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Sexagesimal

Sexagesimal

The sexagesimal (base-sixty) is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BC, and was transmitted to the Babylonians: see Babylonian numerals. Usually, the sexagesimal uses ten numerals (from 0 to 9) according to decimal place system as the sub-base. In this article places are based on decimal, except where otherwise noted. For example, 10 means ten, 60 means sixty.

Sexagesimal in Babylonia

The Sumero-Babylonian version used a digit to represent "one" and another digit to represent "ten", and repeated the symbols in groups up to nine for the former and five for the latter, then used place-position shifting to the left for each power of sixty, with a larger space between one power of sixty and the next — this may be represented schematically here by using sixty and sixty thus: Because there was no symbol for zero with either the Sumerians or the earlier Babylonians, it is not always immediately obvious how a number should be interpreted, and the true value must sometimes be determined by the context; later Babylonian texts used a dot to represent zero. It was later used in its more modern form by Arabs during the Umayyad caliphate.

Usage

60 (sexagesimal) is the product of 3, 4, and 5. 3 is a divisor of 12 (duodecimal), 4 is a common divisor of 12 (duodecimal) and 20 (vigesimal), 5 is a common divisor of 10 (decimal) and 20 (vigesimal). Base-sixty has the advantage that its base has a large number of conveniently sized divisors , facilitating calculations with vulgar fractions. Note that 60 is the smallest number divisible by every number from 1 to 6. Unlike most other numeral systems, sexagesimal is not used so much as a means of general computation or logic, but is used in measuring angles, time and geographic coordinates. The standard unit in sexagesimal is the degree, of which there are 360. The secondary unit is the minute, of which there are 60 minutes in one degree. The tertiary unit is the second, of which there are 60 seconds in one minute. The modern use of sexagesimal corresponds very closely with the modern measurement of time, in which there are 24 hours in a day, 60 minutes in one hour, and 60 seconds in one minute. The modern measurement of time roughly corresponds to the rotation (days) and revolution (years) of the Earth. Units that are smaller than one second are measured using a decimal system. In the Chinese calendar, a sexagenary cycle is commonly used.

Fractions

The sexagesimal system is quite good for forming fractions: 1/2 = 0.30 1/3 = 0.20 1/4 = 0.15 1/5 = 0.12 1/6 = 0.10 1/8 = 0.07:30 1/9 = 0.06:40 1/10 = 0.06 1/12 = 0.05 1/15 = 0.04 1/16 = 0.03:45 1/18 = 0.03:20 1/20 = 0.03 1/30 = 0.02 1/40 = 0.01:30 1/50 = 0.01:12 1/1:00 = 0.01 (1/60 in decimal) but is not very good for simple repeating fractions, because both the neighbours of 60 (i.e. 59 and 61) are prime numbers. 1/7 = 0.08:34:17:08:34:17: recurring

Examples

- The length of a diagonal or a square root in a square of side a = 1, ([http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html YBC 7289 clay tablet]): :: 1.414212... = 30547/21600 = 1.24:51:10 (sexagesimal = 1 + 24/60 + 51/60² + 10/60³), a constant used by Babylonian mathematicians in the Old Babylonian Period (1900 BC - 1650 BC), the actual value for

\sqrt

is 1.24:51:10:07:46:06:04:44...,
- The length of the tropical year in Neo-Babylonian astronomy, (see Hipparchus): :: 365.24579... ^d = 6,5;14,44,51^d ( = 6×60 ＋ 5 + 14/60 + 44/60² + 51/60³), (Note that the average length of a year in the Gregorian calendar is exactly 6,5;14,33^d in sexagesimal notation.)
- The value of π used by Ptolemy: :: 3.141666... = 377/120 = 3.8:30 ( = 3 + 8/60 + 30/60² ).

External link

- [http://jean.nu/view.php/page/CanonSexagesimal Extensive page on Base-sixty] Category:Numeration 60 ja:六十進記数法

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, all representing the same number. This article attempts to explain the various systems of numerals. See also number names. 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 binary numeral for three, the decimal numeral for eleven, or other numbers in other bases. Ideally, a numeral system will:
- Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers)
- Give every number represented a unique representation (or at least a standard representation)
- 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. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.30999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the natural sciences where differing precision is denoted. 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. Elias gamma coding is commonly used in data compression; it includes a unary part and a binary part. 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; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D′ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted. 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 Hindu-Arabic numeral system, borrowed from India, 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). In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-k numeration is also called k-adic notation, not to be confused with p-adic numbers. Bijective base-1 is the same as unary.

History

: See also History of natural numbers and the status of zero. Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, 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 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 Maya of Central America used a base 20/base 18 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this to do advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus. 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. In India, recognizably modern positional numeral systems, passed to the Arabians, probably along with the astronomical tables, was brought to Baghdad by an Indian ambassador around 773. For greater discussion of numeral systems from India, see Arabic numerals and Indian numerals. 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 Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century. 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 is the one most commonly used today. It is assumed to have originated because humans have ten fingers. A base-eight system was devised by the Yuki of Northern California, who used the spaces between the fingers to count. There is also linguistic evidence which suggests that the Bronze Age Proto-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, newan, appears to derive from the word for 'new', newo, suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997). (In French, the word neuf still means both 9 and 'new'.) 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 their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). 60 is a useful base because it has a large number of factors, including all of the first six counting numbers. Base-12 systems were popular because multiplication is easier in them than in base-10 (addition is 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. The Nenets language once used a base 9 system, but has since shifted to decimal under the influence of Russian. The word yúq originally meant 9, but took the value 10 on account of Russian influence; so in current Nenets the word for 9 is xasu-yúq, lit. 'Nenets yúq, whereas 10 is simply yúq, but in Eastern dialects also lúca-yúq, lit. 'Russian yúq. Switches (and their electronic successors, built of vacuum tubes, or later of transistors) have only two possible states: "open" and "closed". Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. (In modern transistors, it is more accurate to say that the voltages are high and low instead of 'on' and 'off'). This binary system is the basis for digital computers. It is used to perform integer arithmetic in almost all digital 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 — some of it may be treated as text and program data. Real numbers (which include numbers other than integers) are usually stored and treated as floating point numbers, which have different rules of arithmetic. The bases that were used in past or used today are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 20, 60.

