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A numeral is a symbol or group of symbols that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article treats the various systems of numerals.
A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the roman numeral for two, the binary numeral for three or the decimal numeral for eleven.
Ideally, a numeration system will represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers), will give every number represented a unique representation (or at least a standard representation) and will reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic.
Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article.
Types of numeral systems
The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol "|" is chosen, for example, then the number seven would be represented by |||||||. The unary system is normally only useful for small numbers; it has some uses in theoretical computer science.
The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if | stands for one, @ for ten and # for 100, then the number 304 can be compactly represented as ### |||| and number 123 as #@@||| . The ancient Egyptian system is of this type, and the Roman system is a modification of this idea.
More useful still are systems which employ special abbreviations for repetitions of symbols; using our ordinary digits for these abbreviations, we could then write 3# 4| for the number 304. The numeral system of English is of this type ("three hundred four"), as are those of virtually all other languages: Chinese, Japanese, and Greek.
More elegant is a positional system: again working in base 10, we use ten different digits 0,...,9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3*100 + 0*10 + 4. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Arabic numeral system is a positional base 10 system; it is used today throughout the world.
Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).
History
- See also History of natural numbers and the status of zero.
The earliest known written tallies appear in the ruins of the Sumerian empire, using clay tablets impressed with a sharp stick and baked. The Sumerians had quite an exotic system based on counts to 60, used in astronomical and other calculations. This system was imported and used by every Mediterranean nation that used astronomy, including the Greeks, Romans and Egyptians. We still use it to count time (minutes per hour), and angle (degrees).
In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. 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 in Digital signal processing.
The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman system remained in common use in Europe until positional notation came into common use in the 1500s.
The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.
Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional records seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle of a place empty of beads.
From China, both the abacus and written tallies may have moved to India, perhaps via Chinese traders and businesses. In India, recognizably modern positional numeral systems, used for astronomy and accounting, appeared in the Mogul empire.
From India, the thriving trade between Islamic Moguls and Africa carried the concept to Cairo. Arabic mathematicians extended the system to decimal fractions, and al-Khwarizmi wrote an important work about it in the 9th century. The system was introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisas Liber Abaci of 1201.
The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications.
Bases used
The base-10 system, the one most commonly used by humans today, originated because we have ten fingers, thus allowing for simple counting. A base-eight system was devised by (at least) the Yuki Pomo of Northern California, who used the spaces between the fingers to count. The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base 20, (possibly originating from the number of a person's fingers and toes). Base 60 was used by the Sumerians and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds). Base-12 systems were popular because multiplication is easier in them than in base-10, (addition just as easy) and because the year has twelve months; we still have a special word for "dozen" and use 12 hours for every night and day.
Electronic components (first vacuum tubes, then transistors) may have only 2 possible states: concat(1) and closed (0). Because this is exactly the set of binary digits, and because arithmetics in a binary system are the easiest to describe electronically (using Boolean algebra), the binary system became natural for electronic computers. It is used to perform integer arithmetic in almost all electronic computers (the only exception being the exotic base-3 and base-10 designs that were discarded very early in the history of computing hardware). Note however that a computer does not treat all of its data as integers. Thus, some of it may be treated as texts and program data. Real numbers (numbers that can be not whole) are usually written down in the floating point notation, that has different rules of arithmetic.
Positional systems in detail
In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b.
For example, in the decimal system (base 10), the numeral 4327 means (4×10^{3}) + (3×10^{2}) + (2×10^{1}) + (7×10^{0}), noting that 10^{0} = 1.
In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a_{1}b^{k} + a_{2}b^{k-1} + a_{3}b^{k-2} + ... + a_{k+1}b^{0} and writing the digits a_{1}a_{2}a_{3} ... a_{k+1} in order. The digits are natural numbers between 0 and b-1, inclusive.
If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base is added in subscript to the right of the number, like this: number_{base}. Numbers without subscript are considered to be decimal.
By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×2^{1}+ 0×2^{0} +1×2^{-1} +1×2^{-2} = 2.75.
Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3_{10} = 0.0100110011001..._{2}). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all bases. Thus, for example in base 2, &pi = 3.1415926..._{10} can be written down as the unperiodic 11.001001000011111..._{2}.
If b=p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers.
Specific numeral systems
- b=2: Binary
- b=3: Ternary
- b=4: Quaternary
- b=5: Quinary
- b=6: Senary
- b=7: Septimal
- b=8: Octal or Octonary
- b=9: Novenary
- b=10: Decimal or Denary (see Arabic numerals, Armenian numerals, Chinese numerals, Greek numerals, Hebrew numerals, Indian numerals, Japanese numerals, Roman numerals)
- b=12: Duodecimal (used in the Chepang language of Nepal)
- b=16: Hexadecimal
- b=20: Vigesimal (see Mayan numerals)
- b=27: Base 27
- b=60: Sexagesimal (see Babylonian numerals)
- b=-3: Negaternary
- b=-2: Negabinary
- b=&phi: Golden mean base
- b=2i: Quater-imaginary base
- b=sqrt(2)i: Binary square-root-2 times i base
- b=i - 1: Binary i-1 base
- b=Fibonacci number: Fibonacci coding
- b varies: Mixed radix