ervin (1998).The art of computer programming. Reading, Mass.: Addison-Wesley Pub. Co.ISBN0-201-03809-9.OCLC823849.The advantages of a modular representation are that addition, subtraction, and multiplication are very simple
^Woodhead, A. G. (Arthur Geoffrey) (1981).The study of Greek inscriptions (2nd ed.). Cambridge: Cambridge University Press. pp. 109–110.ISBN0-521-23188-4.OCLC7736343.
^Chrisomalis, Stephen (2010).Numerical notation : a comparative history. Cambridge: Cambridge University Press. p. 157.ISBN978-0-511-67683-3.OCLC630115876.The first safely dated instance in which the use of Hebrew alphabetic numerals is certain is on coins from the reign of Hasmonean king Alexander Janneus(103 to 76 BC)...
^Swami, Devamrita (2002).Searching for Vedic India. The Bhaktivedanta Book Trust.ISBN978-0-89213-350-5.Maya astronomy finely calculated both the duration of the solar year and the synodical revolution of Venus
^Deming, David (2010).Science and technology in world history. Volume 1, The ancient world and classical civilization. Jefferson, N.C.: McFarland & Co. p. 86.ISBN978-0-7864-5657-4.OCLC650873991.