Positional systems in detail

Also see Positional notation. 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 (4×10³) + (3×10²) + (2×10¹) + (7×10⁰), 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 (itself represented in base 10) is added in subscript to the right of the number, like this: number_base. Unless specified by context, 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 1×2¹+ 0×2⁰ +1×2^-1 +1×2^-2 = 2.75. In general, numbers in the base b system are of the form: :

(a_na_...a_1a_0.c_1c_2c_3...)_b = 
\sum_^n a_kb^k + \sum_^\infty c_kb^

The numbers b^k and b^-k are the weights of the corresponding digits. 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 0.3₁₀ = 0.0100110011001...₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 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.

Change of radix

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7:

1020304 / 7 = 145757 r 5
 145757 / 7 =  20822 r 3
  20822 / 7 =   2974 r 4
   2974 / 7 =    424 r 6
    424 / 7 =     60 r 4
     60 / 7 =      8 r 4
      8 / 7 =      1 r 1
      1 / 7 =      0 r 1   => 11446435

E.g., 10110111 base 2 into base 5:

10110111 / 101 = 100100 r 11  (3)
  100100 / 101 =    111 r  1  (1)
     111 / 101 =      1 r 10  (2)
       1 / 101 =      0 r  1  (1)  => 1213

To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately a terminating fraction in one base may not terminate in another. E.g., 0.1A4C base 16 into base 9:

0.1A4C × 9 = 0.ECAC
0.ECAC × 9 = 8.520C
0.520C × 9 = 2.E26C
0.E26C × 9 = 7.F5CC
0.F5CC × 9 = 8.A42C 
0.A42C × 9 = 5.C58C  => 0.082785...

Generalized variable-length integers

More general is using a notation (here written little-endian) like a₀a₁a₂ for a₀ + a₁b₁ + a₂b₁b₂, etc. This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a-z and 0-9, representing 0-25 and 26-35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (1) then a (0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b-9 (1-35), therefore the weight b₁ is 35 instead of 36. Suppose the threshold values for the second and third digit are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence: a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc. Note that unlike a regular base-35 numeral system, we have numbers like 9b where 9 and b each represent 35; yet the representation is unique because ac and aca are not allowed. The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes. The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are nonzero.

Reference

- D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".
- J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.

External links

- [http://www.elfqrin.com/baseconv.html Numeric Base Converter]
- [http://www.kwiznet.com/p/showCurriculum.php?curriculumID=22 Number Sense & Numeration Lessons] Category:Systems ko:기수법 ja:位取り記数法

60 (number)

60 (Sixty) is the natural number following 59 and preceding 61.

In mathematics

Sixty is a composite number with divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, making it also a highly composite number. Because 60 is the sum of its unitary divisors (excluding itself), it is a unitary perfect number. And because it is divisible by the sum of its digits in base 10, it is a Harshad number. 60 is the smallest number divisible by the numbers 1 to 6. (There is no smaller number divisible by the numbers 1 to 5). 60 is the smallest number with exactly 12 divisors. This number is the sum of a pair of twin primes (29 + 31), as well as the sum of four consecutive primes (11 + 13 + 17 + 19). The smallest non-abelian simple group has order 60. In normal space, the 3 interior angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. The Babylonian number system had a base of sixty. A number system with base-sixty is called the sexagesimal (original meaning of sexagesimal is sixtieth). The sexagenary cycle plays a role in Chinese calendar and numerology.　(Originally, sexagenary means sixty each, also centenary means one hundred each.)

In astronomy

- Messier object M60, a magnitude 10.5 galaxy in the constellation Virgo
- The New General Catalogue object NGC 60, a spiral galaxy in the constellation Pisces
- The Saros number of the solar eclipse series which began on -1020 May 18 and ended on 260 June 26. The duration of Saros series 60 was 1280.1 years, and it contained 72 solar eclipses. The Saros number of the lunar eclipse series which began on -700 March 8 and ended on 616 May 7. The duration of Saros series 60 was 1316.2 years, and it contained 74 lunar eclipses.

In other fields

Sixty is also:
- the atomic number of neodymium, a lanthanide. The neodymium following element with the ordinal number 61 (promethium) has in contrast to the neighbour elements no stable isotopes.
- in the measurement of time, the number of seconds in a minute, and the number of minutes in an hour.
- in geometry, the number of seconds in a minute, and the number of minutes in a degree.
- the number of the European route E60 from Brest, France to Constanta, Romania
- a common speed limit, in miles per hour, for freeways in many US states.
- a common speed limit, in kilometers per hour, in urban areas Russian Federation.
- in years of marriage, the diamond wedding anniversary.
- the maximum number of marbles (game pieces) in Chinese checkers.
- the code for international direct dial calls to Malaysia.
- The year AD 60, 60 BC, or 1960. Category:Integers ko:60 ja:60

Sumerians

Sumer (or Shumer, Egyptian Sangar, Bib. Shinar, native ki-en-gir) formed the southern part of Mesopotamia from the time of settlement by the Sumerians until the time of Babylonia. The oldest tablets thus far discovered containing Sumerian pre-cuneiform script date to around 3500 BC.

