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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 tradegoods.
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 quipulike 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.
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 (4x10^{3}) + (3x10^{2}) + (2x10^{1}) + (7x10^{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^{k1} + a_{3}b^{k2} + ... + 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 b1, 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.
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 0.3_{10} = 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 base10 system, the one most commonly used by humans today, originated because we have ten fingers, thus allowing for simple counting. A baseeight 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 PreColumbian 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). Base12 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 base3 and base10 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 basep numerals whose expansion to the left never stops; these are called the padic numbers.
Positional systems
See also: Computer numbering formats
External Resources D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. AddisonWesley. pp.194213, "Positional Number Systems"
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