Numeral system

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, but they all represent the same number. This article treats differing systems of numerals representing the same system of numbers. The system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., may be called different number systems, and those are not the topic of this article.

Table of contents 1 History 2 Introduction and notation 3 Additional information 4 Specific numeral systems 5 External Resources

History Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including the american indians, used tallies for gambling with horses, slaves. personal services and trade-goods.

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 to 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 quire 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 use 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 record seem to be tallies of abacus results in China around 400AD. In particular, zero was crrectly described by Chinese mathematicians around 932AD, 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 business. In India, recognizably modern numerals appeared in Mogul empire, used for astronomy and accounting.

From India, the thriving trade between Islamic Moguls and Africa carried the concept to Cairo, where Al Kwairzmi wrote the latinate document that popularized positional notation for Europe.

Introduction and notation

In a positional numeral system of base b, 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. In this way, with only finitely many different symbols, every number can be expressed. This is unlike systems which uses different symbols for different orders of magnitude, like the system of Roman numerals or the number names in spoken languages.

For example, when 4327 is written in the decimal system (base 10), it actually means (4x10³) + (3x10²) + (2x10¹) + (7x10⁰), noting that 10⁰ = 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 a₁b^k + a₂b^k-1 + a₃b^k-2 + ... + a_k+1b⁰ and writing the digits a₁a₂a₃ ... 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.

The term scale of notation is also used for a number system.

Additional information

Note that no matter in which base, numerals have terminating or repeating expansions if and only if they are rational. A number that terminates in one base may repeat in another (thus 0.3₁₀ = 0.0100110011001...₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all bases. Thus, for example in base 2, π = 3.1415926...₁₀ can be written down as the unperiodic 11.001001000011111...₂.

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 a people (the name of which I will insert here once I have tracked it down) that used the spaces between the fingers to count. The Maya 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 mainly 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). 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.

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

Positional systems

• b=-3: Negaternary[?]
• b=-2: Negabinary
• 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: Nonary[?]
• 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=φ: 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 varies: Mixed radix

Other systems

• Unary
• Roman numerals

See also: Computer numbering formats

External Resources D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194-213, "Positional Number Systems"