Background

The term "Sumerian" is an exonym (a name given by another group of people), first applied by the Akkadians. The Sumerians described themselves as "the black-headed people" (sag-gi-ga) and called their land ki-en-gir, "place of the civilized lords". The Akkadian word Shumer possibly represents this name in dialect. The Sumerians, with a language, culture, and, perhaps, appearance different from their Semitic neighbors and successors were at one time believed to have been invaders, but the archaeological record shows cultural continuity from the time of the early Ubaid period (5200-4500 BC C-14, 6090-5429 [http://www.calpal-online.de calBC]) settlements in southern Mesopotamia. The challenge for any population attempting to dwell in Iraq's arid southern floodplain was to master the Tigris and Euphrates river waters for year-round agriculture and drinking water. In fact, the Sumerian language is replete with terms for canals, dikes, and reservoirs, indicating that Sumerian speakers were farmers who moved down from the north after perfecting irrigation agriculture there. The Ubaid pottery of southern Mesopotamia has been connected via 'Choga Mami Transitional' ware to the pottery of the Samarra period culture (5700-4900 BC C-14, 6640-5816 [http://www.calpal-online.de calBC]) in the north, who were the first to practice a primitive form of irrigation agriculture along the middle Tigris river and its tributaries. The connection is most clearly seen at Tell Awayli (Oueilli/Oueili) near Larsa, excavated by the French in the 1980s, where 8 levels yielded pre-Ubaid pottery with affinities to Samarran ware. Sumerian speakers spread down into southern Mesopotamia because they had developed a social organization and a technology that enabled them, through their control of the water, to survive and prosper in a difficult environment where, other than a possible indigenous hunter-gatherer population in the marshlands at the head of the Arabo-Persian Gulf and seasonal pastoralists, they had no competition. A distinctive style of painted pottery spread throughout Mesopotamia in the Ubaid period, when the ancient Sumerian cult-center of Eridu was gradually surpassed in size by the nearby city of Uruk. The archaeological transition from the Ubaid period to the Uruk period is marked by a gradual shift from painted pottery domestically-produced on a slow wheel, to a great variety of unpainted pottery mass-produced by specialists on a fast wheel. The date of this transition, from Ubaid 4 to Early Uruk, is in dispute, but calibrated radiocarbon dates from Tell Awayli would place it as early as 4500 BC. By the time of the Uruk period (4500-3100 BC calibrated), the volume of trade goods being inexpensively transported along the canals and rivers of southern Mesopotamia facilitated the rise of many large temple-centered cities where centralized administrations could afford to employ specialized workers. It is fairly certain that it was during the Uruk period that Sumerian cities began to make use of slave labor, and there is ample evidence for captured slaves as workers in the earliest texts. Artifacts, and even colonies of this Uruk civilization have been found over a wide area - from the Mediterranean sea in the west, to the Taurus mountains in Turkey, and as far east as Central Iran. The Uruk period civilization, exported by Sumerian traders and colonists, had a stimulating and influential effect on surrounding peoples, who gradually evolved their own comparable, competing economies. The cities of Sumer could not maintain remote, long-distance colonies purely by military force; the domestic horse did not appear in Sumer until the Ur III period - one thousand years after the Uruk period ended. The end of the Uruk period coincided with a dry period from 3200-2900 BC that marked the end of a long wetter, warmer climate period from ca. 9,000 to 5,000 years B.P. called the Holocene climatic optimum. When the historical record opens, the Sumerians seem to be limited to southern Mesopotamia, although very early rulers such as Lugal-Anne-Mundu are indeed recorded as expanding to neighboring areas as far as the Mediterranean, Taurus and Zagros, and not long after legendary figures like Enmerkar and Gilgamesh, who are associated in mythology with the historical transfer of culture from Eridu to Uruk, were supposed to have reigned. The term 'Sumerian' applies to speakers of the Sumerian language. The Sumerian language is generally regarded as a language isolate in linguistics because it belongs to no known language family; Akkadian belongs to the Afro-Asiatic languages.

History

Main article: History of Sumer. In the earliest known period Sumer was divided into several independent city-states, whose limits were defined by canals and boundary stones. Each was centered on a temple dedicated to the patron god or goddess of the city and ruled over by a priest or king, who was intimately tied to the city's religious rites. Some of the major cities included Eridu, Kish, Lagash, Uruk, Ur, and Nippur. As these cities developed, they sought to assert primacy over each other, falling into a millennium of almost incessant warfare over water rights, trade routes, and tribute from nomadic tribes. The Sumerian king list contains a traditional list of the early dynasties, much of it probably mythical. The first name on the list whose existence is authenticated through archaeological evidence, is that of Enmebaragesi of Kish, whose name is also mentioned in the Gilgamesh epics. This has led some to suggest that Gilgamesh really was a historical king of Uruk. The dynasty of Lagash is well known through important monuments, and one of the first empires in recorded history was that of Eannatum of Lagash, who annexed practically all of Sumer, including Kish, Uruk, Ur, and Larsa, and reduced to tribute the city-state of Umma, arch-rival of Lagash. In addition, his realm extended to parts of Elam and along the Persian Gulf. Lugal-Zage-Si, the priest-king of Umma, overthrew the primacy of the Lagash dynasty, took Uruk, making it his capital, and claimed an empire extending from the Persian Gulf to the Mediterranean. He is the last ethnically Sumerian king before the arrival of the Semitic conqueror, Sargon of Akkad.

Downfall

As the local states grew in strength, the Sumerians began to lose their political hegemony over most parts of Mesopotamia. The Amorites conquered Sumer and founded Babylon. The Hurrians established the empire of Mitanni in northern Mesopotamia around 1595 BC, while the Babylonians controlled the south. Both groups defended themselves against the Egyptians and the Hittites. The Hittites defeated Mitanni but were repulsed by the Babylonians; but the Kassites defeated the Babylonians in 1460 BC. The Kassites were in turn defeated by the Elamites around 1150 BC.

Agriculture and hunting

The Sumerians grew barley, chickpeas, lentils, millet, wheat, turnips, dates, onions, garlic, lettuce, leeks and mustard. They also raised cattle, sheep, goats, and pigs. They used oxen as their primary beasts of burden and donkeys as their primary transport animal. Sumerians hunted fish and fowl. Sumerian agriculture depended heavily on irrigation. The irrigation was accomplished by the use of shadufs, canals, channels, dykes, weirs, and reservoirs. The canals required frequent repair and continual removal of silt. The government required individuals to work on the canals, although the rich were able to exempt themselves. Using the canals, farmers would flood their fields and then drain the water. Next they let oxen stomp the ground and kill weeds. They then dragged the fields with pickaxes. After drying, they plowed, harrowed, raked thrice, and pulverized with a mattock. Sumerians harvested during the dry fall season in three-person teams consisting of a reaper, a binder, and a sheaf arranger. The farmers would use threshing wagons to separate the cereal heads from the stalks and then use threshing sleds to disengage the grain. They then winnowed the grain/chaff mixture.

Architecture

Main article: Sumerian architecture The Tigris-Euphrates plain lacked minerals and trees. Sumerian structures comprised plano-convex mudbrick, not fixed with mortar or with cement. Mud-brick buildings eventually deteriorate, and so they were periodically destroyed, levelled, and rebuilt on the same spot. This constant rebuilding gradually raised the level of cities, so that they came to be elevated above the surrounding plain. The resultant hills are known as tells, and are found throughout the ancient Near East. The most impressive and famous of Sumerian buildings are the ziggurats, large terraced platforms which supported temples. The Biblical Tower of Babel may have been built in a similar manner. Sumerian cylinder seals also depict houses built from reeds not unlike those built by the Marsh Arabs of Southern Iraq until recent years. Sumerian temples and palaces made use of more advanced materials and techniques, such as buttresses, recesses, half columns, and clay nails.

Culture

Sumerian culture may be traced to two main centers, Eridu in the south and Nippur in the north. But the streams of civilization that flowed from them were in strong contrast. The deity Enlil, around whose sanctuary Nippur had grown up, was considered lord of the ghost-land, and his gifts to mankind were said to be the spells and incantations that the spirits of good or evil were compelled to obey. The world he governed was a mountain; the creatures whom he had made lived underground. Eridu, on the other hand, was the home of the culture god Enki (absorbed into Babylonian mythology as the god Ea), the god of light and beneficence, ruler of the freshwater depths beneath the earth (the Apsû), a healer and friend to humanity who was thought to have given us the arts and sciences, the industries and manners of civilization; the first law-book was considered his creation. Eridu had once been a seaport, and it was doubtless its foreign trade and intercourse with other lands that influenced the development of its culture. Its cosmology was the result of its geographical position: the earth, it was believed, had grown out of the waters of the deep, like the ever widening coast at the mouth of the Euphrates. Long before history is recorded, however, the cultures of Eridu and Nippur had coalesced. While Babylon seems to have been a colony of Eridu, Ur, the immediate neighbour of Eridu, may have been a colony of Nippur, since its moon god was the son of Enlil of Nippur. But in the admixture of the two cultures, the influence of Eridu was predominant. Historian Alan Marcus says: "Sumerians held a rather dour perspective on life" One Sumerian wrote: "Tears, lament, anguish, and depression are within me. Suffering overwhelms me. Evil fate holds me and carries off my life. Malignant sickness bathes me." Another wrote, "Why am I counted among the ignorant? Food is all about, yet my food is hunger. On the day shares were allotted, my allotted share was suffering." Though women were protected by late Sumerian law and were able to achieve a higher status in Sumer than in other contemporary civilizations, the culture was male-dominated.

Economy and trade

Discoveries of obsidian from far-away locations in Anatolia and lapis lazuli from northeastern Afghanistan, beads from Dilmun (modern Bahrain), and several seals inscribed with the Indus Valley script suggest a remarkably wide-ranging network of ancient trade centered around the Persian Gulf. The Epic of Gilgamesh refers to trade with far lands for goods such as wood that were scarce in Mesopotamia. In particular, cedar from Lebanon was prized. The Sumerians used slaves, although they were not a major part of the economy. Slave women worked as weavers, pressers, millers, and porters. Sumerian potters decorated pots with cedar oil paints. The potters used a bow drill to produce the fire needed for baking the pottery. Sumerian masons and jewelers knew and made use of alabaster (calcite), ivory, gold, silver, carnelian and lapis lazuli.

Military

lapis lazuli On Left: The Standard of Ur, the bottom panel of the War Side City walls defended Sumerian cities. The Sumerians engaged in siege warfare between their cities, and the mudbrick walls failed to deter foes who had the time to pry out the bricks. Sumerian armies consisted mostly of infantry. Light infantrymen carried battle-axes, daggers, and spears. The regular infantry also used copper helmets, felt cloaks, and leather kilts. The Sumerian military used carts harnessed to onagers. These early chariots functioned less effectively in combat than did later designs, and some have suggested that these chariots served primarily as transports, though the crew carried battle-axes and lances. The Sumerian chariot comprised a four-wheeled device manned by a crew of two and harnessed to four onagers. The cart was composed of a woven basket and the wheels had a solid three-piece design. Sumerians used slings and simple bows. (the recurve bow is a later invention.)

Religion

Main article: Sumerian mythology It can be difficult to speak of a 'Sumerian religion' as such, since practices and beliefs varied widely through time and distance, with each city having its own twist on mythology and theology. It might be said to be henotheistic. The Sumerian were the first recorded beliefs and the source for much of later Mesopotamian mythology, religion, and astrology. The Sumerians worshipped An as the primary god, equivalent to heaven (indeed the word "an" in Sumerian means "sky"). An's closest cohorts were Enki in the south, Enlil in the north and Inana, the deification of Venus, the morning (eastern) and evening (western) star. The sun was Utu, the moon was Nanna, Nammu or Namma was the Mother Goddess, probably considered to be the original matrix; there were hundreds of minor deities. The Sumerian gods (Sumerian dingir, plural dingir-dingir or dingir-a-ne-ne) each had associations with different cities, and their religious importance often waxed and waned with the political power of the associated cities. The gods were said to have created human beings from clay for the purpose of serving them. The gods often expressed their anger and frustration through earthquakes and storms: the gist of Sumerian religion was that humanity was at the mercy of the gods. Sumerians believed that the universe consisted of a flat disk enclosed by a tin dome. The Sumerian afterlife involved a descent into a vile nether-world to spend eternity in a wretched existence as a Gidim (ghost). Sumerian temples consisted of a central nave with aisles along either side. Flanking the aisles would be rooms for the priests. At one end would stand the podium and a mudbrick table for animal and vegetable sacrifices. Granaries and storehouses were usually located near the temples. After a time the Sumerians began to place the temples on top of multi-layered square constructions built as a series of rising terraces: the ziggurats.

Technology

Examples of Sumerian technology include: the wheel, saws, leather, chisels, hammers, braces, bits, nails, pins, rings, hoes, axes, knives, lancepoints, arrowheads, swords, glue, daggers, waterskins, bags, harnesses, boats, armor, quivers, scabbards, boots, sandal (footwear), harpoons, and beer brewing. The Sumerians had three main types of boats:
- skin boats comprised reeds and animal skins
- sailboats featured bitumen waterproofing
- wooden-oared ships, sometimes pulled upstream by people and animals walking along the nearby banks

Language and writing

Main article: Sumerian language. Sumerian is a language isolate, meaning that it is unrelated to any other known languages. There have been many failed attempts to connect Sumerian to other languages group. It is an Agglutinative language; in other words, morphemes (word-units) are stuck together to create words. Sumerians invented picture-hieroglyphs that developed into later cuneiform, and theirs is the oldest known written human language. An extremely large body of hundreds of thousands of texts in the Sumerian language have survived, the great majority on clay tablets. Known Sumerian texts include personal and business letters and transactions, receipts, lexical lists, laws, hymns and prayers, magical incantations, and scientific texts including mathematics, astronomy, and medicine. Monumental inscriptions and texts on different objects like statues or bricks are also very common. Many texts survive in multiple copies because they were repeatedly transcribed by scribes-in-training. Sumerian continued to be the language of religion and law in Mesopotamia long after Semitic speakers had become the ruling race. Understanding Sumerian texts today can be problematic even for experts. Most difficult are the earliest texts, which in many cases don't give the full grammatical structure of the language.

Legacy

The Sumerians are perhaps remembered most for their many inventions. Many authorities credit them with the invention of the wheel and the potter's wheel. Their cuneiform writing system was the first we have evidence of (with the possible exception of the highly controversial Old European Script), pre-dating Egyptian hieroglyphics by at least seventy five years. They were among the first formal astronomers. They came up with the concept of dividing the hour into 60 minutes and the minute into 60 seconds. They may have invented military formations. Perhaps most importantly, the Sumerians ushered in the age of intensive agriculturalism in Ancient Mesopotamia. Einkorn and Emmer wheat, barley, sheep (starting as mouflon) and cattle (starting as aurochs) were foremost among the species cultivated and raised for the first time on a grand scale. These inventions and innovations easily place the Sumerians among the most creative cultures in human pre-history and history.

External links

History
- [http://ancientneareast.tripod.com/Sumer.html The History of the Ancient Near East] Language
- [http://www.sumerian.org/ Sumerian Language Page], perhaps the oldest Sumerian website on the web (it dates back to 1996), features compiled lexicon, detailed FAQ, extensive links, and so on.
- [http://etcsl.orinst.ox.ac.uk/ ETCSL: The Electronic Text Corpus of Sumerian Literature] has complete translations of more than 400 Sumerian literary texts.
- [http://psd.museum.upenn.edu/ PSD: The Pennsylvania Sumerian Dictionary], while still in its initial stages, can be searched on-line, from August 2004. ko:수메르 ja:シュメール

Babylonian numerals

Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their astrological observations and calculations (aided by their invention of the abacus), used a sexagesimal (base-60) positional numeral system inherited from the Sumerian and also Akkadian civilizations. Neither of the predecessors was a positional system (having a convention for which ‘end’ of the numeral represented the units). This system first appeared around 1900BC to 1800BC. It is also credited as being the first known place-value numeral system, in which the value of a particular digit depends both on the digit itself and its position within the number. This was an extremely important development, because prior to place-value systems people were obliged to use unique symbols to represent each power of a base (ten, one-hundred, one thousand, and so forth), making even basic calculations unwieldy. Since their system clearly had an internal decimal system and they used 60 as the second smallest unit instead of 100 as we do today, it is more appropriately considered a mixed-radix system of bases 10 and 6. A large value to have as a base, sixty is the smallest number that can be wholly divided by two, three, four, five, and six, hence also ten, fifteen, twenty, and thirty. Six and ten were also used as sub-bases. Only two symbols used in a variety of combinations were used to denote the 59 numbers. A space was left to indicate a zero, although they later devised a sign to represent an empty place. Sexagesimals still survive to this day, in the form of degrees (360° in a circle), minutes, and seconds in trigonometry and the measurement of time. A common theory is that sixty was chosen due to its prime factorization 2×2×3×5 which makes it divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. Integers and fractions were represented identically - a radix point was not written but rather made clear by context.

Numerals

Image:Babylonian_numerals.jpg

External link

- [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html Babylonian numerals]
- [http://it.stlawu.edu/%7Edmelvill/mesomath/Numbers.html Cuneiform numbers]
- [http://mathforum.org/alejandre/numerals.html Babylonian Mathematics] Category:Babylonia Category:Babylonian numerals Category:Numeral system Category:Sexagesimal

Decimal

The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, probably because humans normally have a total of ten fingers on their hands.

Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) to represent numbers. These digits are frequently used with a decimal point which indicates the start of a fractional part, and with one of the sign symbols + (plus) or − (minus) to indicate sign. The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc. The position of each digit conveys the multiplier (a power of ten) to be used with that digit. Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius < Lat.) means the unit of ten. However, some cultures do or used to use other numeral systems, including the Tzotzil, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal systems, the Babylonians, who used sexagesimal, and the Yuki, who reportedly used octal. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures. Computers commonly use a different system, binary, internally. For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. For most purposes however, binary values are converted by the computer to the equivalent decimal values for presentation to humans. Nevertheless, sometimes computers do use internal representations which are equivalent to decimal for doing arithmetic. Frequently this arithmetic is done on data in the form of binary-coded decimal, but there are other decimal representations in use (see IEEE 754r). Decimal arithmetic is used in computers so that fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations [http://www2.hursley.ibm.com/decimal/decifaq.html].

Fractional numbers

Decimal fractions

A decimal fraction is a vulgar fraction where the denominator is a power of ten. Decimal fractions can be expressed without a denominator, the decimal point being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. E.g. 8/10, 833/100, 83/1000, 8/10000 and 80/10000 are expressed thus: 0.8, 8.33, 0.083, 0.0008 and 0.008. Numbers which can be expressed in this way are called decimal numbers or regular numbers. The integer and fractional parts of a decimal number are separated by a decimal point. In this article, as in most of the English speaking world, a dot (.) is used. It is usual for a decimal number which is less than one to have a leading zero. Trailing zeroes after the decimal point are not necessary, although in science, engineering and statistics they can be retained to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are mathematically the same number, in engineering 0.080 suggests an error of up to 1 part in a thousand, while 0.08 suggests an error of up to 1 in a hundred (see Significant figures).

Other rational numbers

Any rational number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with recurring decimals. Ten is the product of the first and third prime numbers, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions: :1/2 = 0.5 :1/3 = 0.333333… (with 3 recurring) :1/4 = 0.25 :1/5 = 0.2 :1/6 = 0.166666… (with 6 recurring) :1/8 = 0.125 :1/9 = 0.111111… (with 1 recurring) :1/10 = 0.1 :1/11 = 0.090909… (with 09 recurring) :1/12 = 0.083333… (with 3 recurring) :1/81 = 0.012345679012… (with 012345679 recurring) Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13. That a rational must produce a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only (q-1) possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q-1. For instance to find 3/7 by long division: .4 2 8 5 7 1 4 ... 7 ) 3.0 0 0 0 0 0 0 0 2 8 30/7 = 4 r 2 2 0 1 4 20/7 = 2 r 6 6 0 5 6 60/7 = 8 r 4 4 0 3 5 40/7 = 5 r 5 5 0 4 9 50/7 = 7 r 1 1 0 7 10/7 = 1 r 3 3 0 2 8 30/7 = 4 r 2 (again) 2 0 etc The converse to this observation is that every recurring decimal represents a rational number p/q. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance, :

0.0123123123\cdots = \frac \sum_^\infty 0.001^k = \frac\ \frac = \frac = \frac

The real numbers

Every real number has a (possibly infinite) decimal representation, i.e. it can be written as :

x = \mathop(x) \sum_ a_i\,10^i

where
- sign() is the sign function,
- a_i ∈ for all i ∈ Z, are its decimal digits, equal to zero for all i greater than some number (the common logarithm of |x|). Such a sum always makes sense (i.e. converges), even if there is an infinite number of a_i (with negative indices), which is the case for all reals which are not decimal numbers, according to what precedes. The representation is unique, if one excludes representations that end in a recurring 9. Indeed, consider rational numbers which can be written as p/(2^a5^b) (i.e. the only prime factors in denominator are 2 and 5). In this case there is a terminating decimal representation. For instance 1/1=1, −1/2=−0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999…, −1/2=−0.499999…, etc. Rational numbers p/q with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation. This leaves the irrational numbers. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur. Naturally, the same trichotomy holds for other base-n positional numeral systems:
- Terminating representation: rational where the denominator divides some n^k
- Recurring representation: other rational
- Non-terminating, non-recurring representation: irrational and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.

History

Decimal writers

- c. 3500 - 2500 BC Elamites of Iran possibly use early forms of decimal system. [http://www.chn.ir/english/eshownews.asp?no=1622] [http://www.mpiwg-berlin.mpg.de/Preprints/P183.PDF]
- c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc.).
- c. 2600 BC Indus Valley Civilization, earliest known physical use of decimal fractions in ancient weight system: 1/20, 1/10, 1/5, 1/2. See Ancient Indus Valley weights and measures.
- c. 1400 BC Chinese writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts.
- c. 598–670 Brahmagupta – decimal integers, negative integers, and zero
- c. 790–840 Abu Abdullah Muhammad bin Musa al-Khwarizmi – first to expound on algorism outside India
- c. 920–980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions
- 1548/49–1620 Simon Stevin – author of De Thiende ('the tenth')
- 1561–1613 Bartholemaeus Pitiscus– (possibly) decimal point notation
- 1550–1617 John Napier– decimal logarithms

External links

- [http://www2.hursley.ibm.com/decimal/decifaq.html Decimal arithmetic FAQ]
- Tests: [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1352 Decimal Place Value] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1353&CurriculumID=5 Sums] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=739&CurriculumID=5 Fractions]
- [http://www.mathsisfun.com/worksheets/decimals.php Practice Decimal Arithmetic with Printable Worksheets]
- [http://www.mathsisfun.com/converting-decimals-fractions.html Converting Decimals to Fractions] Category:Elementary arithmetic Category:Numeration Category:Fractions 10 ko:십진법 ja:十進記数法 th:เลขฐานสิบ

10 (number)

10 (ten) is the natural number following 9 and preceding 11.

In mathematics

Ten is a composite number, its proper divisors being , and . Ten is the smallest noncototient, a number that can not be expressed as the difference between any integer and the total of coprimes below it. Ten is a Harshad number and a semi-meandric number. Ten is the sum of the first three prime numbers. A polygon with ten sides is a decagon, and 10 is a decagonal number. But it is also a triangular number and a centered triangular number. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. Ten is the first two-digit number in decimal and thus the lowest number where the position of a numeral affects its value. Any integer written in the decimal system can be multiplied by ten by adding a zero to the end (e.g. 855
- 10 = 8550). The reason for the choice of ten is assumed to be that humans have ten fingers (digits). Ten is the number of n-Queens Problem solutions for n = 5.

In numeral systems

The Roman numeral for ten is X (which looks like two V's [the Roman numeral for 5] put together); it is thought that the V for five is derived from an open hand (five digits displayed). The Chinese word numeral for ten is 十, which resembles a cross.

Base	Numeral system
1	unary	- - -
2	binary	1010
3	ternary	101
4	quaternary	22
5	quinary	20
6	senary	14
7	septenary	13
8	octal	12
9	novenary	11
10	decimal	10
over 10 (hexadecimal)		A

In science

- The atomic number of neon
- The number of carbon atoms in decane, a hydrocarbon.
- The number of spacetime dimensions in superstring theory. In astronomy, : Messier object M10, a magnitude 7.5 globular cluster in the constellation Ophiuchus. : The New General Catalogue [http://www.ngcic.org/ object] NGC 10, a magnitude 12.5 spiral galaxy in the constellation Sculptor :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the solar eclipse series which began on -2467 February 28 and ended on -1169 April 18. The duration of Saros series 10 was 1298.1 years, and it contained 73 solar eclipses. :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -2454 June 17 and ended on -1138 August 15. The duration of Saros series 10 was 1316.2 years, and it contained 74 lunar eclipses.

In money

- There are ten cents in a U.S. or Canadian dime, itself one tenth of a dollar. The word was shortened from decime.
- The denomination of Canadian paper money bearing a portrait of Sir John A. Macdonald, Canada's first Prime Minister.

In music

- The interval of a major or minor tenth is an octave plus a major or minor third.
- The title of quite a few albums, including recordings by Pearl Jam and LL Cool J. See Ten (album).

In religion

- People were traditionally tithed one tenth of their produce.
- The number of Commandments from God to man in Mosaic law. (Exodus 20:2-17, Exodus 34:12-26, and Deuteronomy 5:6-21)
- There are said to be ten "Lost Tribes" of Israel.
- There are ten sephiroth in the Tree of Life of Jewish mysticism.

In sports

- In chess, each side can promote all eight pawns to have a total of ten bishops, knights, or rooks.
- In rugby union the fly-half wears the 10 shirt.
- In blackjack, the Ten, Jack, Queen and King are all worth 10 points.

In technology

- Driving a racing car at ten-tenths is driving as fast as possible, on the limit.
- Ten-codes are commonly used on emergency service radio systems.
- ASCII and Unicode code point for line feed
- In MIDI, Channel 10 is reserved for unpitched percussion instruments.
- In the Rich Text Format specification, all language codes for regional variants of the Spanish language are congruent to 10 mod 256.
- In Mac OS X, the F10 function key tiles all the windows of the current application and grays the windows of other applications.

In other fields

- A collection of ten items (most often ten years) is called a decade.
- A decapod crustacean has ten limbs.
- To reduce something by one tenth is to decimate. (In ancient Rome, the killing of one in ten soldiers was the punishment for mutiny.)
- It takes ten to make a minyan.
- With ten being the base of the decimal system, a scale of 1 to 10 is often used to rank things, as a smaller version of a 1-to-100 scale (as is used in percentages and wine-tasting).
- Blake Edwards' 1979 movie 10.
- Something that scores perfectly is "a perfect ten". A person who is attractive and physically flawless is often said to be "a ten", from the idea of ranking that person's appearance and sex-appeal on a 1-to-10 scale. E.g. In the movie 10 Bo Derek is supposedly a "10" on that scale.
- Counting from one to ten before speaking is often done in order to cool one's temper.
- In astrology, Capricorn is the 10th astrological sign of the Zodiac.
- The ordinal adjective is denary. Ten is:
- Number of kingdoms in Five Dynasties and Ten Kingdoms Period
- House number of 10 Downing Street
- The designation of United States Interstate 10, a freeway that runs from California to Florida.
- A Japanese comic: Ten (comic)
- Number of dots in a tetractys.
- Historical years: 10 A.D., 10 B.C., or 1910 1 0 ko:10 ja:10

1 (number)

: This article discusses the number one. For the year AD 1, see 1. For other uses of 1, see 1 (disambiguation) 1 (one) is a number, numeral, and the name of the glyph representing that number. It is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity, and unit is sometimes used as an adjective in this sense. (For example, a line segment of "unit length" is a line segment of length 1.)

History

Some Ancient Greeks did not consider one as a number: they considered it to be the unit, two being the first proper number as it represented a multiplicity.

In mathematics

For any number x: :x·1 = 1·x = x (This expresses the fact that 1 is the multiplicative identity.) As a consequence of this, 1 is a 1-automorphic number in any place-based numbering system. :x/1 = x (see division) :x¹ = x, 1^x = 1, and for nonzero x, x⁰ = 1 (see exponentiation) :x↑↑1 = x and 1↑↑x = 1 (see tetration). Using ordinary addition, we have 1 + 1 = 2; depending on the interpretation of the symbol "+" and the numeral system used, the expression can have many different meanings, listed at one plus one. One cannot be used as the base of a positional numeral system in the ordinary way. Sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as other positional numeral systems. Related to this, one cannot take logarithms with base 1, since the "exponential function" with base 1 is the constant function 1. In the Von Neumann representation of natural numbers, 1 is defined as the set . This set has cardinality 1 and hereditary rank 1. Sets like this with a single element are called singletons. In a multiplicative group or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einheit, unity) is more traditional. However, "1" is especially common for the multiplicative identity of a ring. (Note that this multiplicative identity is also often called "unity".) One is its own factorial, and its own square and cube (and so on, as 1 × 1 × ... × 1 = 1). As a consequence of its being its own square, one is also a Kaprekar number. One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few. It is also the first and second numbers in the Fibonacci sequence, and is the first number in a lot of mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that didn't already have it, and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases. One is the empty product. One is a harmonic divisor number. One is most often used for representing 'true' as a Boolean datatype in computer science. One is currently considered neither a prime number, nor a composite number - although it used to be considered prime. Defining a prime as a number that is only divisible by one and itself, one is a prime. However, for purposes of factorization and especially the fundamental theorem of arithmetic, it is more convenient to not think of one as a prime factor, or to think of it as an implicit factor that's always there but need not be written down. To exclude the number one from the list of prime numbers, primality is defined as a number having exactly two distinct divisors, one and itself. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899, although Carl Sagan included one in a list of prime numbers in his book Contact in 1985. One is one of three possible return values of the Möbius function. Passed an integer that is square-free with an even number of distinct prime factors, the Möbius function returns one. One is the only odd number that is in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2. One is the only 1-perfect number (see multiply perfect number). One is equal to the sum of its digits in any place-based numbering system, making it an all-Harshad number. One is the number of n × n magic squares for n = 1, 3. One is the number of n-queens problem solutions for n = 1. One is a meandric number, a semi-meandric number, and an open meandric number. By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix. One is the value of the sine and cosine at π/2 and 0 radians, respectively. One is the most common leading digit in many sets of data, a consequence of Benford's law. See also -1.

The Arabic glyph

Image:Evolution1glyph.png The glyph used today in the Western world to represent the number 1, a vertical line, often with a little serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmin Indians, who wrote 1 as a horizontal line (in Chinese today this is the way it is written). The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right, but kept the circle small. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. In fonts with text figures, 1 is typically the same height as a lowercase X, for example, Image:TextFigs148.png.

In science

One is:

- set equal to celerity (c), the speed of light, in Heaviside notation to simplify calculations.
- the factor in ratios for unit conversions.
- the total density ratio for a flat universe.
- The atomic number of hydrogen In astronomy, : Messier object M1, a magnitude 7.0 supernova remnant in the constellation Taurus, also known as the Crab Nebula. : The New General Catalogue [http://www.ngcic.org/ object] NGC 1, a 13th magnitude 13 spiral galaxy in the constellation Pegasus :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -2872 June 4 and ended on -1592 July 11 . The duration of Saros series 1 was 1280.1 years, and it contained 73 solar eclipses. :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -2588 March 2 and ended on -1272 April 30. The duration of Saros series 1 was 1316.2 years, and it contained 74 lunar eclipses.

In human society

Many human cultures have given the concept of one-ness symbolic meanings. Many religions consider God to be a perfect example of one-ness. See monad for a detailed discussion of other types of one-ness. One represents unity, togetherness, and absence of separation or discrimination, e.g. "We are all one" or "everyone". Something is unique if it is the only one of its kind. More loosely and exaggeratingly (especially in advertising) the term is used for something very special. One is also an (archaic) expression of the first person singular ("one is not amused") and of the second person singular ("does one take sugar?"). In Western culture, it is believed by many that the maximum number of girlfriends or boyfriends one may have at one time is 1. Also, it is strongly believed that you can be married to only 1 person at any time - this is called monogamy. Being married to more than one person at any time is called bigamy or polygamy. This is illegal in many Western societies. Among children, or when otherwise calling for subtlety, the phrase "number 1" can refer to the act of urination. This can derive from a traditional U.S. elementary school practice of holding up one or two fingers to indicate the approximate time of a requested absence. "Number 1" can also refer to oneself, or that something is first in its class, the latter being used often as a cheer in sports games. On a clock, 1:00 signals that one full hour has passed since the last change of the "AM" or "PM" meridian.

In music

In harmonic analysis of tonal music, the tonic chord is referred to as I. "One is the loneliest number that you'll ever do", according to the first line of "One" by Three Dog Night. The number appears in the title of songs by Metallica, U2, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, Three Dog Night, and the Bee Gees. Also the title of a best-selling compilation album of all Beatles songs that reached number 1 in the UK or US charts (see List of Number 1 Hits (USA)). The song Green grow the rushes, O has the line "One and all alone and evermore shall be so."

In religion

There is one god according to monotheism (see also: tawhid). There is one surat al-Fatiha in the Qur'an.

In sports

In some sports, one is the number of a specific position: in rugby union, the number of the loosehead prop; in baseball, the number representing the pitcher's position; in football, the number of the goalkeeper. In 2004, fans of the Philadelphia Eagles NFL team used the phrase "One" to show support for the team as they inched closer to the Super Bowl. The full text of the phrase was "One Team. One City. One Dream."

In technology

One is the DVD region of the United States and Canada. In the DOS Shell and many Windows programs, the function key F1 calls up online help. On most standard phones, the 1 key is not associated with any letters the way other number keys are, but on the BlackBerry, 1 is also the key for the letters E and R. Some cellular phones associate the "1" key with various symbols (i.e. the pound sign, the ampersand, etc.) when users engage in text messaging. In the Rich Text Format specification, 1 is the language code for the Arabic language. All codes for dialects of Arabic are congruent to 1 mod 256. 1 is a punctuation mark indicating exclamation, or the letter "l" in "leetspeak". 1, in binary, stands for 'yes'.

In other fields

One is:
- the denomination of U.S. dollar bill with George Washington's portrait, and the denomination of coin with Sacagawea's portrait. It is also the denomination of the older Eisenhower and Susan B. Anthony tender coins and the American Silver Eagle bullion coin.
- the denomination of Canadian dollar coin with a swimming loon on the reverse, hence its universally employed nickname, the "loonie"
- in cents of a U.S. dollar, the denomination of coin with Abraham Lincoln's portrait, commonly known as a penny.
- in cents of a Canadian dollar, the denomination of coin with two maple leaves on the reverse, also known as a penny.
- the code for international direct dial phone calls to countries participating in the North American Numbering Plan, such as the United States and Canada.
- the designation of many roads, listed at Route 1.
- as Air Force One is the callsign of any United States Air Force aircraft carrying the President of the United States.
- the house number of Number One Observatory Circle, the US Vice-President's residence.
- the address of Apsley House, known simply as Number 1, London.
- a subway service in New York City. See

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1 (number)

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The Arabic glyph

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